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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X_1$ be a smooth geometrically connected projective curve over the finite field of $q$-elements $\mathbb{F}_q$. Let $S_1\subseteq X_1$ be a reduced divisor consisting of $N_1$ closed points. Set $\mathbb{F}$ to be the algebraic closure of $\mathbb{F}_q$. Let $(X,S):=(X_1, S_1)\otimes_{\mathbb{F}_q}\mathbb{F}$, $l$ a prime number not dividing $q$. There is an equivalence of categories between the category of $\bar{\mathbb{Q}}_l$-lisse sheaves and the category of continous finite dimensional $\bar{\mathbb{Q}}_l$-representations of $\pi_1^{\text{ét}}(X-S,x)$, where $x\in (X-S)(\mathbb{F})$ is a geometric point on which the equivalence functor depends. Since $X-S$ is obtained from $X_1-S_1$ by base change, the Galois group ${\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$ acts on the scheme $X-S$ and hence on the set of isomorphism classes of $\bar{\mathbb{Q}}_l$-lisse sheaves, or equivalently on the set of isomorphism classes of continous finite dimensional $\bar{\mathbb{Q}}_l$-representations of $\pi_1^{\text{ét}}(X-S,x)$, via transport of structures. There are several equivalent ways to describe this action. One way is to look at the fundamental exact sequence of the étale fundamental groups \[ 1\to \pi_1^{\text{ét}}(X-S,x)\to \pi_1^{\text{ét}}(X_1-S_1,x) -^{{\phi}}\rightarrow {\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\to 1. \] For any $\sigma\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$ we choose an element $\tau\in \phi^{-1}(\sigma)$, then for any continous finite dimensional representation $\rho: \pi_1^{\text{ét}}(X-S,x)\to \text{GL}(V)$ we have $\sigma(\rho)=( g\mapsto \rho(\tau^{-1}g\tau))$ for all $g\in\pi_1^{\text{ét}}(X-S,x)$. As an action of ${\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$ on the \textit{isomorphism classes} of $\bar{\mathbb{Q}}_l$-lisse sheaves, the so defined action does not depend on the choice of the element ${\tau\in \phi^{-1}(\sigma)}$. Let $F_1$ be the fraction field of $X_1$, $F=F_1\otimes_{\mathbb{F}_q}\mathbb{F}$ be the fraction field of $X$, $s\in S$. Then the choice of a place $\bar{s}$ of $\bar{F}$ above $s$ defines an inertia group $I_s\subset {\text{Gal}}(\bar{F}/F)$. A $\bar{\mathbb{Q}}_l$-lisse sheaf is said to have ``\textit{Principal unipotent local monodromy at $s$}'' if the composition $I_s\subset {\text{Gal}}(\bar{F}/F)\twoheadrightarrow \pi_1^{\text{ét}}(X-S,x)-^{\rho}\rightarrow {\text{GL}}(V)$ factors through the largest pro-$l$ quotient ($\cong \mathbb{Z}_l$) of $I_s$ with an element of $I_s$ with image $a$ in $ \mathbb{Z}_l$ acting on $V$ as \text{exp}$(aN)$, where $N$ is nilpotent with one Jordan block. Let $\mathcal{T}^{(n)}(X,S)$ be the set of isomorphism classes of rank $n$ irreducible $\bar{\mathbb{Q}}_l$-smooth sheaves on $X-S$, with principal unipotent local monodromy at each $s\in S$. $\mathcal{T}^{(n)}(X,S)$ as a subset of the isomorphism classes of $\bar{\mathbb{Q}}_l$-lisse sheaves is stable under the ${\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$-action. Let $T(X_1,S_1,n)$ denote the number of fixed points of $\mathcal{T}^{(n)}(X,S)$ by the geometric Frobenius $\text{Frob}\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$. For each $m\geq 1$, let $(X_m,S_m):=(X_1,S_1)\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}$, then $T(X_1,S_1,n,m):= T(X_m,S_m,n)$, where $(X_m,S_m)$ is viewed as a pair over $\mathbb{F}_{q^m}$. The aim of the article under review is to give a computation of the number $T(X_1,S_1,n,m)$. It starts with a formula for $T(X_1,S_1,n)$, under the assumption that $n$ and $N_1$ are $\geq 2$, in terms of $N_1$, $n$, $q$, the degrees $\deg(s)$ for $s\in S_1$ and the coefficients of the polynomial $f(t):= \det(1-\text{Frob}\cdot t, H^1(X))$, where ${\text{Gal}}(\mathbb{F}/\mathbb{F}_q)$ acts on the $l$-adic cohomology group $H^1(X)$ by transport of structures. However, the first formula is not helpful to understand how the number of fixed points varies with $m$. One problem is that when one replaces $(X_1,S_1)/\mathbb{F}_q$ by $(X_m,S_m)/\mathbb{F}_{q^m}$, the number $(n/S_1):=\{\text{the largest divisor of }n$ that is prime to all $\deg(s)$ for $s\in S_1$ local systems; $\ell$-adic smooth sheaf; Lefschetz fixed point formula; automorphic representations; function fields; $\mathrm{GL}(n)$; principal unipotent local monodromy; trace formula P. Deligne and Y. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921-982. Étale and other Grothendieck topologies and (co)homologies, Cohomology of arithmetic groups, Finite ground fields in algebraic geometry, Modular and Shimura varieties, Structure of families (Picard-Lefschetz, monodromy, etc.) Counting local systems with principal unipotent local monodromy	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let A be an abelian variety over ${\mathbb{C}}$ and let $X\subset A$ be an algebraic curve; denote by T the sub-group of torsion points of A(${\mathbb{C}})$; if X is not an elliptic curve (hence $genus(X)\geq 2$ in this case), then $T\cap X({\mathbb{C}})$ is a finite set. This is the main theorem proved in this paper. Note that this was called ''the Manin- Mumford conjecture''. It is interesting what Manin wrote (in 1966) when talking about such problems: ''As Lang remarked, Mordell's conjecture is equivalent to the assertion that a curve in an abelian variety has only a finite number of points in common with any subgroup of finite type of variety. This formulation contains no explicit references to the ground field. Nevertheless, it seems probable that the proof of Mordell's conjecture cannot be achieved without using the arithmetic (in the wide sense of the word) of the ground field. If that is so, then our result can be regarded as a reduction of the general problem to the case when the ground field is an algebraic number field: a case which has so far remained inviolate to known methods'' [cf. \textit{Yu. I. Manin}, Am. Math. Soc., Translat., II. Ser. 50, 189-234 (1966), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)]. Note that Bogomolov proved that the n-primary part $T(n^{\infty})\cap X({\mathbb{C}})$ is finite for every integer $n>1$ (\textit{F. A. Bogomolov}, C. R. Acad. Sci. Paris, Sér. A 290, 701-703 (1980; Zbl 0457.14020)]. Note that together with the proof of the Mordell conjecture given by Faltings [\textit{G. Faltings}, ''Endlichkeitssätze für abelsche Varietäten über Zahlkörpern'', Invent. Math. 73, 349-366 (1983); Erratum, Invent. Math. 75, 381 (1984)], this result by the author proves a conjecture by S. Lang [cf. \textit{S. Lang}, Ann. Mat. Pura Appl., IV. Ser. 70, 229-234 (1965; Zbl 0151.274)]. For the proof the author first remarks that it suffices to prove the theorem in case $X\subset A$ is defined over a number field (under specialization $X\cap T$ ''cannot become smaller''). Note that an analogous statement over the algebraic closure of a finite field is not correct. Thus it is understandable that in the proof of the theorem a careful study is made what happens under reduction modulo $p^ 2$, where p is a carefully chosen prime number. This study of the torsion points in the local case is the main (and technically difficult) point of the proof. - We like to stress that the problem studied can be formulated purely over ${\mathbb{C}}$, but at the moment we have no proof which does not use arithmetic properties. Does there exist a proof using analytic tools? - For a related theorem, see VI.4 in the book edited by \textit{G. Faltings} and \textit{G. Wüstholz}, ''Rational points'' (Seminar Bonn/Wuppertal, 1983/84). This is a nice paper, the methods of proof are ingeneous and convincing. algebraic curve on abelian variety; Mordell conjecture; rational points on algebraic curves; Manin-Mumford conjecture; torsion points M. Raynaud, Courbes sur une variété abélienne et points de torsion. \textit{Inventiones} \textit{Mathematica 71}(1983), 207--233.Zbl 0564.14020 MR 0688265 Arithmetic ground fields for abelian varieties, Rational points, Global ground fields in algebraic geometry, Special algebraic curves and curves of low genus Courbes sur une variété abélienne et points de torsion	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $C$ be a nonsingular projective curve, and $J(C)$ be its jacobian. To a divisor $D$ of degree zero on $C$ is associated a point $\gamma (D)$ of the jacobian by integration, and so-called Abel's theorem states that the image of $D$ in the jacobian vanishes if and only if $D$ is rationally equivalent to zero, i.e., $D$ is a divisor of a rational function of $C$, in other words, it gives an algebraic condition for $\gamma (D)=0$, whereas the jacobian is defined complex-analytically. The following conjecture of \textit{S. Bloch} [``Lectures on algebraic cycles'', Duke Univ. Math. Ser. 4 (1980; Zbl 0436.14003)] can be regarded as the weight 2 counterpart of the above equivalence: For any smooth projective variety $V$ over $\mathbb{C}$, there exists a filtration on the Chow group of 0-cycles $\text{CH}_ 0(V)$. Let $S$ be a surface and let $z$ be a cycle on $V \times S$ with $\dim z=m=\dim V$. Then $z$ induces a map $z:\text{CH}_ 0 (V) \to \text{CH}_ 0 (S)$. - - We get also $[z]:gr^ \bullet \text{CH}_ 0(V) \to gr^ \bullet \text{CH}_ 0(S)$. Conjecture (\textit{S. Bloch}). The map $[z]$ depends only upon the cohomology class $\{c\} \in H^ 4(V \times S)$. Metaconjecture. There is an equivalence of category between a suitable category of polarized Hodge structures of weight 2 and a category built up from $gr^ 2\text{CH}_ 0 (S)$. The aim of this article is two-fold: to give a condition for the vanishing of cycles in the intermediate jacobian, and to construct filtrations on the Chow groups which satisfy the above conjectures. polarized Hodge structures of weight 2; vanishing of cycles in the intermediate jacobian; filtration on the Chow group Saito, H.: Generalization of Abel's theorem and some finiteness properties of 0-cycles on surfaces. Compositio math. 84, 289-332 (1992) Parametrization (Chow and Hilbert schemes), Algebraic cycles, Picard schemes, higher Jacobians, Jacobians, Prym varieties, Surfaces and higher-dimensional varieties Generalization of Abel's theorem and some finiteness property of zero- cycles on surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives a method of computing topological fundamental groups of complex algebraic varieties by expressing them as fibres of good fibrations and relating their fundamental groups to those of the total space of the fibration. Let $Y$ and $B$ be nonsingular algebraic varieties over $\mathbb{C}$. Let $p: Y\to B$ be a dominant morphism with general fibre $Y_0$ connected. Assume that $p$ has a section $s$ over a classical open set $U\subset B$. Under certain conditions on $s$ and $U$, the author determines the kernel of the morphism $\pi_1 (Y_0, s(0))\to \pi_1 (p^{-1} (U), s(0))$. He also gives a set of sufficient conditions for this kernel to be trivial. This is used to prove the main theorem: Let $F$ be a nonsingular connected quasi-projective variety of dimension $\geq 2$ and $\overline {F}$ its nonsingular projective compactification such that $\overline {F}- F$ is of codimension $\geq 2$ in $\overline {F}$. Let $W\subset F\times \mathbb{A}^n$ be a reduced (possibly reducible) effective divisor, $W_a=$ fibre over $a\in \mathbb{A}^n$. Assume that the locus of $a\in \mathbb{A}^n$ such that $W_a$ is not a reduced divisor in $F$ is of codimension $\geq 2$ in $\mathbb{A}^n$. Then the natural homomorphism $\pi_1 (F-W_a)\to \pi_1 ((F\times \mathbb{A}^n- W)$ is an isomorphism for a general $a\in \mathbb{A}^n$. Two applications of the theorem are given. For the first, consider general enough hyperplanes $V_i\subset \mathbb{P}^n$, degree $V_i= d_i$, $i=1, \dots, m$. Let $U$ be the subset of the Grassmannian of lines in $\mathbb{P}^n$ consisting of all lines which intersect $V_1\cup \dots \cup V_m$ in distinct $d_1+ \cdots+ d_m$ points and $U_p$ the subset of $U$ consisting of lines passing through a fixed point $p\in \mathbb{P}^n$. The author shows that for $n\geq 3$ and $\sum d_i\geq 3$, the natural monodromy representation of $\pi_1 (U)$ in the generalized mapping class group is surjective with kernel an abelian group generated by at most $m$ elements contained in the centre of $\pi_1 (U)$. Further, if $p$ is also general, then this representation is an isomorphism. The second application is the determination of the fundamental group of the complement of a projective plane curve $f^q+ g^p =0$, $p\geq 2$ and $q\geq 2$ being degrees of the two general homogeneous polynomials $f$ and $g$. Such results are due to \textit{O. Zariski} [Am. J. Math. 51, 305-328 (1929)], \textit{M. Oka} [Math. Ann. 218, 55-65 (1975; Zbl 0335.14005) and J. Math. Soc. Japan 30, 579-597 (1978; Zbl 0387.14004)]\ and \textit{A. Nemethi} [Math. Proc. Camb. Philos. Soc. 102, 453-457 (1987; Zbl 0679.14004)]. Zariski problem; topological fundamental groups of complex algebraic varieties; good fibrations; mapping class group; fundamental group of the complement of a projective plane curve DOI: 10.1016/0040-9383(94)00045-M Homotopy theory and fundamental groups in algebraic geometry, Fundamental group, presentations, free differential calculus Fundamental groups of open algebraic varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this very clearly written paper is to provide a tool for construction of explicit examples of hyperelliptic curves of genus two such that its Jacobian variety admits real multiplication. Let $K$ be a field of characteristic $p\neq 2$. Let $f(x)\in K[x]$ be an irreducible separable polynomial of degree six. Let $C_f$ denote the hyperelliptic curve defined by $y^2=f(x)$ and $J(C_f)$ its Jacobian. Assuming that the Galois group of $f$ is isomorphic to the alternating group of degree five and that $\text{End}^0_K(J(C_f))$ contains a real quadratic field $D$, the main theorem proves that $\text{End}_K(J(C_f))$ is isomorphic to an order of $D$ with discriminant $\equiv 5 \pmod 8$. For the proof, the notion of very smallness of a group representation, which is introduced in [\textit{Yu. G. Zarhin}, Prog. Math. 195, 473--490 (2001; Zbl 1047.14015)] plays a crucial role. Armed with this notion, the author proves the theorem by employing the representation theory of finite groups and algebras. Furthermore, in the last two sections he gives quite a few concrete examples of hyperelliptic curves satisfying the hypotheses of the theorem. hyperelliptic curve; real multiplication; Jacobian variety Jacobians, Prym varieties, Abelian varieties of dimension $> 1$, Curves of arbitrary genus or genus $\ne 1$ over global fields, Arithmetic ground fields for abelian varieties Hyperelliptic Jacobians with real multiplication	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, the author gives a generalization of a result of \textit{B. van Geemen} [Invent. Math. 78, 329--349 (1984; Zbl 0568.14015)] for any field $k$ of characteristic different from $2$. Namely, let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g \geq 2$. Let $\mathfrak{M}_g$ be the moduli space of smooth curves of genus $g$. Let $\mathfrak{I}_{g,k}$ be the closed subscheme of $\mathcal{A}_g \otimes k$ defined as the scheme theoretic image of $\mathfrak{M}_g \otimes k$ via the Torelli map. Let $\mathfrak{S}_{g,k}^{\text{small}}$ be the closed subscheme of $\mathcal{A}_g \otimes k$ defined by relations analogous to the classical Schottky-Jung relations. Theorem. The irreducible component of $\mathfrak{S}_{g,k}^{\text{small}}$ containing $\mathfrak{I}_{g,k}$ is equal to $\mathfrak{I}_{g,k}$. The main difficulty and the main achievement of the paper is to carry out a purely algebraically proof of the result. In order to define the Schottky relations for instance, the author uses the formalism of Mumford's theta function. The second novelty is the use of results of [\textit{G. Faltings} and \textit{C.-L. Chai}, ``Degeneration of abelian varieties'', Ergebnisse der Mathematik und ihrer Grenzbiete, 3. Folge, Band 22, Springer-Verlag (1990 ; Zbl 0744.14031)] on toroidal compactifications and the use of Fourier-Jacobi expansion of theta functions that enables to go deeper in the boundary. Schottky-Jung relations; theta functions; Mumford's uniformization Theta functions and curves; Schottky problem The small Schottky-Jung locus in positive characteristics different from two.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper contains two parts. In the first one, the following result is proved: ``Let M be a Riemann surface of genus $g,$ G(n,m) the Grassmann manifold of n-dimensional quotients of ${\mathbb{C}}^ m$ and k a positive integer with $k\geq n-2g$. Then the inclusion $Hol_ d(M,G(n,m))\to Map_ d(M,G(n,m))$ induces cohomology isomorphisms up to dimension $k- 2m^ 2g$, provided that $d\geq d_ 0(k,n,g)$. - Here $Hol_ d$ (resp. $Map_ d)$ stands for holomorphic (resp. continuous) maps of degree $d.$ The bound $d_ 0$ is done explicitly. For maps in the projective space, the result was shown by \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] who conjectured more general results, as the above ones. The proof is done by induction, starting with Segal's result, and uses extensively holomorphic bundles on M. One uses all this, and previous technique of the author [``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020)], to rederive in the second part of the paper the formula given by \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond., A 308, 523-615 (1982; Zbl 0509.14014)] for the cohomology of the moduli spaces of holomorphic bundles on curves. holomorphic maps; Riemann surface; Grassmann manifold; quotients; cohomology of the moduli spaces of holomorphic bundles on curves Frances Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. 24 (1986), no. 2, 221-275. Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Geometric invariant theory, Homogeneous spaces and generalizations On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $f:X\rightarrow C$ be a family of curves of genus $g$ with smooth general fiber. When $g=1$, \textit{K. Kodaira} proves a fundamental formula in [Ann. Math. (2) 78, 1--40 (1963; Zbl 0171.19601)] which represents the global invariants of the surface $X$ by the local invariants of the singular fibers together with the modular invariant. The purpose of this paper is to generalize Kadaira's formula to the higher genus cases. In the author's previous paper [Math. Z. 222, No. 4, 655--676 (1996; Zbl 0864.14016)], he defines the Chern numbers $c_{1}^{2}(F),c_{2}(F),\chi_{F}$ of a singular fiber $F$, and shows that, if $g=1$, then $c_{1}^{2}(F)=0$ and $c_{2}(F)$ is exactly the coefficient in Kodaira's formula according to the type of the singular fiber. His generalization is formulated with the help of these local invariants together with some natural modular invariants constructed from a holomorphic map from $C$ to the moduli space of semistable curves of genus $g$. Chern numbers; family of curves; singular fiber Chen, Z., Lu, J. and Tan, S.-L., On the modular invariants of a family of non-hyperelliptic curves of genus 3, http://math.ecnu.edu.cn/preprint/2010-006.pdf Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Chern numbers of a singular fiber, modular invariants and isotrivial families of curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Grothendieck's anabelian conjecture for smooth, hyperbolic curves over finite fields states that, given any two hyperbolic curves $U$ and $V$ over finite fields, any isomorphism between the arithmetic fundamental groups $\pi_1(U)$ and $\pi_1(V)$ arises from a unique isomorphism $U \to V$. In particular, if $U$ is a smooth hyperbolic curve over a finite field, the fundamental group of $U$ determines the isomorphism class of $U$. This was proven in \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)] in the affine case and in \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47 (2007; Zbl 1143.14305)] in the proper case. The current paper proves a stronger, ``prime-to-$p$'' version of this result: For any smooth curve $X$ over a finite field $k_X$, there is a natural inclusion $\pi_1(\overline{X} := X \times_{\text{Spec } k_X} \text{Spec } \overline{k_X}) \hookrightarrow \pi_1(X)$. Let $\Delta_X$ be the maximal prime-to-$p$ quotient of $\pi_1(\overline{X})$, and let $S$ be the kernel of the canonical map $\pi_1(\overline{X}) \to \Delta_X$. Write $\Pi_X := \pi_1(X)/S$. Then, if $U$ and $V$ are smooth hyperbolic curves defined over finite fields, any isomorphism $\Pi_U \to \Pi_V$ arises from a unique isomorphism $U \to V$. In particular, $\Pi_U$ determines the isomorphism class of $U$. A similar birational result is also proved for absolute Galois groups of function fields, strengthening a result of \textit{K. Uchida} [Ann. Math. 106 (1977; Zbl 0372.12017)]. The outline of the proof is similar to that of \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)]. Given $\Pi_X$, one first addresses the ``local theory'', attempting to recover the closed points of $X$ by recovering the conjugacy classes of decomposition groups of closed points of $X$ as subgroups of $\Pi_X$. In the case treated by the current paper, the points of $X$ can only be recovered up to a finite set $E_X$, the ``exceptional set''. Then, one uses Kummer theory to recover the group $\mathcal{O}_{E_X}^{\times}$ of rational functions on $X$ with support away from $E_X$. Lastly, one recovers the additive structure on the set of rational functions $k_X(X)$. The problem of the exceptional set $E_X$ is dealt with by a trick allowing one to pass to an infinite extension of $k_X$ over which $X$ has infinitely many points, followed by a descent argument. \bibSaidi-Tamagawa1article label=Saïdi-Tamagawa1, author=Saïdi, Mohamed, author=Tamagawa, Akio, title=A prime-to-$p$ version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic $p>0$, journal=Publ. Res. Inst. Math. Sci., volume=45, number=1, pages=135--186, date=2009, doi=10.2977/prims/1234361157, issn=0034-5318, review=\MR2512780, Finite ground fields in algebraic geometry, Coverings of curves, fundamental group, Arithmetic ground fields for curves, Curves over finite and local fields A prime-to-$p$ version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic $p>0$	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the equivalence of three ``points of view'' on the notion of a $G$-torsor when the base scheme is a Dedekind scheme. The proof is based on author's generalization of Chevalley's theorem on semi-invariants in [\textit{A. Borel}, Linear algebraic groups. 2nd enlarged ed. Graduate Texts in Mathematics, 126. New York etc.: Springer-Verlag. (1991; Zbl 0726.20030)] and a Tannakian description of $G$-torsors given [\textit{N. Saavedra Rivano}, in: Catégories Tannakiennes. Lecture Notes in Mathematics. 265. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0241.14008)] and by \textit{M. V. Nori} [Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)]. This interesting paper includes as an application that the fibered category of $G$-torsors on a regular proper curve over a field $k$ is an Artin stack locally of finite presentation over $k$. In the introduction, the author fixes notation and presents main results of the paper under review. Let $X$ be a Dedekind scheme, $T$ the $X$-scheme, $V$ a vector bundle on $X$, $L \subset V$ a locally split line bundle. In this notation let $\underline{Aut}(V,T)$ be the representable functor whose $T$-points are automorphisms $f$ of $V\otimes{\mathcal O}_T$ such that $f(L\otimes {\mathcal O}_T) = L\otimes {\mathcal O}_T $. Let $V$ be a representation of a flat, affine group scheme $G$ of finite type over $X$. Let $\{X_1, \ldots , X_r \}$ be the (nonempty) connected components of $X$ and ${\mathbf i}=(i_1, \ldots , i_r)$ a sequence of natural numbers. Let $\bigwedge^{\mathbf i} V$ be the vector bundle such that $\bigwedge^i V\| X_k = \bigwedge^{i_k} V\| X_k $ for $k = 1, \ldots , r$ and let $ t(V)$ be some finite iteration of the operations $\otimes, \bigwedge^i, \mathrm{Sym}^j, \bigoplus, (\cdot)^* $. The author calls such an iteration a tensorial construction. Theorem 1.1. Let $G$ be a flat algebraic group over a Dedekind scheme $X$. There is a representation $V$ of $G$, a tensorian construction $ t(V)$, and a locally split line bundle $L \subset t(V)$ such that $G \simeq \underline{Aut}(V,T)$. Theorem 1.2. Let $G$ and $X$ be as above. Let $Y$ be a scheme faithfully flat over $X$. There is a natural equivalence that is functorial in $Y$ of the following groupoids: (i) The groupoid of $G_Y$-torsors; (ii) The groupoid of tensor functors $F: {\mathbf{Rep}}\; G \to {\mathbf{Bun}}_Y$, that on each fiber over $X_{\mathrm{Zar}}$ are faithful and exact. The second section is devoted to the application of Theorem 1.1 to the stack of $G$-torsors over a curve. Let $k$ be a field, let $GTor_X$ denotes the fibered category that assigns to a $k$-scheme $T$ the groupoid of $G_{X_T}$-torsors. Theorem 2.1. The fibered category $GTor_X$ is an Artin stack, locally of finite presentation over $k$. In section 3 algebraic groups over Dedekind schemes are dealt with. Here, the author proves Theorem 1.1. The last section treats the Tannakian viewpoint. It begins with a treatment of the category of representations ${\mathbf{Rep}} G $ of a flat algebraic group $G$ over a Dedekind scheme and for $X$-scheme $Y$ the fibered category ${\mathbf{Bun}}_Y$, where for an object $U$ in $X_{\mathrm{Zar}}$, ${\mathbf{Bun}}_Y(U) = \mathbf{Bun}_{Y_U}$ is the category of all finite rank vector bundles on $ Y(U)$. Let $P \to Y$ be a $G_Y$-torsor, $F_P$ denotes the tensor functor from ${\mathbf{Rep}} G$ to ${\mathbf{Bun}}_Y$, that on each fiber over $X_{\mathrm{Zar}}$ is faithful and exact (Lemma 4.5). The converse is true. Let $F: {\mathbf{Rep}} G \to {\mathbf{Bun}}_Y$ be a tensor functor that on each fiber over $X_{\mathrm{Zar}}$ is faithful and exact. The author shows that there is a natural equivalence $F \simeq F_P$ for a uniquely defined $G_Y$-torsor $P$. Following Nori and generalizing Nori`s approach to the investigated case the author of the paper under review extends (uniquely) the functor $F$ to a tensor functor from the fibered category ${\mathbf{Rep}}^` G$ over $X_{\mathrm{Zar}}$, where for each $U$ in $X_{\mathrm{Zar}}$ , ${\mathbf{Rep}}^` G(U) = {\mathbf{Rep}}_{U}^` G$ is the category of flat quasicoherent ${\mathcal O}_U $-modules to the fibered category ${\mathbf{QCoh}}_Y$, where for each $U$ in $X_{\mathrm{Zar}}$, ${\mathbf{QCoh}}_Y(U) = \mathrm{QCoh}_{Y_U}$ is the category of quasicoherent ${\mathcal O}_{Y_U} $-modules. The main theorem of the section is Theorem 4.8. It`s prove gives Theorem 1.2. Torsor; Dedekind scheme; fibered category; Artin stack; tensorial construction; groupoid; tensor functor; category of representations Broshi, M., $G$-torsors over a Dedekind scheme, J. Pure Appl. Algebra, 217, 11-19, (2013) Group schemes, Linear algebraic groups over finite fields $G$-torsors over a Dedekind scheme	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In {\S}1 the author develops a general theory of divisors and linear systems on an integral projective curve with Gorenstein singularities. A (generalized) divisor is synonymous with a fractional ideal, i.e. a subsheaf of the constant sheaf of the function field of the curve C which is coherent as ${\mathcal O}_ C$-module. The sum of two divisors is defined only if one of them is a Cartier divisor. The notion of complete linear systems, Riemann-Roch theorem and Serre duality theorem remain valid. As an example the author gives a new proof of a theorem of Fujita which says that the canonical system $\| K\|$ has no base points if $p_ a(C)>0$, and is very ample if $p_ a(C)>1$. - In {\S}2 he applies the results of {\S}1 to prove a theorem about an irreducible plane curve of degree $k.$ If Z is a subscheme of finite length d of C, it gives a precise upper bound for $h^ 0({\mathcal O}_ C(Z))$ in terms of k and d and tells when the bound is attained. The proof is by induction on k, and even when C is non-singular (the case Max Noether considered) one has to deal with singular curves in the course of induction, and this is the point ignored by Noether. subscheme of irreducible plane curve; linear systems; integral projective curve with Gorenstein singularities; Cartier divisor R. Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), 375--386. Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special algebraic curves and curves of low genus Generalized divisors on Gorenstein curves and a theorem of Noether	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this thesis we prove cohomological descent between the (small) étale site and the logarithmic sites for the sheaf of logarithmic differential forms. The existing techniques in the literature allow us to prove cohomological descent for a broader class of sheaves. Specifically, we compare the Kummer log étale cohomology with the étale cohomology for quasi-coherent sheaves defined on the Kummer log étale site. By using this, we prove that morphisms of log schemes where the underlying morphism of schemes is affine have trivial higher direct images for ket locally classical quasi-coherent sheaves on the Kummer log étale site. In the same spirit, we compare the log étale cohomology with the étale cohomology for classical vector bundles on the log etale site. We also give a proof for the fact that log regular log schemes are rationally singular, something which is considered known. The case of Zariski log schemes has been already treated by \textit{K. Kato} [Am. J. Math. 116, No. 5, 1073--1099 (1994; Zbl 0832.14002)]. We provide a proof for the case of étale log schemes. As a final application we define algebraic log de Rham cohomology for fs log schemes using the log étale site. Along these lines, we prove with a different method the theorem of A. Ogus, [\textit{A. Ogus}, Lectures on logarithmic algebraic geometry. Cambridge: Cambridge University Press (2018; Zbl 1437.14003), Theorem V.4.2.5(1)], regarding the generalization of algebraic de Rham cohomology to schemes with toric singularities. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Generalizations (algebraic spaces, stacks) Cohomological descent for logarithmic differential forms in the log etale topology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1993 \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] gave a ``GIT-free'' construction of the moduli space of semistable vector bundles on smooth projective curve $X$ over an algebraically clased field $k$. This construction was presented by him at German-Spanish Workshop on Moduli spaces of Vector Bundles, Essen, February 2007. The present notes give a self-contained and clear introduction to the moduli spaces of vector bundles on a curve and the generalized $\Theta$-divisor. For the sake of simplicity, all the exposition is devoted to the rank 2 case with determinant isomorphic to $\omega_X$. Also there is a brief discussion of generalizations the arbitrary rank and degree, supplied with necessary references. Notes contain following sections. -- Outline of the construction. -- Background and notation. -- A nice over-parameterizing family. -- The generalized $\Theta$-divisor. -- Raynaud's vanishing result for rank two bundles. -- Semistable limits. -- Positivity. -- The construction. -- Prospect to higher dimension. moduli space; vector bundles on a curve; generalized Theta divisor Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Faltings' construction of the moduli space of vector bundles on a smooth projective curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be a smooth projective curve of genus $g\ge 2$. For all integers $d$ and $r\ge 1$ let $U(r,d)$ be the moduli space of rank $r$ and degree $d$ stable vector bundles on $X$. For each positive integer $k$ set $B_{r,d}^k:= \{E\in U(r,d)\mid h^0(E)\ge k\}$ (the Brill-Noether loci for rank $r$) [\textit{I. Grzegorczyk} and \textit{M. Teixidor i Bigas}, Lond. Math. Soc. Lect. Note Ser. 359, 29--50 (2009; Zbl 1187.14038)]. The Brill-Noether loci $B_{r,d}^k$ and their twisted version [\textit{S. Casalaina-Martin} and \textit{M. Teixidor i Biga}, Math. Nachr. 284, No. 14--15, 1846--1871 (2011; Zbl 1233.14025); \textit{G. H. Hitching} et al., ``Nonemptyness and smoothness of twisted Brill-Noether loci'', Ann. Mat. Pura ed Appl., (to appear)] are very interesting and give Torelli-type theorems. He study the tangent cones to $B_{r,d}^k$ and give a rank $r$ generalization of the Riemann-Kempf singularity theorem. To prove it fr any rank $r$ vector bundle $V$ over $X$ the author defines a surjective rational map from the Hilbert scheme $\mathrm{Hilb}^d(\mathbb {P}V)$ to the Quot-scheme $\mathrm{Quot}^{0,d}(V^\ast)$. As a corollary if $k=r$ and $h^0(X,E) =r+n$ he proves that the $n$-th secant variety of the rank $1$ locus of $\mathbb{P}\mathrm{End}E$ is contained in the tangent cone of $B_{r,d}^r$. vector bundles; Brill-Noether theory for vector bundles; singularities; stable vector bundles; Riemmann-Kempf singularity theorem Vector bundles on curves and their moduli, Special divisors on curves (gonality, Brill-Noether theory) A Riemann-Kempf singularity theorem for higher rank Brill-Noether loci	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the present paper, we discuss the homotopy sequences for varieties over curves. We prove that, for instance, for a morphism from a normal variety to a certain smooth curve over a field of characteristic zero, it holds that the induced outer homomorphism between the étale fundamental groups satisfies some group-theoretic conditions if and only if the homotopy sequence for the morphism of varieties is exact. Moreover, we also give a refinement of a known result concerning the Grothendieck conjecture for hyperbolic polycurves of dimension two by means of our study of homotopy sequences. Homotopy theory and fundamental groups in algebraic geometry Homotopy sequences for varieties over curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In an earlier paper [\textit{A. Moriwaki}, J. Am. Math. Soc. 11, No. 3, 569--600 (1998; Zbl 0893.14004)] the second author proved the relative Bogomolov's inequality for algebraic varieties over an algebraically closed field of characteristic zero. This capacious paper under review consisting of an introduction, 10 sections and an appendix is devoted to prove the arithmetic analogues of the mentioned result and of Cornalba-Harris-Bost's inequality [\textit{M. Cornalba} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455--475 (1988; Zbl 0674.14006)] in the case of semistable families of arithmetic varieties. Since the usual push-forward for arithmetical cycles is insufficient for the authors' purposes, they need to introduce the notion of arithmetic $L^1$-cycle and to extend the usual Chow groups defined by Gillet-Soulé [\textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. In section 1 of the paper the investigation of locally integrable forms is done. Section 2 is devoted to the study of the properties of three variants of arithmetic Chow groups. In section 3 the authors introduce the notion of weak positivity of arithmetic divisors. In sections 4 and 5 the authors prove the Riemann-Roch theorems for generically finite morphisms and for stable curves. Sections 6 and 7 provide respectively an asymptotic upper bound of analytic torsion and formulae for arithmetic Chern classes. The first main result of the paper consists of a relative Bogomolov inequality in the arithmetic case and appears in section 8. The second main result of the paper is a relative Cornalba-Harris-Bost inequality (theorem 10.1.4). Finally, a comparison of theorems 8.1 and 10.1.4 is given (each of them has its advantage). This is a substantial paper which contains many deep results on the subject. vector bundles; relative Bogomolov's inequality; semistable families of arithmetic varieties; arithmetic Chow group; Riemann-Roch theorems; locally integrable forms [23] Kawaguchi (S.) and Moriwaki (A.).-- Inequalities for semistable families of arithmetic varieties, J. Math. Kyoto Univ. 41 (2001), no.~1, 97-182. &MR~18 \| &Zbl~1041. Arithmetic varieties and schemes; Arakelov theory; heights, (Equivariant) Chow groups and rings; motives, Topological properties in algebraic geometry Inequalities for semistable families of arithmetic varieties.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Generalizing a construction by G. Humbert, the author constructs a family of curves of genus $g>0$ whose Jacobian admit real multiplication by $2\cdot\cos(2\pi/(2g+1))$. This generalized construction is used to establish the existence of a regular extension of the rational function field ${\mathbb{Q}}(T)$ having the Galois group $PSL_ 2({\mathbb{F}}_{p^ 2})$ provided that $p\equiv \pm 2$ mod 5. Examples of 2-parameter families of hyperelliptic curves over ${\mathbb{Q}}$ of genus $g=7$ (resp. 19) are given whose Jacobian is isogenous over ${\mathbb{Q}}$ (resp. an extension of ${\mathbb{Q}})$ to a product of 7 (resp. 19) elliptic curves. Proofs are outlined only or omitted. complex multipliction; isogenous Jacobian; real multiplication; rational function field; hyperelliptic curves Mestre, J. F.: Courbes hyperelliptiques à multiplications réelles. C. R. Acad. sci. Paris, ser. I math. 307, 721-724 (1988) Elliptic curves, Algebraic functions and function fields in algebraic geometry, Complex multiplication and abelian varieties, Galois theory Courbes hyperelliptiques à multiplications réelles. (Hyperelliptic curves with real multiplications)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be an integral projective curve defined over the complex numbers and $D_ 1,\dots,D_ n$ mutually disjoint effective Cartier divisors on $X$. A parabolic structure on a torsion free coherent sheaf $E$ on $X$ consists on a filtration of the vector space $H^ 0(E\mid D_ j)$ for each $j=1,\dots,n$, and on some weights associated to these filtrations allowing one to define a parabolic sheaf and the notion of (semi-)stable generalized parabolic sheaf. This technical notion occurs naturally when one studies vector bundles on singular curves by pulling them back to the desingularization of the curve. The author proves existence and studies the moduli spaces of semi-stable generalized parabolic sheaves of a certain type (for which the filtrations have only 2 steps). As an application, she obtains partial desingularizations of moduli spaces of torsion free sheaves on a nodal curve. effective Cartier divisors on integral projective curve; vector bundles on singular curves; desingularization; moduli spaces of semi-stale generalized parabolic sheaves Vector bundles on curves and their moduli, Singularities of curves, local rings, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) Generalized parabolic sheaves on an integral projective curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a translation of the book ``Le problème des modules pour les branches planes'' of \textit{O. Zariski} [Hermann, Paris (1973; Zbl 0317.14004)] by Ben Lichtin. It is based on notes from a course of Zariski at Centre de Mathématiques de l'École Polytechnique 1973 and contains an appendix by B. Teissier considering the moduli problem from the point of view of deformation theory. Zariski's aim is to study the space of isormorphism classes of plane curve singularities (analytically irreducible curve germs) of given equsingularity type, i.e. the moduli space $M(\Gamma)$ of plane curve singularities with fixed semigroup $\Gamma$. His ideas and results were the basis for further research in this direction [cf. for example \textit{O. A. Laudal} and the reviewer, ``Local moduli and singularities'', Lect. Notes Math. 1310 (1988; Zbl 0657.14005)]. The first three chapters of the book introduce the basic notions, especially invariants as the semigroup, the conductor, the characteristic of a branch, short parametrizations, etc. Then the moduli space $M(\Gamma)$ is studied. It is proved that $M(\Gamma)$ is not seperated in general. The structure of $M(\Gamma)$ is analyzed for special examples. The dimension of the generic component of the moduli space for $\Gamma=\langle n, n+1\rangle$ is computed. This was generalized later on by C. Delorme. The appendix of Teissier is based on the idea that each plane curve singularity with semigroup $\Gamma$ appears as a deformation of the associated monomial curve, more precise in its negatively weighted part. It turns out that every plane branch has a miniversal equisingular deformation, over a smooth base, with an equisingular section. Among several results a ``natural'' compactification of $M[\Gamma)$ is given, as well as an interpretation of the generic component. moduli problem; plane curve singularity; semi group; deformation Zariski, O., \textit{The Moduli Problem for Plane Branches (with an Appendix by Bernard Teissier)}, 39, (2006), AMS, Providence RI Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry The moduli problem for plane branches. With an appendix by Bernard Teissier. Transl. from the French by Ben Lichtin	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book gives a full account of the theory of moduli for elliptic curves order ${\mathbb{Z}}$ from a scheme-theoretical point of view. In order to give representable moduli problems elliptic curves need to be rigidified by introducing supplementary structures not preserved by the automorphism group. The $\Gamma$ (N)-problem, or full level-N problem, is determined by a choice of generators of the group of division points with order dividing N. For elliptic curves over an arbitrary base ring this has to be stated in terms of subgroup schemes exactly generated by two divisors. There are similar statements for the $\Gamma_ 0(N)-$, $\Gamma_ 1(N)$-problems and their variants. The authors introduce functorial machinery to treat these problems in a uniform way and avoid algebraic stacks by the device of relative representability. The first four chapters give a self-contained account of the elementary theory of elliptic curves over a scheme, their finite subgroup and moduli problems. The reader is referred to other sources for deeper results such as the Serre-Tate theorem on deformations and the construction of Tate curves. The subsequent chapters deal with the intricate details involved in completing the moduli scheme at infinity and describing its structure at points corresponding to super-singular curves. In particular two chapters cover problems specific to characteristic p and Igusa curves. The culminating result is a general good-reduction theorem for Jacobians. Altogether this book is an essential complement to the many works on the transcendental theory of elliptic curves. full level-N problem; elliptic curves; moduli scheme at infinity; characteristic p; Igusa curves; good-reduction theorem for Jacobians 26. Katz, Nicholas M. and Mazur, Barry \textit{Arithmetic moduli of elliptic curves}Annals of Math. Studies, vol. 108, Princeton Univ. Press, Princeton, NJ, 1985 Math Reviews MR772569 Families, moduli of curves (algebraic), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fine and coarse moduli spaces, Elliptic curves, Algebraic moduli problems, moduli of vector bundles, Modular and automorphic functions, Special algebraic curves and curves of low genus Arithmetic moduli of elliptic curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this third lecture of these proceedings the author introduces the Picard functor of a morphism $f:X\to Y$ of schemes and discusses the usual representability criteria due to Grothendieck, Serre, Murre, Oort and others. The case of the Picard functor of a proper flat curve $f:X\to S$, locally of finite presentation, is discussed in details: its neutral component, its unipotent part and its torical and abelian component. Picard functor; Picard scheme; relative effective divisors; flat curve Picard groups, Arithmetic ground fields for curves, Local ground fields in algebraic geometry The Picard functor	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $\mathcal{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points. The Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the geometric fundamental groups of the ``Teichmüller tower'' of $\{ \mathcal{M}_{g,n} \}$ (i.e. the collection of all the stacks and the natural maps between them). Ihara showed that this action extends to a faithful action of the profinite Grothendieck-Teichmüller group $\widehat{\mathrm{GT}}$, which contains $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. \par The authors of the paper under review prove that $\widehat{\mathrm{GT}}$ is actually equal to the group of homotopy automorphisms of the Teichmüller tower in genus $g = 0$. They use an operadic model $\mathcal{M}$ of the genus $0$ Teichmüller tower: they replace marked point by boundary circles, which allows to glue curves along boundary components and to fill boundary components. These operations yield maps between the corresponding moduli spaces, and the operad obtained this way is equivalent to the classical framed little disks operad. The authors' first main theorem states more precisely that $\widehat{\mathrm{GT}}$ is the group of homotopy automorphisms of the profinite completion of $\mathcal{M}$. The difficult part of this theorem is showing that the obvious faithful action of $\widehat{\mathrm{GT}}$ on each $\mathcal{M}(n)$ is compatible with the operad structure (which is related to [\textit{A. Hatcher} et al., J. Reine Angew. Math. 521, 1--24 (2000; Zbl 0953.20030)]) and that all homotopy automorphisms are obtained this way. \par The authors' second main theorem deals with the compactification $\overline{\mathcal{M}}_{g,n}$ of $\mathcal{M}_{g,n}$, obtained by allowing nodal singularities in the curves. These moduli spaces admit a natural operadic structure by gluing curves along marked points. The operad $\mathcal{M}$ maps to $\overline{\mathcal{M}}_{0,\bullet+1}$ (the latter being a homotopy quotient of the former under an $\mathrm{SO}(2)$-action [\textit{G. C. Drummond-Cole}, J. Topol. 7, No. 3, 641--676 (2014; Zbl 1301.55005)]). The authors prove that the action of $\widehat{\mathrm{GT}}$ on the profinite completion of $\mathcal{M}$ extends to a nontrivial action on the profinite completion of $\overline{\mathcal{M}}_{0,\bullet+1}$. Note that actions of $\widehat{\mathrm{GT}}$ are usually constructed on schemes whose associated complex analytic spaces are $K(\pi,1)$, whereas $\overline{\mathcal{M}}_{0,n+1}$ is simply connected. infinity operads; Grothendieck-Teichmüller group; absolute Galois group; moduli space of curves , Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Loop space machines and operads in algebraic topology, Abstract and axiomatic homotopy theory in algebraic topology Operads of genus zero curves and the Grothendieck-Teichmüller group	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Grothendieck-Riemann-Roch theorem plays an essential role in the study of algebraic geometry. Roughly speaking, it describes how far the Chern character map between the algebraic $K$-theory and the algebraic Chow-theory to be commutative with the push-forward operations w.r.t. proper morphisms. This theorem vastly generalizes the classical Riemann-Roch theorem on compact Riemann surfaces, and hence it can be viewed as an Atiyah-Singer type index theorem. In previous work, the Grothendieck-Riemann-Roch theorem was extended by several mathematicians to an arithmetic setting in the sense of Arakelov. In this situation, the generators of algebraic $K$-groups and algebraic Chow groups are equipped with additional Hermitian structures so that the fibers of an arithmetic variety over the infinite places can be treated as the same footing as the finite ones. Historically, \textit{H. Gillet} and \textit{C. Soulé} [Invent. Math. 110, No. 3, 473--543 (1992; Zbl 0777.14008)] proved a degree one version of the arithmetic Grothendieck-Riemann-Roch theorem, they used the theory of Ray-Singer analytic torsion to define a reasonable push-forward on the level of arithmetic $K$-groups. Whence \textit{J.-M. Bismut} and \textit{K. Köhler} [J. Algebr. Geom. 1, No. 4, 647--684 (1992; Zbl 0784.32023)] developed the analytic torsion theory to higher degree case, a complete proof of the arithmetic Grothendieck-Riemann-Roch was given by Gillet, Rössler and Soulé [\textit{H. Gillet} et al., Ann. Inst. Fourier 58, No. 6, 2169--2189 (2008; Zbl 1152.14023)]. But, due to technical reason, these two works had to restrict to push-forward by morphisms which are smooth over generic fibers. The aim of the article under review is to break this limit, namely is to generalize the works of Gillet-Soulé and Gillet-Rössler-Soulé to arbitrary projective morphisms, not necessarily smooth over the generic fibers. This forces the authors to enlarge the arithmetic $K$-groups and the arithmetic Chow groups to afford corresponding push-forward functorialities. The strategy the authors followed was replacing smooth differential forms in the theory of Gillet-Soulé by currents with possibly non-empty wave front sets. As a consequence, by explaining the relationship between the topological correction term and the generalized analytic torsion classes, the authors obtained all the possible forms of the arithmetic Grothendieck-Riemann-Roch theorem. Arakelov theory; Grothendieck-Riemann-Roch theorem; projective morphism Burgos, J. I.; Freixas, G.; Litcanu, R., The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms, Ann. Fac. Sci. Toulouse Math. (6), 23, 3, 513-559, (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Riemann-Roch theorems The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A classical theorem of \textit{K. Brauner} [Abh. Hamburg 6, 1--55 (1928; JFM 54.0373.01)] gives a formula for the generators and relations of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular curve in $\mathbb C^2$. These elegant formulas depend only on the characteristic pairs of a Puiseux series expansion of the curve. Zariski states this theorem as part of his discussion on Puiseux series in the chapter on resolution of singularities in his book ``Algebraic surfaces''. Berlin: Springer (1935; Zbl 0010.37103)]. The arithmetic analogue of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular plane curve is thus the algebraic fundamental group $\pi_1(\text{Spec}(R)-V(f))$, where $R=k[[x,y]]$ is a power series ring over an algebraically closed field $k$, and $f\in R$ is irreducible. In positive characteristic, Puiseux series expansions do not always exist. In this paper we prove an arithmetic analogue of Brauner's theorem, valid in arbitrary characteristic. The generators and relations in our theorem for the prime to $p$ part of the algebraic fundamental group coincide with those of Brauner's theorem. Main theorem. Let $R=k[[x,y]]$ be a power series ring over an algebraically closed field $k$ of characteristic $p\geq 0$. Suppose that $f\in R$ is irreducible. Let $U=\text{Spec}(R_f)= \text{Spec}(R)-V(f)$. Let $(m_i,n_i)$, $1\leq i\leq g$, be the characteristic pairs of $f$. Then $\pi_1^{(p)}(U)$ is isomorphic to the prime to $p$ part of the pro-finite completion of the free group on the symbols $Q_0,\dots,Q_g,P_1, \dots,P_g$ with the relations \[ Q_i^{m_i}= P_i^{\overline n_i} Q_{i-1}^{m_{i-1}m_i},\;1\leq i\leq g,\quad Q_0=1, \] \[ P_{i+1} P_i^{y_i}Q_{i-1}^{m_{i-1} x_i}=Q_i^{x_i},\;1\leq i\leq g-1, \] $x_i$ and $y_i$ are integers such that $x_i\overline n_i=y_im_i+1$, $\overline n_i=n_i-n_{i-1}m_i$. The prime to $p$ part of a fundamental group $\pi_1$ is the quotient $\pi_1^{(p)}$ of $\pi_1$ by the closed normal subgroup generated by its $p$-Sylow subgroups. Let $C=V(f)$, $m$ be the maximal ideal of $R$. The basic strategy of the proof is to construct an embedded resolution of singularities $\tau:X\to\text{Spec}(R)$ of $C$ such that the preimage of $C$ is a simple normal crossing divisor. Then Abhyankar's lemma [\textit{S. Abhyankar}, Am. J. Math. 81, 46--94 (1959; Zbl 0100.16401)] shows that $\pi_1^{(p)} (\text{Spec} (\widehat{\mathcal O}_{X,x})- \tau^{-1}(C))$ has an extremely simple form for any $x\in\tau^{-1}(m)$. We further compute the prime to $p$ part of the algebraic fundamental group of the complement of $\tau^{-1}(C)$ in the formal completion of $X$ along each of the exceptional curves of $X\to\text{Spec}(R)$. curve singularity; algebraic fundamental group; arbitrary characteristic; power series ring Cutkosky, S. D.; Srinivasan, H.: The algebraic fundamental group of a curve singularity. J. algebra 230, 101-126 (2000) Coverings of curves, fundamental group, Singularities of curves, local rings, Formal power series rings, Algebraic functions and function fields in algebraic geometry, Germs of analytic sets, local parametrization, Homotopy theory and fundamental groups in algebraic geometry The algebraic fundamental group of a curve singularity	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The starting point for the paper under review is a series of (mostly unpublished) investigations by Faber and Zagier into relations between tautological classes on the moduli space of curves. Faber and Zagier empirically discovered an infinite sequence of polynomial expressions in the tautological classes on $M_g$, governed by two curious power series, which as far as they could check on computer would all vanish in the tautological ring. They conjectured that this would indeed always be the case, which was later proved by \textit{R. Pandharipande} and \textit{A. Pixton} [``Relations in the tautological ring'', \url{arXiv:1101.2236}]. Later, Pixton ingeniously wrote down an extension of their formulas to the spaces $M_{g,n}$, and then to the Deligne-Mumford boundary $\overline{M}_{g,n}$. Pixton conjectured that not only should these extended formulas also always produce relations; moreover, all tautological relations on all spaces $\overline{M}_{g,n}$ should arise in this way. We remark that this conjecture contradicts the Faber conjecture on $M_g$, so Faber and Zagier did not expect that all relations would arise from this construction. The main result of this paper is that Pixton's extended Faber-Zagier relations are indeed true in cohomology for all $g$ and $n$. After this paper was written, \textit{F. Janda} [``Tautological relations in moduli spaces of weighted pointed curves'', \url{arXiv:1306.6580}] proved that these relations are in fact true also in the Chow rings. The main tool used here is the Givental-Teleman classification of semisimple CohFTs. This is the reason their proof only works in cohomology -- the Givental-Teleman classification is only known in cohomology, due to the fact that it uses the Madsen-Weiss theorem. We remark that the Madsen-Weiss theorem, hence also the Givental-Teleman classfication, is expected to remain valid also in Chow. In any case, the authors consider the behavior of Witten's 3-spin CohFT as they move towards a non-semisimple point of the Frobenius manifold, in terms of the action of the R-matrix. Applying the R-matrix gives what appears to be a divergent expression. Knowing that this expression must in fact converge, it follows that all coefficients before negative exponents are actually zero, which produces relations between tautological classes. The relations they obtain in this way are exactly Pixton's relations. \textit{F. Janda} [``Relations in the Tautological Ring and Frobenius Manifolds near the Discriminant'', \url{arXiv:1505.03419}] has recently proved a striking universality result: all relations obtained by this method, for \textit{any} generically semisimple CohFT, follow from the 3-spin relations (i.e. Pixton's). moduli of curves; tautological rings; cohomological field theories R. Pandharipande, A. Pixton, and D. Zvonkine, Relations of $\overline{M}_{g,n}$ via $3$-spin structures, J. Amer. Math. Soc. 28 (2015), 279--309. Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Relations on $\overline{\mathcal{M}}_{g,n}$ via $3$-spin structures	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Grothendieck abelian categories play the role of models of possibly noncommutative schemes [\textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002); \textit{J. T. Stafford} and \textit{M. Van den Bergh}, Bull. Am. Math. Soc., New Ser. 38, No. 2, 171--216 (2001; Zbl 1042.16016; \textit{M. Kontsevich} and \textit{A. L. Rosenberg}, in: The Gelfand Mathematical Seminars, 1996--1999. Dedicated to the memory of Chih-Han Sah. Boston, MA: Birkhäuser. 85--108 (2000; Zbl 1003.14001)], which is motivated by the Gabriel-Rosenberg reconstruction theorem claiming that a quasi-separated scheme can be reconstructed, up to isomorphism of schemes, solely from the abelian category of quasi-coherent sheaves on the scheme, which is a Grothendieck category [\url{https://stacks.math.columbia.edu/}]. The theorem was initially established for noetherian schemes by \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323--448 (1962; Zbl 0201.35602)] and generalized to quasi-separated schemes by \textit{A. Rosenberg} [\url{https://archive.mpim-bonn.mpg.de/id/eprint/1516/}]. The Gabriel-Popescu theorem [\textit{N. Popesco} and \textit{P. Gabriel}, C. R. Acad. Sci., Paris 258, 4188--4190 (1964; Zbl 0126.03304)] allows of interpreting Grothendieck categories as a linear version of Grothendieck topoi [\textit{W. Lowen}, J. Pure Appl. Algebra 190, No. 1--3, 197--211 (2004; Zbl 1051.18007)]. Different $2$-categories obtain, depending on the choice of morphisms. \begin{itemize} \item[$\mathsf{Grt}$] the $2$-category of Grothendieck categories and left adjoints as morphisms \item[$\mathsf{Grt}_{\flat}$] the $2$-category of Grothendieck categories and left exact left adjoints as morphisms \end{itemize} The $2$-category $\mathsf{Grt}_{\flat}$\ is the main object of study in this paper. It is shown that $\mathsf{Grt}_{\flat}$\ can be endowed with a monoidal structure, where the exponentiable objects are characterized. From an algebro-geometric standpoint, this can be seen as a contribution to the understanding of exponentiable schemes or Hom-schemes when restricted to the flat case. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] aims to $\mathsf{Grt}_{\flat}$\ can simulate flat algebraic geometry via a collection of examples. \item[\S 3] shows that the monoidal structure $\boxtimes$\ on $\mathsf{Grt} $\ [\textit{W. Lowen} et al., Int. Math. Res. Not. 2018, No. 21, 6698--6736 (2018; Zbl 1408.18024)] nicely restricts to $\mathsf{Grt}_{\flat}$, which is easy on the level of objects, but a highly non-trivial task on the level of morphisms. The problem of exponentiability is introduced. It is shown that the category of linear presheaves $\mathrm{Mod}\left( \mathfrak{a}\right) $\ are exponentiable (Proposition 3.15). \item[\S 4] investigates the properties of the forgetful functor \[ :\mathsf{Grt}_{\flat}^{\circ}\rightarrow\mathsf{Cat}_{k} \] showing that it is representable (Prposition 4.2). \item[\S 5] introduces and investigates quasi-injective Grothendieck categories (\S 5.1), continuous linear categories (\S 5.2) and then connect the two concepts (\S 5.3). These are technical tools for the main theorem. \item[\S 6] gives the main theorem: Theorem 6.1. A Grothendieck category is exponentiable in $\mathsf{Grt}_{\flat}$ iff it is continuous. In particular, every finitely presentable Grothendieck category is exponentiable. \item[\S 7] is a collecction of examples and instances of the main theorem. The most relevant is the following: Proposition 7.2. Let $X$ be a quasi-compact quasi-separated scheme over $k$, then $\mathsf{Qcoh}\left( X\right) $\ is exponentiable. \end{itemize} Grothendieck categories; noncommutative algebraic geometry; flatness; monoidal structures; exponentiability; continuous categories; quasi-coherent sheaves Noncommutative algebraic geometry, Sheaves in algebraic geometry, Topoi, Accessible and locally presentable categories, Grothendieck topologies and Grothendieck topoi, Monoidal categories, symmetric monoidal categories, Abelian categories, Grothendieck categories Exponentiable Grothendieck categories in flat algebraic geometry	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $Y$ be an integral Noetherian scheme. Let $f : X \rightarrow Y$ be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension $d$ and with geometrically connected fibres. Assume that $X$ is reduced. Let $\omega ^d_{X/Y}$ be the sheaf of relative regular differential forms. The aim of the paper is to study when the direct images of this sheaf are torsion-free. The question being local, it is assumed that $Y = \text{Spec } D$ for some complete discrete valuation ring $D$ with residue class field $k$ algebraically closed and $X = \text{Proj } S$ for some graded reduced $D$-algebra $S$. The authors give a set of equivalent conditions for $R^if_\omega^d_{X/Y}$ to be torsion-free. Using them the following results are proved. (1) Suppose that $X/Y$ is arithmetically $S_{k+2}$ (Serre's condition). Then $R^{d-i}f_\omega^d_{X/Y}$ is torsion-free and commutes with base change for all $i\leq k$. (2) If $R^{d-i}f_\omega^d_{X/Y}$ is torsion-free for all $i\leq k$ then $R^if_{\mathcal O}_X$ is torsion-free for all $i\leq k$. (3) Suppose that $X/Y$ is globally a complete intersection. Then the sheaf $R^if_\omega^d_{X/Y}$ is torsion-free for all $i$. (4) If $X$ is a surface (i.e. $d=1)$ then $R^1f_\omega ^1_{X/Y}$ is torsion-free if and only if there exists a graded Cohen-Macaulay algebra $S/D$ such that $X=\text{Proj }S$. Specialise now to the case of arithmetic surfaces. Thus $D$ is further assumed to be a discrete valuation ring with quotient field of characteristic zero and residue field of positive characteristic, $X$ is a regular scheme of dimension $2$ and $f$ is proper, flat and with connected fibres. Let $X_y$ denote the special fibre of $f$. Write $X_y = \sum n_i F_i$, $F_i$ being reduced and irreducible curves, $n = \text{g.c.d.}(n_1,\cdots ,n_m)$. Theorem. With the above notations assume that $p \nmid n$. (1) Then $h^0(X,{\mathcal O}_{X_y}) = 1,$ and $R^1f_\omega ^1_{X/Y}$, $R^1f_{\mathcal O}_X$ are torsion-free. (2) Assume that $X/Y$ is projective. Then there exists a graded Cohen- Macaulay algebra $S/D$ such that $X =\text{Proj} (S)$. Finally the authors examine the effects of base extension on $R^1f_{\mathcal O}_X$ and $R^1f_\omega ^1_{X/Y}$. \textit{J. Kollár} [Ann. Math., II. Ser. 123, 11-42 (1986; Zbl 0598.14015)] had shown that if $X$ is a complex projective manifold and $Y$ is a reduced variety then $R^if_*\omega _X$ is torsion-free for all $i$. relative regular differential forms; torsion freeness of direct image of sheaf; dualizing sheaves; arithmetic surfaces DOI: 10.1090/S0002-9939-98-04499-2 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Modules of differentials On the cohomology of regular differential forms and dualizing sheaves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In these notes we give a reasonably self contained proof of three of the main theorems of the Diophantine geometry of curves over function fields of characteristic zero. Let $F$ be a function field of dimension one over the field of the complex numbers $\mathbb{C}$ i.e. a field of transcendence degree one over $\mathbb{C}$. Let $X_F$ be a smooth projective curve over $F$. We prove that: \begin{itemize} \item[--] If the genus of $X_F$ is zero then it is isomorphic, over $F$, to the projective line $\mathbb{P}^1$. \item [--] If the genus of $X_F$ is one and $X_F$ is not isomorphic (over the algebraic closure of $F)$ to a curve defined over $\mathbb{C}$, then the set of $F$ -- rational points of $X_F$ has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil). \item[--] If the genus of $X_F$ is strictly bigger than one and $X_F$ is not isomorphic (over the algebraic closure of $F)$ to a curve defined over $\mathbb{C}$, then the set of $F$ -- rational points of $X_F$ is finite (former Mordell Conjecture). \end{itemize} The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [\textit{R. Hartshorne}, ``Algebraic geometry'', Graduate Texts in Mathematics, 52, 496 p. (1977; Zbl 0367.14001)] or \textit{P. Griffiths} and \textit{J. Harris} [Principles of algebraic geometry. New York, NY: John Wiley \& Sons Ltd. (1994; Zbl 0836.14001)]. arithmetic over function fields; arithmetic of algebraic curves; Mordell Weil theorem; Mordell conjecture Rational points, Global ground fields in algebraic geometry, Arithmetic ground fields for curves, Heights, Elliptic curves over global fields, Curves of arbitrary genus or genus $\ne 1$ over global fields, Arithmetic theory of algebraic function fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Diophantine geometry on curves over function fields	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\mathrm{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson's co-simplicial ring of adèles $\mathbb{A}_X^\bullet $, we have an equivalence $\mathsf{Perf}(X)\simeq\|\mathsf{Perf}(\mathbb{A}_X^\bullet)\|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_X^\bullet$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand-Naimark's reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest. adèles; vector bundles; perfect complexes; moduli stacks Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Stacks and moduli problems Adelic descent theory	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to present an algebro-geometric approach to the study of the geometry of the moduli space of stable bundles on a smooth projective curve defined over an algebraic closed field $k$, of arbitrary characteristic. One of the basic ideas is to consider a notion of divisor of higher rank and a suitable Abel-Jacobi map generalizing the classical notions in rank one. Let ${\mathcal O}_C$ be the structure sheaf of the curve $C$ and let $K$ be its field of rational functions, considered as a constant ${\mathcal O}_C$-module. We define a divisor of rank $r$ and degree $n$, an $(r,n)$-divisor for short, to be any coherent sub-${\mathcal O}_C$-module of $K^r= K^{\oplus r}$ having rank $r$ and degree $n$. Since $C$ is smooth, these submodules are locally free and coincide with the matrix divisors defined by \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)]. Denote by $\text{Div}^{r,n}_{C/k}$ the set of all $(r,n)$-divisors. This set can be identified with the set of rational points of an algebraic ind-variety ${\mathcal D} iv^{r,n}_{C/k}$ that may be described as follows. For any effective ordinary divisor $D$ set $\text{Div}^{r,n}_{C/k} (D)= \{E\in \text{Div}^{r,n}_{C/k}\mid E\subseteq {\mathcal O}_C (D)^r\}$ where ${\mathcal O}_C (D)^r$ is considered as a sub ${\mathcal O}_C$-module of $K^r$. The elements of the set $\text{Div}^{r,n}_{C/k} (D)$ can be identified with the rational points of the scheme $\text{Quot}^m_{{\mathcal O}_C (D)^r/X/k}$, $m=r\cdot \deg D-n$, parametrizing torsion quotients of ${\mathcal O}_C (D)^r$ having degree $m$. It is natural to stratify the ind-variety ${\mathcal D} iv^{r,n}_{C/k}$ according to Harder-Narasimhan type ${\mathcal D} iv^{r,n}_{C/k}= ({\mathcal D} iv^{r,n}_{C/k} )^{ss} \cup \bigcup_{P\neq ss} {\mathcal S}_P$ where $({\mathcal D} iv^{r,n}_{C/k} )^{ss}$ is the open ind-subvariety of semistable divisors. The cohomology of each stratum stabilizes and this stratification is perfect. (Here cohomology means $\ell$-adic cohomology for a suitable prime $\ell$.) In particular, there is an identity of Poincaré series \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= P\bigl( ({\mathcal D} iv^{r,n}_{C/k} )^{ss}; t\bigr)+ \sum_{P\neq ss} P({\mathcal S}_P, t)\cdot t^{2d_r}, \] where $d_P$ is the codimension of ${\mathcal S}_P$. Let $r$ and $n$ be coprime. Then the notations of stable and semistable bundle over $C$ coincide, and the moduli space $N(r,n)$ of stable vector bundles having rank $r$ and degree $n$ is in this case a smooth projective algebraic variety. It is natural to define, by analogy with the classical case, Abel-Jacobi maps $\vartheta: ({\mathcal D} iv^{r,n}_{C/k} )^{ss}\to N(r,n)$, by assigning to a divisor $E$ its isomorphism class as a vector bundle. In order to find the Betti numbers of $N(r,n)$ it suffices to know those of $({\mathcal D} iv^{r,n}_{C/k} )^{ss}$. This computation reduces, for arbitrary $r$ and $n$, to the calculation of those of ${\mathcal D} iv^{r,n}_{C/k}$. The varieties $\text{Div}^{r,n}_{C/k} (D)$ are analogous to Grassmannians and share with them the property of having a decomposition into Schubert ``strata''. One obtains \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= {{\prod^r_{j=1} (1+t^{2j-1} )^{2g}} \over {(1-t^{2r}) \prod^{r-1}_{j=1} (1- t^{2j})^2}}. \] moduli space of stable bundles; divisor of rank $r$ and degree $n$; algebraic ind-variety; semistable divisors; stratification; Poincaré series; Abel-Jacobi maps \beginbarticle \bauthor\binitsE. \bsnmBifet, \bauthor\binitsF. \bsnmGhione and \bauthor\binitsM. \bsnmLetizia, \batitleOn the Abel-Jacobi map for divisors of higher rank on a curve, \bjtitleMath. Ann. \bvolume299 (\byear1994), page 641-\blpage672. \endbarticle \OrigBibText \beginbarticle \bauthor\binitsE. \bsnmBifet, \bauthor\binitsF. \bsnmGhione and \bauthor\binitsM. \bsnmLetizia, \batitleOn the Abel-Jacobi map for divisors of higher rank on a curve, \bjtitleMath. Ann. \bvolume299 (\byear1994), page 641-\blpage672. \endbarticle \DOI10.1007/BF01459804 \endOrigBibText \bptokaddids, structpyb \endbibitem Mathematical Reviews (MathSciNet): Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Abel-Jacobi map for divisors of higher rank on a curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0747.00028.] The purpose of this paper is to determine explicitly the local structure of the Hilbert scheme of curves in $\mathbb{P}^ 3$ at certain points. Actually, the main body of this work is devoted to a quite detailed exposition of the various deformation theories of a closed subscheme $Y$ of a projective scheme $X = \text{Proj}(S)$, where $K$ is a field and $S$ is a graded Noetherian $K$-algebra, with $S_ 0 = K$. The types of deformations are: the deformations of $Y$ as a subscheme of $X$, the deformations of the ideal sheaf ${\mathcal I}_ Y$ as a sheaf of ${\mathcal O}_ X$-modules, the deformations of the ideal $I(Y)$ as a homogeneous $S$-module, and ``conical deformations'' of $Y$. This exposition builds on previous work by \textit{O. A. Laudal} [in Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 218-240 (1986; Zbl 0597.14010)] and \textit{J. O. Kleppe} [Math. Scand. 45, 205-231 (1979; Zbl 0436.14004)], but the material is presented here with a view towards explicit calculations. Various conditions are worked out under which these deformation theories are isomorphic. The theory is then applied to the calculation of the deformation theory of particular space curves, yielding explicit local equations for some singular points of the Hilbert scheme and explicit examples of obstructed curves (i.e. corresponding to singular points of the Hilbert scheme) of maximal rank. The same examples were found independently by \textit{Bolondi}, \textit{Kleppe} and \textit{Miró-Roig}. The question remains whether curves with seminatural cohomology (i.e. curves $C$ such that for each $n$, the sheaf ${\mathcal J}_ C(n)$ has at most one nonzero cohomology group) are unobstructed. isomorphic deformation theories; deformation theory of space curves; Hilbert scheme of curves; deformation theories of a closed subscheme Charles H. Walter, Some examples of obstructed curves in \?³, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324 -- 340. Plane and space curves, Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory Some examples of obstructed curves in $\mathbb{P}^ 3$	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first goal of this paper is to generalize in local complex geometry the main idea in \textit{W. L. Chow} and \textit{B. L. van der Waerden}' construction [Math. Ann. 113, 692-704 (1937; Zbl 0016.04004)]: associating the ``incidence divisor'' of a cycle with a given analytic family of cycles (the linear projective varieties of suitable dimension in the case of $\mathbb{P}_N (\mathbb{C}))$. Under very general hypotheses we associate a Cartier divisor on the parameter space in which arbitrary singularities are allowed to a local complete intersection. The second goal is to study the singularities of functions obtained by integration of meromorphic cohomology classes on an analytic family of cycles. Our result enables us to control the pole order of the meromorphic functions on the parameter space from the pole order of the given cohomology class. local complex geometry; Cartier divisor; local complete intersection; singularities of functions; integration of meromorphic cohomology classes D. Barlet - J. Magnusson , Integration de classes de cohomologie méromorphes et diviseur d'incidence , Ann. Sci. École Norm. Sup. ( 4 ) 31 ( 1998 ), 811 - 842 . Numdam \| MR 1664218 \| Zbl 0963.32019 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Complete intersections Integration of meromorphic cohomology classes and incidence divisors	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a Weil cohomology theory for smooth proper varieties over a field. Grothendieck's ``conjecture C-plus'' asserts the Künneth projectors are algebraic. His ``conjecture D'' asserts that in the group of algebraic cycles numerical equivalence agrees with homological equivalence. Differential graded categories are regarded as ``noncommutative schemes'' by the specialists; the differential graded category associated with an ordinary scheme being the category of perfect complexes. In the noncommutative world, noncommutative analogues of Grothendieck's conjectures have been formulated. Let $k$ be a perfect field of characteristic $p$. In the article under review, the author proves that these noncommutative conjectures (for ``smooth proper $k$-linear dg categories'') are equivalent to the corresponding commutative conjectures (for smooth proper $k$-varieties). Here, one uses the rational crystalline cohomology in the conjectures C-plus and D, and the topological periodic cyclic homology in the noncommutative versions of these conjectures. The author gives some applications. From the abstract: ``As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of `low-dimensional' orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot's cohomological interpretation of the classical zeta function and of Grothendieck's conditional approach to 'half' of the Riemann hypothesis.'' Reviewer's remark: There is a citation with broken hyperlink ``[?book]'' on p. 5052 in the published version of the article. It should point to the author's book [Noncommutative motives. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1333.14002)]. noncommutative algebraic geometry; motive; crystalline cohomology Noncommutative algebraic geometry, Applications of methods of algebraic $K$-theory in algebraic geometry, $p$-adic cohomology, crystalline cohomology, $K$-theory and homology; cyclic homology and cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Chain complexes (category-theoretic aspects), dg categories A note on Grothendieck's standard conjectures of type $\mathrm{C}^+$ and $\mathrm{D}$ in positive characteristic	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The notion of a ``Galois point'' of an algebraic curve was introduced by Yoshihara in 1996 and can be described as follows. Let $C$ be an irreducible smooth projective curve defined over an algebraically closed field $K$ of characteristic $p \geq 0$. Let $K(C)$ be the corresponding function field. Let $P$ be a point in $\mathbb{P}^2$ and $\varphi$ a rational map for $C$ to $\mathbb{P}^2$ being birational onto its image. A point $P \in \mathbb{P}^2$ is a Galois point for $\varphi(C)$ if the function field extension $K(\varphi(C))/{\pi_P}^*K(\mathbb{P}^1)$ induced by the projection $\pi_P$ is Galois. If $P$ is a smooth point of $\varphi(C)$ (resp. contained in $\mathbb{P}^2 \setminus \varphi(C)$) then it is said to be inner (resp. outer). If $P$ is a Galois point, then the corresponding Galois group will be denoted with $G_P$. One of the most interesting problems in studying Galois points is to determine the number of Galois points that a plane curve can have. For smooth plane curves, this number was determined by \textit{H. Yoshihara} [J. Algebra 239, No. 1, 340--355 (2001; Zbl 1064.14023)] and \textit{S. Fukasawa} [Rend. Semin. Mat. Univ. Padova 129, 93--113 (2013; Zbl 1273.14066)]. Not so many examples of singular plane curves $C$ having two Galois points are known, but following the works of Yoshihara, Takahashi and Terasoma characterizations are known for $C$ to have $2$ Galois points. In this paper applications of these results for rational and elliptic curves are presented. More precisely it is shown that if $p \neq 2$, then there exist a morphism $\varphi: \mathbb{P}^1 \rightarrow \mathbb{P}^2$ which is biratonal onto its image, such that {\parindent=0.7cm \begin{itemize}\item[--] $\deg(\varphi(C))=5$, $\varphi(C)$ admits two Galois points $\varphi(P_1)$ and $\varphi(P_2)$ such that $G_{\varphi(P_i)}$ is cyclic of order $4$, for $i=1,2$ and $p \neq 3$; \item[--] $\deg(\varphi(C))=5$, $\varphi(C)$ admits two Galois points $\varphi(P_1)$ and $\varphi(P_2)$ such that $G_{\varphi(P_i)} \cong (\mathbb{Z}/4\mathbb{Z})^{\oplus 2}$, for $i=1,2$; \item[--]$\deg(\varphi(C))=5$, $\varphi(C)$ admits two Galois points $\varphi(P_1)$ and $\varphi(P_2)$ such that $G_{\varphi(P_1)}$ is cyclic of order $4$ and $G_{\varphi(P_2)} \cong (\mathbb{Z}/4\mathbb{Z})^{\oplus 2}$; \item[--] $\deg(\varphi(C))=6$, $\varphi(C)$ admits two Galois points $\varphi(P_1)$ and $\varphi(P_2)$ such that $G_{\varphi(P_i)}$ is cyclic of order $5$, for $i=1,2$. \end{itemize}} As an application, it is shown in Theorem 3 that if $p \neq 3$ and $E \subset \mathbb{P}^2$ is the curve defined by $X^3+Y^3+Z^3=0$, then there exists a morphism $\varphi: E \rightarrow \mathbb{P}^2$ which is birational onto its image with $\deg(\varphi(C))=4$, such that $\varphi(E)$ admits two inner Galois points. Galois points; plane curves; Galois groups; automorphism groups Fukasawa, S., A birational embedding of an algebraic curve into a projective plane with two Galois points, preprint Plane and space curves, Algebraic functions and function fields in algebraic geometry, Automorphisms of curves A birational embedding of an algebraic curve into a projective plane with two Galois points	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the main theorem of this paper: if $f: P\to X$ is a flat morphism of algebraic varieties over an algebraically closed field $k$ of arbitrary characteristic, with $X$ a non-singular projective curve and the geometric generic fiber of $f$ is a normal, separably rationally connected variety, then $f$ has a section. Here, a normal variety is said to be separably rationally connected if there is a morphism $C\to V$, with $C$ a smooth rational curve, whose image does not contain any singular point of $V$ and such that the pull-back via $f$ of the tangent bundle of $V$ is ample on $C$. This result easily implies the statement of the title of the present article. The main theorem is closely related to a question posed by Kollar, Miyaoka and Mori, who asked whether the analagous statement, but using a different notion of rationally connected variety, is true. Both notions agree if $V$ admits a resolution of singularities. When $k$ is the field of complex numbers, the main theorem was proved by using a not purely algebraic argument in [\textit{T. Graber}, \textit{J. Harris} and \textit{J. Starr}, J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)]. The proof given in the present paper, partially based on that of the mentioned article, is purely algebraic and rather convoluted. By means of a normalized base change via a suitable finite, generally étale morphism $Y\to X$ there is a reduction to the case where all the fibers of $f: P\to X$ are reduced. In this situation, a key ingredient in the proof is a theorem saying that given a finite morphism $Y\to X$ of irreducible smooth projective curves over $k$ there is a family $\{Y_t\}_{t\in X}$ of curves such that (among other properties) for suitable points $0$, $\infty$ in $X$ we have that $Y_0$ is nodal, with $Y$ as normalization, and $Y_\infty$ has an irreducible component which is naturally isomorphic to $X$. Techniques from the theory of Hilbert schemes also play an important role in the proof. These and other auxiliary results and methods discussed in the paper may be of independent interest. rational point; section; rationally connected variety; family of curves; Hilbert scheme de Jong, A.J., Starr, J.: Every rationally connected variety over the function field of a curve has a rational point. Am. J. Math. 125(3), 567--580 (2003) Rational points, Schemes and morphisms, Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry Every rationally connected variety over the function field of a curve has a rational point	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be an arbitrary smooth complex projective model of the fibre product $X_1\times_C X_2$, where $X_1\rightarrow C$ is an elliptic surface over a smooth projective curve $C$ and $X_2\rightarrow C$ is a family of $K3$ surfaces. Such a family is said to be with semistable degenerations of rational type, if, for every singular fibre, the Picard-Lefschetz transformation $\gamma$ satisfies the condition $(\log(\gamma))^2\neq 0$. In the present paper the author proves that the Grothendieck standard conjecture $B(X)$ of Lefschetz type holds for $X$ under the assumption that $X_2\rightarrow C$ is a family of $K3$ surfaces with semistable degenerations of rational type such that \text{rank NS}$(X_{2s})\neq 18$ for a generic geometric fibre $X_{2s}$. Furthermore he proves that $B(X)$ holds for every smooth projective compactification $X$ of the Néron minimal model of an abelian scheme of relative dimension 3 over an affine curve if the generic fibre is an absolutely simple abelian variety with reductions of multiplicative type at all infinite places. algebraic cycle; elliptic variety; standard conjecture of Lefschetz type; $K3$ surface; abelian scheme Tankeev, S G, On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models, Izv. Ross. Akad. Nauk Ser. Mat., 78, 181-214, (2014) Algebraic cycles, Variation of Hodge structures (algebro-geometric aspects), Classical real and complex (co)homology in algebraic geometry, $4$-folds On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be an algebraic curve of genus $g\geq 2$, defined over an algebraically closed field $k$ of characteristic zero, let $S_{n,d}(X)$ be the variety of all stable vector bundles $E$ on $X$ of rank $n$ such that $\wedge^nE$ is isomorphic to a fixed line bundle of degree $d$, $n$ and $d$ are relatively prime. The analogue of Torelli's theorem for $S_{n,d}$ is the assertion: \[ \text{If } S_{n,d}(X_1) \cong S_{n,d}(X_2), \text{ then } X_1\cong X_2. \tag{} \] For $n=1$ this is Torelli's classical theorem. The case $k=\mathbb C$, $g=2$, $n=2$ was studied by \textit{P. E. Newstead} [Topology 7, 205--215 (1968; Zbl 0174.52901)], who described the complete structure of $S_{2,d}$, showed a projective embedding of $S_{2,d}$ in $\mathbb P_5$ and proved that the variety of lines of $S_{2,d} \subset \mathbb P_5$ is isomorphic to the Jacobian $J(X)$ of $X$. In the case $k=\mathbb C$, $g\geq 2$, $n=2$ \textit{D. Mumford} and \textit{P. E. Newstead} [Am. J. Math. 90, 1200--1208 (1968; Zbl 0174.52902)] proved () and showed an isomorphism of the intermediate Jacobian $J^2(S_{2,d}(X)$ to $J(X)$. The author [Math. USSR, Izv. 3 (1969), 1081--1101 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1149--1170 (1969; Zbl 0225.14009)] gave the following geometric reconstruction of $X$ from $S_{2,d}(X)$: if $V$ denotes the base variety of the family of lines of some type on $S_{n,d}$, $\varphi: V \to\mathrm{Alb}(V)$ is the Albanese's application and $\gamma: J(X)\to J(X)$ is the endomorphism of doubling, then $\mathrm{Alb}(V)\cong \bar J(X)$ and $X\cong \gamma(\varphi(V))$. In the present paper it is proved (*) in the general case and pointed out the geometric way of reconstructing of $X$ from $\prod S_{n,d}(X)$. The procedure is the following. Let $M$ be the set of varieties of form $\prod S_{n,d}(X)$ $((n,d)$ runs through a finite set of pairs $(n,d)$, $n$ and $d$ are coprime). For $V\in M$ let $\bar B_n(V)$ denote a connected component of the base variety of projective spaces $\mathbb P_n\subset V$ such that $(-K_V)\cdot \mathbb P_n= 2 \mathbb P_{n-1}$, where $K_V$ is the canonical class on $V$ and $\mathbb P_{n-1}$ is a hyperplane in $\mathbb P_n$ (all connected components of the base variety are actually isomorphic to each other). Let $B_n(V)$ be a fibre of Albanese's application $\bar B_n(V)\to\mathrm{Alb}\bar B_n(V))$ (for $V\in M$ the fibres are isomorphic to each other). The pair $(n,d)$ defines a sequence of natural numbers $\{m_1,m_2,\dots,m_N\}$ such that \[ S_{2,1}(X) =B_{m_N}(B_{m_{N-1}}(\ldots B_{m_1}(S_{n,d}(X))\ldots)). \] For example the pair $(n,1)$ gives the sequence $\{(n-1)(g-1), (n-2) (g-1),\dots, 2(g-1)\}$. By examination of projective subspaces of $S_{n,d}$ were found very useful the new notions of a rigid pair of bundles and rigid bundle: a pair $(E_1,E_2)$ is rigid if $\text{rk}\,E_1 \deg E_2 - \text{rk}\,E_2 \deg E_1=1$ and a bundle $E$ is rigid if it is representable as an extension $0\to E_1\to E\to E_2\to 0$ where $(E_1,E_2)$ is a rigid pair. Tjurin, A.: Analogs of Torelli's theorem for multi-dimensional vector bundles over an arbitrary algebraic curve. Izv. Akad. Nauk SSSR, Ser. Mat. Tom34 (2), 338--365 (1970), Math. USSR Izvestija Vol.4 (2), 343--370 (1970). Torelli problem, Vector bundles on curves and their moduli Analogs of Torelli's theorem for multidimensional vector bundles over an arbitrary algebraic curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review gives a survey of recent results on rational points of algebraic varieties over number fields, mainly obtained by the fibration method, and thus can be viewed as a natural continuation of an earlier author's publication [in: Number theory. Proc. int. cong. discr. math. number theory, India 1996, Contemp. Math. 210, 19--39 (1998; Zbl 0910.14008)]. Note however that a substantial part of mentioned results is given with complete proofs (especially in chapter 1); moreover, chapter 3 contains original results due to the author. Here are some details on the content of the paper. The introduction contains a brief account of basic notions necessary to introduce the reader into the context of problems under consideration: the Brauer-Manin pairing and its use for describing obstructions to the Hasse principle and weak approximation, particularly in the special case when the variety under investigation can be fibred over the projective line so that all geometric fibres contain a component of multiplicity one. Chapter 1 is focused on the descent method for non-complete varieties (``open descent''). First, the author presents a simplified proof of the ``formal lemma'' of \textit{D.~Harari} [Duke Math. J. 75, No. 1, 221--260 (1994; Zbl 0847.14001)] which is indispensable while working with the Brauer pairing on non-complete varieties. Second, he gives a summary of the descent method explaining the relationship between torsors and the Brauer group. Finally, he discusses a beautiful example of \textit{D.~ R. Heath-Brown} and \textit{A.~N.~Skorobogatov} [Acta Math. 189, 161--177 (2002; Zbl 1023.11033)] when this method (combined with the circle method) allows one to prove that for certain hypersurfaces of norm type the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. The author gives a slightly simplified version of the proof of this result. (It is worth mentioning that this result has been generalized, using the same method of open descent, by \textit{J.-L.~Colliot-Thélène, D.~Harari, and A.~N.~Skorobogatov} [Lond. Math. Soc. Lecture Notes Ser. 303, 69--89 (2003; Zbl 1087.14016)]. Chapter 2 is devoted to a new method invented by \textit{P. Swinnerton-Dyer} [in: Abelian Varieties, Walter de Gruyter, Berlin, 273--292 (1995; Zbl 0849.14009)] and improved by \textit{J.-L.~Colliot-Thélène, A.~N.~Skorobogatov} and \textit{P.~Swinnerton-Dyer} [Invent. Math. 134, 579--650 (1998; Zbl 0924.14011)] which ensures the existence of many rational points on a variety fibred over the projective line into principal homogeneous spaces under an abelian variety. The author describes in some detail the main features and gives an overview of the main results obtained by the method. The emphasis is made on a recent result by \textit{P.~Swinnerton-Dyer} [in: Ann. Sci. Éc. Norm. Sup. 34, 891--912 (2001; Zbl 1003.11028)] establishing the Hasse principle for all diagonal cubic hypersurfaces of dimension at least 3 (as well as for some classes of diagonal cubic surfaces) defined over the field of rational numbers (under the assumption of finiteness of the 3-primary part of the Tate-Shafarevich group of certain elliptic curves). In chapter 3 the author proves an (unconditional) analogue of the above mentioned result of Swinnerton-Dyer in the case where the ground field is a function field in one variable over a finite field ${\mathbb F}_q$ ($q$ odd, $q\equiv 2\pmod 3)$. Hasse principle; torsor; descent; Brauer group Jean-Louis Colliot-Thélène, Points rationnels sur les fibrations, Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Society Mathematical Studies 12, Springer, 2003, p. 171-221 Rational points, Arithmetic ground fields (finite, local, global) and families or fibrations, Global ground fields in algebraic geometry Rational points on fibrations	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Logarithmic geometry is a general tool to work with singular varieties like toric varieties or varieties with ''semi-stable reduction'' as well as if they were smooth (over some base). In this theory, the analog of a smooth morphism is what is called a log smooth morphism (eg. the structural morphism of a toric variety). A general setting for this theory was developed by \textit{K. Kato} [in: Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore/MD 1988, 191-224 (1989; Zbl 0776.14004)], including the main aspects of an extension of crystalline cohomology [developed in the classical case by \textit{P. Berthelot} ``Cohomologie cristalline des schemas de caractéristique $p>0$''. Lect. Notes Math. 407 (1974; Zbl 0298.14012)] to log schemes $(X, M)$ over a base with a log structure. Among all the generalizations of the classical results or constructions one can ask in this new setting, the paper under review treats the case of Poincaré duality (this kind of result can potentially be used to prove that the Frobenius is an isomorphism in crystalline cohomology$\otimes \mathbb{Q}$), including the case of coefficients locally free of finite type. The base scheme has to be the finite length Witt vectors $W_{m}$ of a perfect field of positive characteristic with trivial log structure or with the log structure associated to $\mathbb{N} \rightarrow W_{m}$ sending $1$ to $0$ which play a key role in the proof, by \textit{O. Hyodo} and \textit{K. Kato} [in: Périodes $p$-adiques, Sém. 1988, Astérisque 223, 221-268 (1994; Zbl 0852.14004)], of a conjecture of Fontaine, the underlying scheme $X$ of the log scheme $(X,M)$ has to be proper and there is a slight technical assumption about the kind of log structure needed. The Poincaré duality takes therefore, mutatis mutandis, the standard form. The proof follows step by step some constructions and proofs by Hartshorne and Berthelot done in the classical case showing clearly what has to be modified. Poincaré duality; logarithmic crystalline cohomology; logarithmic geometry; Witt vectors Tsuji, T, Poincaré duality for logarithmic crystalline cohomology, Compositio Math., 118, 11-41, (1999) $p$-adic cohomology, crystalline cohomology Poincaré duality for logarithmic crystalline cohomology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a one-hour report of ICM 1998, \textit{C. Deninger} [Doc. Math., J. DMV , 23--46 (1998; Zbl 0899.14001)] conjectured a cohomological formalism for algebraic varieties over rings of integers of number fields which relates to arithmetic zeta functions, particularly to Riemann zeta function; as well-known, here for these zeta functions there exists the Hasse-Weil conjecture. Such a conjectural arithmetic cohomology can be taken as a counterpart of $\ell$-adic cohomology, developed by Grothendieck's school for the construction of a Weil cohomology, for algebraic varieties over finite fields which relates to local zeta functions; also here for these local zeta functions one has the Weil conjecture proved by Deligne. Deninger's conjectural cohomology arises from J.-P. Serre's conjectural approach of local factors of zeta functions attached to algebraic varieties and Deninger has tried to realise it by leafwise cohomology of certain foliated dynamical systems. This is on one hand. On the other hand, \textit{S. Lichtenbaum} [Compos. Math. 141, No. 3, 689--702 (2005; Zbl 1073.14024)] defined a type of Grothendieck topology, called Weil-étale topology, for an arithmetic variety. He conjectured that the Euler characteristic of the Weil-étale cohomology gives the special values of the arithmetic zeta function for a given arithmetic variety. In this paper, the author discusses Deninger's dynamical systems and Weil-étale topology in terms of topoi over finite fields and over rings of integers of number fields, respectively. Then several properties for the interplay between the dynamical systems and the Weil-étale topoi are also obtained. In particular, the author defines a topos morphism which describes a relationship between the Weil-étale topos and the topos associated to a Deninger's dynamical system. Deninger's dynamical system; topos; Weil-étale topology Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta functions and $L$-functions of number fields On the analogy between Deninger dynamical systems and the Weil-étale topos	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let S be a geometrically irreducible smooth scheme over a finitely general ${\mathbb{Z}}$-algebra $R\subset {\mathbb{C}}$. Let $X\to S$ be an abelian scheme. Fix some general point $\eta: Spec {\mathbb{C}}\to S$ and let L be the function field of $S\otimes_ R\bar R$, where $\bar R$ denotes the algebraic closure. - The author gives relations between the Lie algebra $G_{Hod}$ of the special Mumford-Tate group, the Lie algebra $G_{\ell}$ of the Zariski closure of the image of $\pi_ 1(S,\eta)$ in $GL(H^ 1_{et}(X,{\mathbb{Q}}_{\ell}))$ and the Lie algebra of the differential Galois group of the Gauss-Manin connection $(H^ 1_{DR}(X_ L),\nabla_ L)$. He also gives the interrelations between the Hodge and the Tate conjecture for all powers of X and some stronger conditions on X which imply these conjectures. The author announces for certain classes of abelian schemes a proof of the conjectures of Hodge, Tate, and Mumford and Tate (i.e. $G_{Hod}\otimes {\mathbb{Q}}_{\ell}\oplus {\mathbb{Q}}_{\ell}=G_{\ell})$, which is inspired by ideas of Ribet and Tankeev. These abelian varieties also provide interesting examples for other questions. Assume for example that X is a product of simple, polarized abelian varieties A such that the involution on $E=End^ 0A$ is of the first kind and such that E contains a commutative subfield of degree dim(A). The above conjectures are true for X and, moreover, so is a conjecture of Katz, which asserts that the Lie algebra of the differential Galois group is generated by the p-curvature of $\nabla_ L$. Finally, the author indicates a new approach to the Chowla-Selberg formula in a special case. Hodge conjecture; abelian scheme; Mumford-Tate group; differential Galois group of the Gauss-Manin connection; Tate conjecture André, Y. , Sur certaines algèbres de Lie associées aux schémas abéliens , Note C.R.A.S.t. 299 I n^\circ 5 (1984), 137-140. Analytic theory of abelian varieties; abelian integrals and differentials, Connections (general theory), Lie algebras of linear algebraic groups, Arithmetic problems in algebraic geometry; Diophantine geometry Sur certaines algèbres de Lie associées aux schémas abéliens. (On certain Lie algebras associated with abelian schemes)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians From publisher's description: ``This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.'' The topic of this book is the unramified Brauer group and its application to rationality problems and Hasse principles. The theory is developed not as in most other textbooks, but mainly through exercises with many intermediate steps and hints making them very accessible e.g. for graduate students or researchers looking for a quick introduction to the field. Difficult theorems like Merkurjev-Suslin are cited. I think the strategy of this concise book allows one to quickly learn the subject if you like the approach through exercises in the style of Serre's books. The book has a good balance between modern geometric language and explicit equations. You should not take it as a reference containing full proofs. At the end of most chapters, further reading recommendations are given. The Russian original, extended notes of a 2011 reading seminar at the Steklov Mathematical Institute, can be found at [\url{arXiv:1512.00874}]. Other books covering the theory of the Brauer group and the Brauer-Manin obstruction are [\textit{A. Skorobogatov}, Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015); \textit{B. Poonen}, Rational points on varieties. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1387.14004)]. These books contain more proofs. [Zbl 0972.14015] covers more applications of Brauer groups and torsors, but does not contain the basics as the book under review and [Zbl 1387.14004]. The applications to rationality problems (Part III) are unique to this book. Part I ``Preliminaries on Galois cohomology'' is an introduction to group and Galois cohomology. In Part II (``Brauer group'') treats the special case of the Brauer group of a field and its relation to Severi-Brauer varieties in Chapter 3 and then in Chapter 4 the (Azumaya algebra) Brauer group of a scheme first defined by \textit{A. Grothendieck} [in: Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 46--66 (1968; Zbl 0193.21503); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 67--87 (1968); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 88--188 (1968; Zbl 0198.25901)] and the unramified Brauer group $\mathrm{Br}^{nr} (K)$ (cf. the first part of the title of the book) [\textit{J. L. Colliot-Thélène}, Proc. Symp. Pure Math. 58, 1--64 (1995; Zbl 0834.14009)] of a function field $K$ finitely generated over a constant field $k$ of characteristic 0 defined as the intersection of the kernels of the residue maps $\mathrm{res}_v:\mathrm{Br}(K)\to\mathrm{Br}(K_v)\to\mathrm{Hom}(G_{\kappa_v},\mathbb Q/\mathbb Z)$ for all discrete valuations $v$ of $K$ trivial on $k^\ast$ with residue field $\kappa_v$, also with their geometric meaning and the unramified Brauer group of a normal irreducible variety over $k$. The unramified Brauer group of a smooth proper variety is a stably birational invariant. The book restricts to finitely generated function fields of characteristic 0 because they have perfect residue fields. Part III ``Applications to rationality problems'' is the second part of the title and the heart of the book: Chapter 5 produces an example of a quotient of a variety over an al-gebraically closed field of characteristic 0 by a finite group and a proof that is not stably rational using the non-triviality of the unramified Brauer group. Chapter 6 ``Arithmetic of Two-dimensional Quadrics'' is on the discriminant and the Clifford invariant of quadrics and their geometric meaning. Chapter 7 ``Non-rational Double Covers of $\mathbb P^3$'' proves using the unramified Brauer group that certain unirational threefolds are not rational. In Chapter 8 the Weil restriction of scalars and algebraic tori are introduced and applied to rationality problems. Chapter 9 is on an example [\textit{A. Beauville} et al., Ann. Math. (2) 121, 283--318 (1985; Zbl 0589.14042)] of a variety over a perfect field of characteristic $\neq 2$ which is non-rational, but stably rational, hence unirational. In Part IV (``The Hasse principle and its failure'') the Hasse-Minkowski theorem on quadratic forms over global fields is proved by a geometric reduction to quadrics of dimension 1 and the Hasse principle for the Brauer group (Chapter 11). In Chapter 12 the Brauer-Manin obstruction introduced by \textit{Yu. I. Manin} [in: Actes Congr. internat. Math. 1970, No. 1, 401--411 (1971; Zbl 0239.14010)] is defined and applied to explaining the failure of the Hasse principle for the Lind-Reichardt equation $2y^2=x^4-17$. Appendix A contains a short introduction to etale cohomology without proofs or exercises, but references to the standard source [\textit{J. S. Milne}, Étale cohomology. Princeton Mathematical Series. 33. Princeton, New Jersey: Princeton University Press. (1980; Zbl 0433.14012)]. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Brauer groups of schemes, Rationality questions in algebraic geometry, Rational points, Rational and unirational varieties, Galois cohomology Unramified Brauer group and its applications	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By analyzing the Galois action on the étale fundamental group, the author studies continuous $\ell$-adic representations of the geometric fundamental group of a variety $X$ over a field $k$ \[\rho: \pi^{\text{ét}}_1(X_{\bar{k}},\bar{x}) \to \mathrm{GL}(\mathbb{Z}_{\ell}) \] which arise from geometry. Let $X$ be a geometrically connected variety over a field $k$, $\bar{x}$ a geometric point, and $\ell$ a prime. A representation $\rho$ is arithmetic if it is a subquotient of a representation \[ \tilde{\rho}: \pi^{\text{ét}}_1(X_{k'},\bar{x}) \to \mathrm{GL}(\mathbb{Z}_{\ell}) \] where $k'$ is a finite extension of $k$. It is geometric if it is a subquotient of the monodromy representation \[ \pi^{\text{ét}}_1(X_{\bar{k}},\bar{x}) \to \mathrm{GL}((R^i\pi_*\underline{\mathbb{Z}_{\ell}})_{\bar{x}}) \] for some smooth proper morphism $\pi: Y \to X$ and some integer $i\geq 0$. Moreover, if $X$ is normal and $k$ is finitely generated of characteristic zero, the main theorems claim that there exists a positive integer $N=N(X,\ell)$ such that any arithmetic (resp. geometric) representation which is trivial mod $\ell^N$ is unipotent (resp. trivial). The strategy to study the action of $\mathrm{Gal}(\bar{k}/k)$ on $\pi^{\text{ét}}_1(X_{\bar{k}},\bar{x})$ is anabelian methods, and these technical results are the main ingredients of the proof. For Part II, see [the author, Duke Math. J. 170, No. 8, 1851-1897 (2021; Zbl 07369863)]. arithmetic representation; fundamental group; weight; unipotent Homotopy theory and fundamental groups in algebraic geometry, Galois representations, Étale and other Grothendieck topologies and (co)homologies Arithmetic representations of fundamental groups. I.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For part I see \textit{D. S. Nagaraj} and \textit{C. S. Seshadri}, ibid. 107, No. 2, 101-137 (1997; Zbl 0922.14023). Let $X_0$ be an irreducible projective curve of arithmetic genus $g\geq 2$ whose singularity is one ordinary point. The authors give a generalisation of Gieseker's construction for arbitrary rank. They construct a birational model $G(n,d)$ of the moduli space $U(n,d)$ of stable torsion free sheaves in the case $(n,d)=1,$ such that $G(n,d)$ has normal crossing singularities and behaves well under specialization, i.e. if a smooth projective curve specializes to $X_0,$ then the moduli space of stable vector bundles of rank $n$ and degree $d$ on $X$ specializes to $G(n,d)$. This generalizes an earlier work of Gieseker in the rank two case. Theorem 1. There exists a canonical structure of a quasi-projective variety on $G(n,d)$ and a canonical proper birational morphism $\pi _{*}:G(n,d)\rightarrow U(n,d)_s$ onto the moduli space of stable torsion free sheaves on $X_0.$ The singularities of $G(n,d)$ are (analytic) normal crossings. If $(n,d)=1$, $G(n,d)$ is projective, since $U(n,d)_s=U(n,d)$ is projective. projective curves with singularities; moduli space of stable vector bundles Nagaraj, D. S.; Seshadri, C. S., \textit{degenerations of the moduli spaces of vector bundles on curves, II: generalized gieseker moduli spaces}, Proc. Indian Acad. Sci. Math. Sci., 109, 165-201, (1999) Vector bundles on curves and their moduli, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Singularities of curves, local rings Degenerations of the moduli spaces of vector bundles on curves. II. (Generalized Gieseker moduli spaces)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is a global survey of approximation theorems; more precisely, it contains approximation theorems for sections of sheaves ${\mathcal F}\otimes{\mathcal E}_X$, where ${\mathcal F}$ is an analytic (or algebraic) coherent sheaf defined over a coherent real analytic space $X$ (or a real affine variety $X)$ and ${\mathcal E}_X$ is the sheaf of germs of $C^\infty$ functions on $X$. The paper is divided into three parts: in the first one, the analytic case, we prove that the set of sections of ${\mathcal F}$ is dense in the strong (or Whitney) topology in the space of sections of ${\mathcal F}\otimes {\mathcal E}_X$. The second part deals with the algebraic case. In this context we use the classical Weierstrass theorem and obtain approximation in the weak (or compact open) topology. In the real algebraic context, Theorems A and B are not true, so we study two subcategories of coherent sheaves: $A$-coherent and $B$-coherent sheaves. They are studied in \S 6 and \S 7; we prove approximation theorems for sections of ${\mathcal F}\otimes {\mathcal E}_X$ by means of sections of ${\mathcal F}$. The third part of this article is devoted to fiber bundles. Approximation theorems were known also for sections of fiber bundles. In \S 8, via a duality theory, we find new approximation results for sections of fiber bundles of a more general class, which are not, in particular, locally trivial. Locally trivial algebraic vector bundles do not have the usual good behaviour. For example, the fiber is not in general generated by global sections. Bundles having the latter property are called strongly algebraic and are studied in \S 9. We prove that an algebraic vector bundle is strongly algebraic if, and only if, its total space is an affine variety. By applying the approximation results to the sheaf $\Hom({\mathcal F}, {\mathcal G})$, we can extend to general vector bundles the results about the equivalence of the analytic and $C^\infty$ classification. For strongly algebraic vector bundles, if $X$ is compact, we have similar results about $C^\infty$ and algebraic classification. survey; approximation theorems; fiber bundles Tognoli, A.: Approximation theorems in real analytic and algebraic geometry, Lectures in real geometry (Madrid, 1996), de Gruyter Exp. Math., 23, pp. 113-166. Berlin, (1996) Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs, Real-analytic manifolds, real-analytic spaces, Real-analytic and semi-analytic sets Approximation theorems in real analytic and algebraic geometry	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper arises from the extensions of the Manin-Mumford conjecture, where we shall focus on the case of (complex connected) commutative algebraic groups $G$ of dimension 2. This context predicts finiteness for the set of torsion points in an algebraic curve inside $G$, unless the curve is `special', i.e. a translate of an algebraic subgroup of $G$. Here we shall consider not merely the set of torsion points, but its topological closure in $G$ (which equals the maximal compact subgroup). In the case of abelian varieties this closure is the whole space, but this is not so for other groups $G$; actually, we shall prove that in certain cases (where a natural dimensional condition is fulfilled) the intersection of this larger set with a non-special curve must still be a finite set. Beyond this, in the paper we shall briefly review some of the basic algebraic theory of group extensions of an elliptic curve by the additive group $G_a$, which are especially relevant in the said result. We shall conclude by stating some general questions in the same direction and discussing some simple examples. The paper concludes with the reproduction of a letter of Serre (whom we thank for his permission) to the second author, explaining how to obtain explicit projective embeddings of the said group extensions. Pietro Corvaja, David Masser, and Umberto Zannier, Sharpening 'Manin-Mumford' for certain algebraic groups of dimension 2, Enseign. Math. 59 (2013), no. 3-4, 225 -- 269. Abelian varieties of dimension $> 1$, Arithmetic ground fields for abelian varieties, Global ground fields in algebraic geometry Sharpening `Manin-Mumford' for certain algebraic groups of dimension 2	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $K$ be a number field, $X$ a smooth projective curve of genus $g\ge 2$ over $K$, $J$ the Jacobian of $X$ and $r$ the rank of the Mordell-Weil group of $J(K)$. Chabauty's method, made effective by Coleman, produces a way to compute, in many cases, the finite set $X(K)$ of rational points on $X$, provided that $r<g$ [\textit{C. Chabauty}, C. R. Acad. Sci., Paris 212, 882--885 (1941; JFM 67.0105.01); \textit{R. F. Coleman}, Ann. Math. (2) 121, 111--168 (1985; Zbl 0578.14038); \textit{W. McCallum} and \textit{B. Poonen}, Panor. Synth. 36, 99--117 (2012; Zbl 1377.11077)]. Replacing the Jacobian with non-abelian Selmer varieties and the abelian integrals with iterated Coleman integrals, \textit{M. Kim} [Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006); Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)]. proposed a way to extend Chabauty-Coleman approach to curves having $r\ge g$. A different method has been introduced in [\textit{J. S. Balakrishnan} et al., J. Reine Angew. Math. 720, 51--79 (2016; Zbl 1350.11067)] for hyperelliptic curves with a model $y^2=f(x)$ where $f$ is a monic polynomial of odd degree without multiple roots, when $K={\mathbb{Q}}$ and $r=g$. This method involves $p$-adic Arakelov theory and intersection theory on arithmetic surfaces. In the paper under review, the authors extends this method to the case of an algebraic number field $K$. They also consider bielliptic curves of genus $2$, extending to number fields the result proved for $K={\mathbb{Q}}$ in [\textit{J. S. Balakrishnan} and \textit{N. Dogra}, Duke Math. J. 167, No. 11, 1981--2038 (2018; Zbl 1401.14123)]. They illustrate their method by giving several examples, some with $r$ up to $4$, where the approach of [\textit{S. Siksek}, Algebra Number Theory 7, No. 4, 765--793 (2013; Zbl 1330.11043)] fails. They describe the algorithm for elliptic curves and for curves of genus $2$ over quadratic fields. quadratic Chabauty; bielliptic curves; $p$-adic heights Curves of arbitrary genus or genus $\ne 1$ over global fields, Rational points, Arithmetic varieties and schemes; Arakelov theory; heights, Elliptic curves over global fields, Heights Explicit quadratic Chabauty over number fields	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A famous theorem of A. Weil gives an upper bound for the number of rational points on curves over finite fields. A curve is called maximal if it attains this upper bound, the cardinality of the finite field being necessarily a square (say $q^2)$. The genus $g$ of a maximal curve satisfies $g\leq q(q-1)/ 2$. The paper deals with the classification problem for maximal curves of a given genus. The first result on this problem is due to \textit{H. G. Rück} and \textit{H. Stichtenoth} [J. Reine Angew. Math. 457, 185-188 (1994; Zbl 0802.11053)] and they showed that the curve $X^{q+1}+ Y^{q+1}= Z^{q+1}$ is the unique maximal curve with $g= q(q-1)/ 2$. The main tool here is the action of Frobenius on the Jacobian variety of a maximal curve. The main result of the paper under review is the classification of the curve $X^m+ Y^m= Z^m$ with $m= (q+1)/ 2$ as the unique maximal curve with genus $g= (q-1) (q-3)/8$ having a nonsingular plane model. Besides the action of Frobenius on the Jacobian variety, the other main tool here is the theory of Frobenius orders of morphisms developed by \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc. (3) 52, 1-19 (1986; Zbl 0593.14020)]. finite fields; rational points; maximal curves A. Cossidente, J. W. P. Hirschfeld, G. Korchmáros, F. Torres, On plane maximal curves, Compositio Math. Curves over finite and local fields, Arithmetic ground fields for curves, Finite ground fields in algebraic geometry On plane maximal curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theorem of Gross and Zagier relates the Néron-Tate heights of Heegner points on modular curves to the central derivative of certain $L$-functions. The arithmetic Gan-Gross-Prasad conjecture is a generalization of this theorem to higher-dimensional Shimura varieties. The author has proposed an approach to establish the conjecture using an ``arithmetic'' relative trace formula. The paper is a major breakthrough in this approach. The paper proves the fundamental lemma for the relative trace formula, with the assumption that the local field is ${\mathbb Q}_p$ where $p$ is a large enough prime. The argument should work for all local fields with large residue characteristic, in the general case there are just some claims that are expected to be true but need to be checked. A relative trace formula is a distribution which can be decomposed into a sum of orbital integrals. The fundamental lemma are identities for the orbital integrals over a local field, when the orbital integrals are associated to a base function (identity element of Hecke algebra). Once such identities are established, it is strong evidence that the trace formula approach will work, and some partial results can follow with relatively small amount of extra work. The reviewer does not claim to understand the details of the proof of the fundamental lemma, however the brilliance of the idea behind it is clear. Previously Jacquet introduced an idea to use Fourier transform in the proof of fundamental lemma. The current paper further developed the idea, and uses Weil representation (which encodes Fourier transform as Weyl group action) in the proof. This is combined with a local-global argument where the local orbital integral identity is derived from a global one, and an induction argument. The paper also used the same argument to reprove the fundamental lemma for Jacquet-Rallis relative trace formula that is used for Gan-Gross-Prasad conjecture for unitary groups. That fundamental lemma was originally proved using arithmetic geometric method by \textit{Z. Yun} [Duke Math. J. 156, No. 2, 167--227 (2011; Zbl 1211.14039)] for positive characteristic case and extended to any local field with large residue characteristic using a standard argument in logic. In comparison, the proof presented in current paper can be considered an elementary proof. arithmetic fundamental lemma; arithmetic Gan-Gross-Prasad conjecture; Kudla-Rapoport special divisor; relative trace formula; unitary Shimura variety; Weil representation Theta series; Weil representation; theta correspondences, Special values of automorphic $L$-series, periods of automorphic forms, cohomology, modular symbols, $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Algebraic cycles, Modular and Shimura varieties Weil representation and arithmetic fundamental lemma	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book originates from a series of lectures and seminar talks for graduate students at various universities in Japan. It is a research monograph describing research by the author on finite branched coverings of projective complex manifolds in connection with the theory of algebraic functions of several complex variables. The author presents a theory generalizing earlier work in one complex variable, in particular work of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)]. The book has three chapters: 1. ``Branched coverings of complex manifolds'', 2. ``Fields of algebraic functions'', 3. ``Weil-Tōyama theory''. The main result in chapter 1 is a theorem providing a suitable sufficient condition for the existence of a finite Galois covering of a projective manifold, branched over a given divisor. In the case of compact Riemann surfaces, this originates in a problem posed by Fenchel. In addition, chapter 1 contains many well chosen examples of finite branched coverings. -- Chapter 2 studies finite abelian coverings using the theory of currents developed by \textit{G. de Rham} and \textit{K. Kodaira} in ``Harmonic integrals'', Lecture Notes Inst. Adv. Study (Princeton 1950). There are results providing necessary and sufficient conditions for the existence of a finite abelian covering onto a projective complex manifold branched over a given divisor. Furthermore, the set of all isomorphism classes of finite abelian branched coverings of a projective complex manifold is described using the notion of rational divisor classes. The results can be considered as higher dimensional generalizations of results of Iwasawa from 1952. -- Chapter 3 studies similar questions for finite Galois coverings, but according to the author the results obtained are not completely satisfactory. The book should be a good source for inspiration to further work. finite branched coverings of projective complex manifolds; algebraic functions; finite Galois coverings M. NAMBA, Branched coverings - algebraic functions, Pitman Research Notes in Mathematics Series 161. Zbl0706.14017 MR933557 Coverings of curves, fundamental group, Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Low-dimensional topology of special (e.g., branched) coverings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization Branched coverings and algebraic functions	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be an irreducible smooth projective curve of genus $g$ defined over an algebraically closed field $k$. Let $r\geq 2$ be an integer, assume that $r>3$ if $g=2$. Let ${\mathcal N}(r)$ (respectively $N(r)$) denote the moduli stack (respectively moduli space) of stable PGL$(r,k)$-bundles on $X$. The authors prove that the natural map ${\mathcal N}(r) \to N(r)$ is an isomorphism outside a closed subset of codimension three. This identifies the Brauer group of ${\mathcal N}(r)$ with the Brauer group $\mathrm{Br}(N(r))$ of $N(r)$. The moduli space $N(r)$ has $r$ irreducible components $N(r)_i$ parametrized by ${\mathbb Z}/r{\mathbb Z}$. Let $M(r,L)$ be the moduli space of vector bundles of rank $r$ with determinant isomorphic to $L$. Then $N(r)_i$ is the quotient of $M(r,L), d(L)=i$, by the $r$-torsion points of the Jacobian. Using the Leray spectral sequence of the quotient map, the authors show that there is a surjective homomorphism $\mathrm{Br}(N(r)_i) \to \mathrm{Br}(M(r,L))$ and explicitly compute the kernel of this map. It is known that $\mathrm{Br}(M(r,L)) \cong {\mathbb Z}/h{\mathbb Z}$, where $h$ is the greatest common divisor of $r$ and $i$. Brauer group; moduli stack; stable projective bundles I. Biswas and A. Hogadi, \textit{Brauer group of moduli spaces of} PGL($r$)\textit{-bundles over a curve}, arXiv:0904.4640. Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli Brauer group of moduli spaces of PGL$(r)$-bundles over a curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This reproduction of the 1979 edition of André Weil's (1906-1998) Œuvres scientifiques covers more than 1500 pages. The articles and books are quoted in chronological order along the three volumes. For almost all references André Weil adds explicit and enlightening comments, the corresponding manuscript having been read in advance by Jean-Pierre Serre. The periods concerned by the three volumes are respectively 1926-1951, 1951-1964, 1964-1978. The Collected Papers are not the Opera omnia of the great mathematician. In particular they stop at the year 1978. In volume I a list of eleven conferences by Weil at the Bourbaki Seminar is dressed. Most articles are written in French. We tried to feature flashes on the various contributions ranged by general themes in each of the three volumes. Surprisingly Weil claimed himself to be an algebraist. But he could not have accepted Salomon Lefschetz's statement: If it is just turning a winch, it is algebra; if an idea comes in, it is topology. References to Bourbaki are not frequent in the collection. The interested reader should consult the Bourbaki Archives now available at the Paris Academy of Sciences. They also contain indications on Weil's living conditions during the times of World War II. Volume I Geometry / Number Theory [1926] The very first mentioned article is about surfaces with negative curvature. Weil extends a result due to Carleman: For a surface $S$ with boundary $L$ the inequality $S \leq L^2/4\pi $ holds. Weil pays tribute to Hadamard's seminar at Collège de France. [1948c] Weil considers this text to be a mere exercise. For more information he refers to his \textit{Basic number theory}. [1939a] The purpose of this article is to explain analogies between the field of algebraic numbers and the field of algebraic functions, by elementary methods. Algebra [1927c,1928] In order to verify that reliability on earlier great mathematicians may lead to useful results, Weil associates an algebraic curve $C$ to a finite algebraic field {k} of numbers determined by an equation with coefficients in $k$. He is able to obtain a precious tool for the study of diophantian equations. Geometry on algebraic curves is about points or systems of points that remain invariant with respect to birational transformations. Weil relates how he got acquainted with Mordell's achievements. [1929] In 1922 Mordell announced that for a cubic $C$ of gender 1, given by an equation with rational coefficients, the points with rational coordinates located on $C$ may be deduced rationally from a finite number of them. Weil succeeds in replacing that algebraic theorem by a statement of essentially arithmetical flavor. [1932c] From Weil: I had not given up trying to prove the Mordell conjecture; I did not even loose hope to come closer to that aim [\dots] by a careful analysis and a thorough use of the tools I had performed in my thesis. [1934a] The note is written like a letter addressed to Hecke by Weil and Chevalley. It follows methods due to Artin and Herbrand. [1942] This long letter to Artin is introduced by the following lines: I have now reached a stage in my work on correspondences where I feel it will be helpful if I make a general survey of the theory, for you and a few such people. This seems all the more desirable, as I now find that the final writing up is going to involve a recasting of the intersection theory in algebraic geometry (since I am not altogether pleased with v.d.Waerden's treatment of this subject); this means that things may not get ready for quite a while. [1934b,c] For both these articles the framework concerns the homomorphic association of a group of matrices to a given group $G$. One obtains a linear representation of $G$. É. Cartan observes that by the way a remarkable fact is established: A unitary group for which the set of all traces is finite, is by itself finite. [1936i] The article contains remarks made by Weil on technical results due to Chevalley; the considered groups may not satisfy Hausdorff's axiom. [1951b] This is a very technical paper; Weil acknowledges conversations with Chevalley and Shapiro. He regrets the inappropriate terminolgy Weil group, so much more because there could be a confusion with the Weyl group in the theory of semisimple groups. [1935b] As soon as 1926-27 during his stay in Göttingen Weil got acquainted with the theory of ideals in the ring of polynomials and foundations of algebraic geometry. He could improve on his thesis. The Bourbaki group obtained the publication of this article, in memory of Jacques Herbrand. [1946a] From the introduction to the present publication: Algebraic geometry, in spite of its beauty and importance, has long been held in disrepute by many mathmaticians as lacking proper foundations. Our method of exposition will be dogmatic and unhistorical throughout, formal proofs, without references, being given at every step. [1938b] The expository article on different ways to think about the Riemann-Roch property was followed by comments made by Weil. He acknowledged that the subject had been one of his favorite topics at least in its classical form as far as algebraic curves are considered. He states that he really did not take part in later investigations. [1939b] Weil estimates that among finite groups the groups with $p^n$ elements, $p$ being a prime number, range among the most important; they show up in the proof of Sylow's theorem. Analogies with nilpotent Lie groups are not completely understood. Weil announces some new results. In his comments he also mentions that in the text a mistake was detected by Michel Lazare: a famous `One concludes easily'! [1943b] In his comments Weil provides some indications on misleading terminologies. [1940b] Due to circumstances the publication of this article was premature. In the recension it said that except an unproven `important lemma' the article could not provide any information to German algebrists! [1947b] In his comments Weil speaks about the importance for him of results due to de Rham and Hodge. He also mentions the discussions he had with H. Cartan while both of them hold a position at the Strasbourg university. He reports on a story also related by H. Cartan. Unsatisfied by the usual presentation of the general Stokes formula, they agreed on a conclusion: We should write once and for all a correct proof of that formula and then publish it in a book. Was it the birth of Bourbaki? [1949e] The theory of fibre spaces, and in particular the theory of homogenuous functions, was uprising; its cohomological aspect had become of great importance. The Bourbaki committee demanded collaborators for reports on that subject. It was suggested to call Weil algebra a certain graded differential algebra. Weil said he would prefer the teminology universal algebra attached to the Lie algebra $A(G)$ of a group $G$. [1941] Weil describes the state of the arts among mathemticians around Prrinceton. Should one build mathematics on set theory, as done by Bourbaki? Could that be realized for probability theory, differential geometry, algebraic geometry? Weil claimed that any remedy for that dangerous situation could come only from a strict adherence to the principles of algebra. [1948a] After Weil the develoment of geometry on algebraic varieties, the varieties deduced from algebraic curves, did always occupy an outstanding position due to the importance of the particular properties and to the simplicity of the generation method; so studying these objects is a natural introduction to any general theory. [1948b] This paper is the continuation of the preceding one. [1949d] Riemann had informed Hermite that the periods of $2n$ periodical meromorphic functions of $n$ variables satisfying certain relations and inequalities enable the possibility of formulating thêta series. Related work performed by Weierstraß, Picard, Poincaré, Appell contributes to the generality of Abelian varieties. Well announces simpler versions. [1949b] The title may sound elementary. Weil reports that his study starts indeed from a reading of wholly elementary procedures due to Gauß. But he relies on many other mathematicians' achievements and directly on Dolbeault for specific questions. The outcome is very impressive and conjectures remain. [1949c] From the author's introduction: The notion of abstract variety makes it possible for the algebraic geometers to imitate closesly the definitions and procedures which have been so frutfully applied by modern topologists to the theory of fibre bundles. [1950b] This long article contains Weil's address to the 1950 ICM Congress in Cambridge/USA. It may be enlightening to quote from his comments and his talk in a nonchronological order. Weil's wish was to develop algebraic geometry over rings, the ring of integers being the most elementary case. He does not claim to having been the pioneer, but is satisfied to see the outcome in Shimura's work on local rings and the theory of schemes by Grothendieck, his students and successoras. And now the quotation from the beginnning of Weil's speach, expressing a naive observation. The preceding speaker concluded his address with a reference to Dedekind and Weber. It is therefore fitting that I should begin with a homage to Kronecker. There appears to have been a certain feeling of rivalry, both scientific and personal, between Dedekind and Kronecker during their life-time; this developed into a feud between their followers, which was carried on until the partisans of Dedekind, fighting under the banner of the purity of algebra, seemed to have won the field, and to have exterminated or converted their foes. [1951a] Weil's motivation is providing an acount of results due to Northcott, Siegel and himself, and further settle algebraic foundations for the theory of arithmetics on algebraic varieties. [1950a] Weil intends to write a survey combining known facts about algebraic varieties and others likely to be. An Abelian variety is an abstract `complete' variety in which is defined a law of composition making it a group. Analysis [1932a] Weil presents a new proof of Poincaré's theorem concerning a system of curves defined on a ring-shaped surface by a differential equation of first order. In Hadamard's seminar importance of ergodicity is stressed. It recommends the study along the achievemenets due to Poincaré, the physicists Maxwell and Boltzman, as well as von Neumann and G.D. Birkhoff later on. [1932b] Weil became fond of functions on several complex variables, motivated by his famous theorem (going back actually to Cauchy) on the convergence radius of a Taylor series. [1935d] The article is an elaboration of [1932b]. [1935c] Making use of von Neumann's definition of almost periodic functions on a group, Weil was able to extend a homomorphic image of the group to its closure which is a compact group. The relation with Haar's measure will anounce the beginning of abstract Harmonic Analysis. [1935e] The article is insignificant in Weil's eyes. [1936a] The article is a minor contribution by Weil at the first (and last) International Topology meeting held in Moscow in 1935, with Alexandroff as chief organizer. [1936b] The text about arithmetics on algebraic varieties is the written version of talks delivered by Weil after the just mentioned Moscow meeting. He acknowledges being unable to understand one specific page and hence concludes that it must be a mistake! [1935a] This draft appears in its final form [1938a] three years later. [1938a] Weil states that some of the most brilliant advances in modern mathematics have been accomplished in the domains of arithmetics, algebraic varieties and topology. These topics show analogies and very often rely on groups that happen to be Abelian; the latter restriction is frequently the origin of success. In his comments Weil says that nowadays one may claim that this very long article inaugurated the study of fibre vector spaces on an algebraic curve of dimension $\geq$ 1. [1936d] The article is the summary of a talk delivered at the Geneva university. The existence and uniqueness of an invariant measure on a locally compact group is explained. Weil extends the statement to homogeneous spaces. [1946b] Weil had thought being able to detect in Siegel's computing of volumes some opportunities to strengthen his theory of an invariant measure on homogeneous spaces and to find nontrivial examples. [1943a] In this long carefully written technical article, the coauthors rely to a certain extent on ideas expressed by Ahlfors. In the later comments Weil acknowledges that their results did not go so far as Chern's pioneering work in a new area of differential geometry. Their article makes use of embeddings, whereas Chern's formulation is intrinsic. General Topology [1936fg] During his Moscow stay Weil had opportunities to discuss with Pontrjagin and he became aware of the necessity to have a book on fundamentals of a theory on topological groups. This had to be done before the publication of Weil's book on Integration Theory. He introduced for instance the notions of complete spaces, bicompact spaces (later called compact spaces). Yet the most important event was the rise of the Bourbaki group of mathematicians with Weil's influential membership. One of the earliest books by Bourbaki was indeed the issue on Topology. [1937] The section reproduces all the 39 pages of Weil's book with explicit topic stated in the title: \textit{Sur les espaces à structure uniforme et sur la topologie générale.} The comments made by the author are reduced to the following statement: Forty years later one may smile on my eagerness to get rid of the countable situation. Chase it through the door, it will reenter through the window! [1945] Corrections have first been pointed out by H. Samelson. [1936h] Weil got interested in elliptic functions on the in that time popular $p$-adic groups. His results come close to the classical case. Functional Analysis [1927a,b] Weil examines essential differences between the sets of continuous functions and the set of square summable functions (Hilbert spaces). He mentions Volterra \textit{la cara e buona imagine paterno} who launched viewpoints exceeding his own results. It is also reported Courant having attributed to Weil the statement \textit{unproduktiv}. [1940c] The reading of this instructive text should be recommended to historians of mathematics. It provides an extensive survey of the interplay between probability theory and measure theory. More generally it furnishes a clew for the comprehension of modern axiomatized mathematics methods. The pioneering works of Lebesgue and Kolmogoroff are stressed. Yet Weil recognized later that, possibly boosted by his study on the Haar measure for groups, his reduction to locally compact spaces was too restrictive. But it had been more or less adopted by Bourbaki in the first editon of his Theory on Integration. [1940d] This book may be considered to be one of the first classics in Abstract Harmonic Analyis, along with the treatises by Pontrjagin, Loomis, and a few other authors. In his comments Weil reveals related historical facts. He also makes some surprising statements on himself: By temperament I was born arithmetician and algebrist, somehow geometer, not at all analyst. The real analyst, to feel himself comfortable, must possess instruments guaranteeing precision [dots] For me, as soon as one has to do with analysis, my instinct leads me to reduce this kit to an as small as possible number of strong useful tools; after that the algebrist may emerge again. General [1936c] Weil describes personal views on the organisation of studies in India. [1936e] Weil comments on examination procedures of Mathematiques spéciales students. [1938c] Weil explains his personal opinion on \textit{French Science} around 1938. [1940a] On March 26, 1940, André Weil wrote a letter to his sister Simome at the time he wanted to prepare a draft for a historical study of arithmetics and algebra theories starting with Gauß's Disquisitiones. He says to Simone: Maybe you get the feeling of the beginning; you wll surely not understand what comes after. [1947a] The article is an important contribution to the collective volume \textit{Les grands courants de la pensée mathématique} organized and edited by François Le Lionnais. Weil had often tried to avoid delays of that publication. His article examines essentially the possible evolutions of many mathematical topics; towards the end philosophical and epistemological considerations are made. [1949a] This article on applied mathematics is Weil's answer to a problem suggested to him by Claude Lévi-Strauss. Weil; Bourbaki; Artin; Chevalley; Shimura; Taniyama; Hodge; Kähler; Fermat; Ehresmann Collected or selected works; reprintings or translations of classics, History of mathematics in the 20th century, History of number theory, History of algebraic geometry, Proceedings, conferences, collections, etc. pertaining to number theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry Œuvres scientifiques. Collected papers. Vol. I (1926-1951)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Deformation theory in algebraic geometry is a basic tool that is just as old as the idea of classifying algebraic varieties. In his famous memoir on abelian functions from 1857, Riemann initiated the study of deformations of the complex structure of compact one-dimensional complex manifolds of given topological genus $g$, that is of compact Riemann surfaces (or non-singular complex projective curves) of genus $g$. The deformation of complex algebraic surfaces seems to have been considered first by Max Noether in 1888. However, a systematic deformation theory for higher-dimensional complex manifolds could be developed only as late as in 1958 and in the years thereafter, thus even more than 100 years after Riemann's pioneering memoir, and with the modern conceptual framework of sheaf theory, sheaf cohomology, and Hodge theory as a crucial ingredient [cf.: \textit{K. Kodaira}, ``Complex manifolds and deformation of complex structures.'' Grundlehren der Mathematischen Wissenschaften, 283. New York etc.: Springer-Verlag. X, 465 (1986; Zbl 0581.32012)]. This modern approach to complex-analytic deformation theory is well-known as Kodaira-Nirenberg-Spencer-Kuranishi theory, and its immediate effect on the development of classification theory in algebraic geometry has been equally revolutionary. In fact, A. Grothendieck immediately realized the functorial character of the Kodaira-Spencer theory of infinitesimal deformations of complex-analytic manifolds, and he outlined an analogous approach within his newly created foundational framework of algebraic schemes and formal algebraic geometry right away. These fundamental ideas, which eventually led to what is now called algebraic deformation theory, were published in a series of Bourbaki seminar expositions collected in his celebrated ``Fondements de la Géométric Algébrique'' [\textit{A. Grothendieck} ``FGA'', Secretariat Math. Paris (1962; Zbl 0239.14002)]. In the sequel, algebraic deformation theory has rapidly grown into a vast and central topic in modern abstract algebraic geometry, due to its crucial importance in regard to variational problems, including local properties of moduli spaces of varieties, vector bundles, and singularities. In its present state of art, algebraic deformation theory is highly formalized, both conceptually and methodically rather involved, widely ramified within algebraic geometry, and therefore not easily accessible to non-experts in the field. Although being of such importance and ubiquity in variational algebraic geometry, and while still developing rapidly, algebraic deformation theory had so far not yet found an adequate reflection in the relevant textbook literature. This fact made it rather difficult for non-specialists to find a solid orientation in this vast field of contemporary mathematical research, all the more so as numerous subtle technicalities and allegedly well-known results are scattered in the huge literature as ``folklore'', without rigorous and detailed proofs. The book under review is an attempt to partially fill this bothersome gap in the literature, and to provide a largely self-contained and comprehensive account of deformation theory in classical algebraic geometry, with complete proofs of those results and techniques that are needed to fully understand the local deformation theory of algebraic schemes over an algebraically closed field. This includes the careful explanation of those basic tools that are indispensable, for example, in the local study Hilbert schemes, Quot schemes, Picard schemes, and other classifying objects in this context. In this vein, and for the first time in the literature, the author compiles some of the many folklore results scattered in the literature, with detailed and systematic proofs, which must be seen as a just as valuable as rewarding contribution towards a solid foundation of algebraic deformation theory, and as an utmost useful service to the mathematical community likewise. As for the contents, the present book consists of four chapters, each of which is divided into several sections and subsections. In addition, and for the convenience of the non-expert reader, there are five appendices devoted to some basic facts from commutative algebra and algebraic geometry which are used throughout the text. After a brief introduction, in which an outline of the complex-analytic infinitesimal deformation theory à la Kodaira-Nirenberg-Spencer-Kuranishi serves as an explanation of the logical structure of algebraic (and functorial) deformation theory, Chapter 1 introduces the reader to infinitesimal deformations in an elementary fashion. The first section discusses extensions of rings and algebras, together with their generalizations to schemes over a base scheme, whereas the second section explains locally trivial deformations of schemes. This includes infinitesimal deformations of non-singular affine schemes, automorphisms of deformations and their extendability properties, first-oder locally trivial deformations, higher-order deformations, and the related (cohomological) obstruction theory for deformations. Chapter 2 provides the foundations of formal deformation theory. Starting with the concepts of formal smoothness and relative obstruction spaces for ring extensions, the author reconsiders infinitesimal deformations of algebraic schemes via M. Schlessinger's theory of functors of Artin rings, culminating in Schlessinger's famous theorem on the existence of (semi-) universal formal deformations [cf.: \textit{M. Schlessinger}, Trans. Am. Math. Soc. 130, 208--222 (1968; Zbl 0167.49503)]. This is followed by a discussion of deformation functors and local moduli functors in general, including their obstruction spaces as well as concrete applications to the deformation theory of algebraic surfaces in characteristic zero. In this regard, the author provides a largely self-contained treatment of formal deformation theory along the classical approach, that is without introducing cotangent complexes (à la L. Illusie) or methods of differential graded Lie algebras as more recent tools in deformation theory. Chapter 3 illustrates the general theory of deformation functors by several concrete examples. With the single exception of deformations of algebraic vector bundles, which have already been exhaustively treated in several recent research monographs like the book of \textit{D. Huybrechts} and \textit{M. Lehn} [``The geometry of moduli spaces of sheaves''. Aspects Math. E 31 (1997; Zbl 0872.14002)], the author describes in great detail the most important deformation functors in current algebraic geometry, mainly by carefully verifying Schlessinger's conditions for the existence of (semi-)universal deformation spaces in the respective cases. Moreover, the first-order deformations, i.e., the tangent spaces of these functors, as well as the corresponding obstruction spaces are thoroughly analyzed, thereby revealing a similar pattern in almost all of these exemplary cases. As for the concrete examples treated in this chapter, the author discusses the deformation functor of an affine scheme with at most quotient singularities, the local (relative) Hilbert functor of closed subschemes of a scheme, the local Picard functor of a scheme, deformations of sections of an invertible sheaf on a scheme, deformations of various types of morphisms of schemes, deformations of a closed embedding, and (co-)stable subschemes. Chapter 4 gives a thorough introduction to Hilbert schemes, Quot schemes, and flag Hilbert schemes. As the author points out, these objects are needed to construct important examples of global deformations, and to study their local behaviour in the framework of the main theme of the present text. Besides, until now it was rather difficult to give precise references for many results on the geometry of these classifying objects, just like in local deformation theory, and this very fact has been another reason for the author to include that chapter in his book. Together with the very recent monograph ``Fundamental Algebraic Geometry: Grothendieck's FGA Explained'' by \textit{B. Fantechi} et al. [Math. Surv. Monogr. 123 (2005; Zbl 1085.14001)], this chapter in the book under review provides the only (and overdue) reasonably comprehensive exposition on Grothendieck's Hilbert and Quot schemes. Apart from a general introduction to Castelnuovo-Mumford regularity, flattening stratifications, Hilbert schemes, Quot schemes, flag Hilbert schemes, and Grassmannians, together with applications to families of projective schemes, there is a concluding section on plane curves, their equisingular infinitesimal deformations, and their so-called Severi varieties. The author's approach to the proof of existence of nodal curves with an arbitrary number of singularities is very original and apparently new. Based on the use of multiple point schemes, this example strikingly illustrates the power of algebraic deformation theory even in classical curve theory. The five appendices at the end of the book collect some basic standard topics and are titled as follows: A. Flatness; B. Differentials; C. Smoothness; D. Complete intersections; E. Functorial language. Most of the results presented here come with full proofs, which further strengthens the already high degree of self-containedness of the book. There are no explicit exercises or working problems accompanying the text, but there is a wealth of illustrating concrete examples, additional remarks, historical annotations, and hints to further reading. The bibliography includes 190 references, ranging from the very classical up to the most recent articles and books, and both a very carefully compiled list of used symbols and a just as thorough alphabetical index considerably enhance the value of the entire treatise. Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. The author's style of writing captivates by its high degree of comprehensiveness, completeness, rigour, sytematical exposition, creative originality, lucidity, and user-friendliness in a like manner. The entire text is kept at a level that makes it suitable for graduate students with a solid background in commutative algebra, homological algebra, and basic algebraic geometry. But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on. textbook; formal methods; local deformation theory; local moduli; Hilbert schemes; Picard functors E. Sernesi, \textit{Deformations of algebraic schemes}. Springer 2006. MR2247603 Zbl 1102.14001 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Formal methods and deformations in algebraic geometry, Local deformation theory, Artin approximation, etc., Infinitesimal methods in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Deformations of algebraic schemes	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $C$ be a non-singular complex curve of genus $g>1$. For a fixed line bundle $\xi$ over $C$ of given degree $d$, denote by ${\mathcal M}(2,\xi)$ the (coarse) moduli space of isomorphism classes of rank-2 stable vector bundles over $C$ with determinant line bundle isomorphic to $\xi$. By a result by \textit{J.-M. Drezet} and \textit{M. S. Narasimhan} [Invent. Math. 97, No 1, 53-94 (1989; Zbl 0689.14012)], the Picard groups $\text{Pic}{\mathcal M}(2,\xi))$ are freely generated by the line bundle ${\mathcal O_ M}(\Theta)$ associated with an ample divisor $\Theta$ in ${\mathcal M}(2,\xi)$. This divisor $\Theta$ is called the generalized theta divisor of ${\mathcal M}(2,\xi)$, and its global sections $f\in H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))$ are called the generalized theta functions on ${\mathcal M}(2,\xi)$. Recent developments in conformal quantum field theory gave rise to conjectures about the dimension of the spaces $H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)$ of generalized theta functions of order $k$ [cf. \textit{E. Verlinde} and \textit{H. Verlinde}, ``Conformal field theory and geometric quantization'', Prepr. PUPT-89/1149 (1989)]. In the present paper, the author provides a partial verification of these conjectures, i.e., of the so-called Verlinde formulae. More precisely, he proves that in the special case of $k=1$ and $d=\deg\xi$ an odd integer, the conjectured formula \[ \dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))=2^{g-1}\centerdot (2^ g-1) \] indeed holds true. Moreover, using previous results of \textit{A. Beauville} [Bull. Soc. Math. Fr. 119, No. 3, 259-291 (1991)], the author succeeds in exhibiting an explicit base for the vector space $H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))$. The method of proof, in the particular case under investigation, is based upon the handy description of the moduli space $M(2,\xi)$ for hyperelliptic ground curves, which is due to \textit{U. V. Desale} and \textit{S. Ramanan} [Invent. Math. 38, 161-185 (1976; Zbl 0323.14012)], and on a comparison argument with respect to the number of odd theta characteristics on the base curve $C$. A general verification of the conjectured Verlinde formulae, i.e., for $\dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)$ with $k$ and $d=\deg \xi$ arbitrarily chosen, has recently been obtained by \textit{A. Szenes} and \textit{A. Bertram} [cf. ``Hilbert polynomials of moduli spaces of rank-2 vector bundles'', I, II (Harvard-University, September 1991 and November 1991)]. An announcement and sketch of their general results was published by A. Szenes, after the appearance of the present article [cf. \textit{A. Szenes}, Int. Math. Res. Not. 1991, No. 7, 93-98 (1991)]. coarse moduli space of rank-2 stable vector bundles; generalized theta divisor; generalized theta functions; odd theta characteristics Y. Laszlo, La dimension de l'espace des sections du diviseur thêta généralisé , Bull. Soc. Math. France 119 (1991), no. 3, 293-306. Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Theta functions and abelian varieties, Families, moduli of curves (algebraic) Dimension of the space of sections of the generalized theta divisor.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity $C$ and for the so called divisorial filtration for a modification of $({\mathbb C}^2,0)$ by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ $C$ or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious ``motivic version'' of an A'Campo type formula. Campillo, A., Delgado, F., Gusein-Zade, S.: Multiindex filtrations and motivic Poincaré series. Monatshefte. Math. 150, 193-209 (2007) Singularities of curves, local rings, Singularities in algebraic geometry, Complex singularities Multi-index filtrations and generalized Poincaré series	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper consists of an introduction to set the stage and to announce the main result (and some of its shortcomings), six sections and references. The first section gives a brief overview of the construction of abelian schemes over Hilbert modular surfaces. In the second section the main object of study, the fundamental relative motive $\mathcal V^{p,q}$, is introduced. Its various realizations are discussed. In particular, there is a comparison result between the Betti realization and intersection cohomology. The third section deals with the Betti and de Rham realizations. The fourth section gives an overview of $L$-functions. The results of Brylinski-Labesse and Chai are discussed. The exposition is quite detailed and technical. In the fifth section the main result (cf. \textit{infra}) is presented. The proof of this result takes quite some place. As usual an important role is played by Beilinson's Eisenstein map. This is explained in the sixth section. Let $F/{\mathbb Q}$ be a real quadratic field with ring of integers $\mathcal O_F$. Let $G/{\mathbb Q} \hookrightarrow\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}$ such that the maps $G\rightarrow{\mathbb G}_m$, ${\mathbb G}_m\hookrightarrow\text{Res}_ {F/{\mathbb Q}}{\mathbb G}_{m,F}$ and $\text{det}:\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\rightarrow \text{Res}_{F/{\mathbb Q}}{\mathbb G}_{m,F}$ lead to a Cartesian square. The group $G$ gives rise to a Shimura variety $S({\mathbb C})$ with a $G({\mathbb A}_f)$ action. Let $V^{p,q}=\text{Sym}^pV_2^{\vee}\otimes\text{Sym}^qV _2^{\vee}$, where $V_2^{\vee}$ is the dual of the standard representation of $\text{GL}_2$. It is known that intersection cohomology with respect to the Baily-Borel compactification decomposes into $G({\mathbb A}_f)$-isotypic components $H^2_{\text{ét}}(\pi_f)$ which are $\text{Gal} (\overline{\mathbb Q}/{\mathbb Q})$-modules. An important result due to J. L.~Brylinski and J.-P.~Labesse says that, for almost all $p$, the local $L$-functions $L_p(s,H^2_{\text{ét}}(\pi_f))$ can be computed as special values of some automorphic $L$-function. This $L$-function has simple zeros at $s=1-n$, $p+2\leq n\leq p+q+2$ for certain $\pi_f$, assuming $p\geq q>0$. It turns out to be appropriate to work with relative motives. Let $\mathcal A/S$ be the universal abelian scheme over $S$. $\mathcal A$ comes equipped with an $\mathcal O_F$-action. As a matter of fact, one is interested in constructing elements in $H^{p+q+3}_{\mathcal M}(\mathcal A^{p+q},\overline{\mathbb Q}(n))$. A basic role is played by the direct summand $\mathcal V^{p,q}$ of $R(\mathcal A^{p+q}/S)$, built as suitable $\text{Sym}^p\otimes\text{Sym}^q$. In Beilinson's philosophy, for a pure (Chow) motive $M$ with coefficients in $\overline{\mathbb Q}$ and $L$-function $L(s,H^i_{ \text{ét}}(M))$, one has, for $n>i/2+1$, regulator maps \[ r_{\mathcal H}:H^{i+1} _{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\rightarrow H^{i+1}_{\mathcal H}(M\otimes {\mathbb R},{\mathbb R}(n)) \] such that there exist $\overline{\mathbb Q}$-subspaces $\mathcal K(i,n) \subset H^{i+1}_{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))$ such that \[ r_{\mathcal H} (\mathcal K(i,n))=L(n,H^i_{\text{ét}}(M))\det\mathcal{DR}(i,n), \] where $\mathcal{DR} (i,n)$ is Deligne's $\overline{\mathbb Q}$-structure. In the underlying situation, write $r_{\mathcal H}:H^{\bullet}_{!\mathcal M}(\mathcal V^{p,q},{\mathbb Q}())\rightarrow H^{\bullet}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R},{\mathbb R}())$ for Beilinson's regulator map. For $S=S(\pi)\cup\{\ell\}$, let ${L_S(s,H^2_{\text{ét}} (\pi_f)^{\vee}))=\prod_{p\not\in S}L_p(s,H^2_{\text{ét}}(\pi_f)^{\vee})}$. The main result may be stated as follows: Let $\pi\in\text{coh} (V^{p,q})$ be stable or $\varepsilon(\pi_f)=-1$; then there is a $G({\mathbb A}_f)$ -submodule $\mathcal K(p,q,n)\subset H^{p+q+3}_{\mathcal M}(\mathcal V^{p,q}, {\mathbb Q}(n))$ for all $p+2\leq n\leq p+q+2$ and $p\geq q>0$, such that: (i) $r_{\mathcal H}(\mathcal K(p,q,n))\subset H^{p+q+3}_{\mathcal H}(\mathcal V^{p,q}\otimes {\mathbb R},{\mathbb R}(n))$; (ii) $r_{\mathcal H}(\mathcal K(p,q,n))(\pi_f)=L_S(n,H^2_ {\text{ét}}(\pi_f))\cdot\mathcal{DR}(p,q,n)(\pi_f)$ (equality of $\overline{\mathbb Q}$-subvector spaces in ${\mathbb R}\otimes_{\mathbb Q}\overline{\mathbb Q}$). $K(p,q,n)$ can be explicitly constructed. In (ii) $\mathcal{DR}(p,q,n)(\pi_f)$ is Deligne's $\overline{\mathbb Q}$-structure on absolute Hodge cohomology $H^{p+q+3}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R}, {\mathbb R}(n))(\pi_f)$. higher regulators; Hilbert modular surface; motivic cohomology; absolute Hodge cohomology Kings, G., Higher regulators, Hilbert modular surfaces, and special values of \textit{L}-functions, Duke Math. J., 92, 1, 61-127, (1998) Special values of automorphic $L$-series, periods of automorphic forms, cohomology, modular symbols, Automorphic forms on $\mbox{GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Étale cohomology, higher regulators, zeta and $L$-functions ($K$-theoretic aspects) Higher regulators, Hilbert modular surfaces, and special values of $L$-functions	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This note extends the proper base change theorem for torsion sheaves to the following. Given noetherian schemes with Y excellent, a proper morphism $\pi:\quad Y\to X,$ a morphism $f:\quad X'\to X$ whose pullback $f':\quad Y'\to Y$ is normal, and an abelian sheaf F on Y, the base change morphisms $f^(R^{\bullet}\pi_F)\to R^{\bullet}(\pi_'f'{}^F)$ are isomorphisms. The proof uses the original theorem, standard results on étale cohomology, and results from SGA 4 (Sém. Géométrie algébrique 4). proper base change theorem; étale cohomology Deninger, C., A proper base change theorem for non-torsion sheaves in ètale cohomology, J. Pure Appl. Algebra, 50, 231-235, (1988) Étale and other Grothendieck topologies and (co)homologies, Families, fibrations in algebraic geometry A proper base change theorem for non-torsion sheaves in étale cohomology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Shimura-Taniyama-Weil conjecture states that every elliptic curve over rationals appears as a factor of the Jacobian of a modular curve. The Taylor-Wiles (also Diamond, Breuil, Conrad) proof of this conjecture uses the marvelous interaction of local Frobenius elements inside the absolute Galois group to show that the deformation rings are small enough to be isomorphic to Hecke algebras. Most ingredients of Wiles' proof (except for the construction of modular Galois representations and a detailed description of arithmetic moduli of elliptic curves) are covered (to some extent) by the author's book [\textit{H. Hida}, Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, No. 69, Cambridge University Press (2000; Zbl 0952.11014)]. The present book provides the missing ingredients, as well as an outline of some new modularity results concerning two-dimensional Galois representations. Prerequisites for the reader are rather high (for example, some experience with arithmetic algebraic geometry). Complete proofs (or detailed references) of all statements are given, hence the book may be adressed to graduate students working in number theory and arithmetic algebraic geometry. The book is divided into five chapters. Algebro-geometric methods (Cartier duality, cohomology of coherent sheaves, descent etc.) are recalled in the preliminary Chapter 1, which are then used in studying classification problems of elliptic curves. In Chapter 2 the author gives a comprehensive account of the theory of moduli spaces of elliptic curves. The first five sections contain an explanation of the theory of divisors on curves, geometric theory of elliptic curves, elliptic curves over $p$-adic and complex numbers. Elliptic curves with level $N$-structures are studied in section 2.6. Regularity of the moduli schemes (Katz-Mazur theorem) is established in section 2.8 (Theorem 2.8.2). Let $p> 3$ be a prime, $(p,N)=1$. The main result of section 2.9 (Proposition 2.9.6) says that the $p$-ordinary moduli scheme is irreducible. Chapter 3 concerns the so-called ``control theorems''. In section 3.1 the horizontal control theorem (Theorem 3.15) has been proved. This result was used in [\textit{H. Hida}, Modular forms and Galois cohomology (loc. cit.)] for the identification of a (minimal) universal Galois deformation ring with an appropriate Hecke algebra. Vertical control theorems are proved in section 3.2, which describe specialisations of $p$-adic deformation of classical modular forms (Theorem 3.2.15, Lemma 3.2.18). They allow to identify the $p$-ordinary Galois deformation ring with a big $p$-adic Hecke algebra, and are useful in many other arithmetic applications. In Chapter 4 the author constructs modular Galois representations (and studies their ramifications) using Jacobians of modular curves (Theorem 4.2.7). Such representations are first constructed for modular forms of weight 2 by the method employed by \textit{G. Shimura} in his book [Introduction to the arithmetic theory of automorphic functions, Iwanami-Shoten and Princeton University Press (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)], and a general situation is reduced to this case. Let us recall that there are (another) two ways to construct modular Galois representations directly: geometric (Deligne), and analytic (Langlands). In the last Chapter 5 the author discusses modularity problems: the description of the original Wiles' treatment (Theorems 5.2.1 and 5.2.6), and the author's new results concerning elliptic $\mathbb{Q}$-curves (section 5.2.3). control theorems; Shimura-Taniyama-Weil conjecture; elliptic curve; modular curve; deformation rings; Hecke algebras; modular Galois representations; moduli spaces of elliptic curves; modular forms Hida, H.: Geometric modular forms and elliptic curves. World Scientific, Singapore (2000) Galois representations, Elliptic curves over global fields, Research exposition (monographs, survey articles) pertaining to number theory, $p$-adic theory, local fields, Holomorphic modular forms of integral weight, Local ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Geometric modular forms and elliptic curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $N$ be a natural number and let $X_0(N)$ be the modular curve with respect to $\Gamma_0(N)$. Let $J_0(N)$ be the Jacobian variety of $X_0(N)$ and let $\mathcal{C}(N)$ be the $\mathbb{Q}$-rational cuspidal divisor class group. When $N$ is a prime, \textit{B. Mazur} [Publ. Math., Inst. Hautes Étud. Sci. 47, 33--186 (1977; Zbl 0394.14008)] proved that the two groups $\mathcal{C}(N)$ and $J_0(N)(\mathbb{Q})_{\mathrm{tor}} $ coincide. For general cases, however, it is still an open problem whether the two groups are equal. Let $C_0(N)$ be the closed subset of $X_0(N)$ consisting of all cusps and let $\bar{J}_0(N)$ denote the generalized Jacobian of $X_0(N)$ with modulus $C_0(N)$ in the sense of \textit{J.-P. Serre} [Algebraic groups and class fields. Transl. of the French edition. New York etc.: Springer-Verlag (1988; Zbl 0703.14001)]. In this paper the authors investigate the structure of its torsion subgroup $\bar{J}_0(N)(\mathbb{Q})_{\mathrm{tor}}$, and observe that $\bar{J}_0(N)(\mathbb{Q})_{\mathrm{tor}}$ is unexpectedly smaller than $J_0(N)(\mathbb{Q})_{\mathrm{tor}} $ by proving the following result. Let $p$ be a prime number and $n$ be a positive integer. Then (1) $\bar{J}_0(p)(\mathbb{Q})_{\mathrm{tor}}$ is a cyclic group of order 2. (2) If $p\not\equiv 11\pmod {12}$ and $p>5$, then $\bar{J}_0(p^n)(\mathbb{Q})_{\mathrm{tor}}$ is isomorphic to the trivial group up to $2p$-torsion. (3) The statement (2) holds without the assumption $p\not\equiv 11\pmod {12}$ but up to $6p$-torsion. Furthermore they determine the structure of $\bar{J}_0(p^n)(\mathbb{Q})_{\mathrm{tor}}$ more precisely under the hypothesis that $J_0(p^n)(\mathbb{Q})_{\mathrm{tor}}=\mathcal{C}(p^n)$. modular curves; cuspidal divisor classes; generalized Jacobian varieties; torsion points Jacobians, Prym varieties, Elliptic and modular units Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the trace formula approach to the lifting problems of automorphic representations of reductive groups, the matching of local orbital integrals has been reduced to that for the unit element of the Hecke algebra related to relevant groups, which is usually called the fundamental lemma. In general, the fundamental lemma is one of the most difficult open problems in the trace formula approach to the theory of automorphic representations. The fundamental lemma of Jacquet and Ye is the one in the relative trace formula approach to the quadratic base change problem for $\text{GL}(n)$ [\textit{H. M. Jacquet} and \textit{Y. Ye}, Bull. Soc. Math. Fr. 120, 263--295 (1992; Zbl 0785.11032)]. An equivalent formulation of the fundamental lemma of Jacquet and Ye is an identity of two distributions, called Jacquet and Ye's conjecture. One of the distributions is of generalized Kloosterman integral type and is defined over $\text{GL}(n,E)$, and the other distribution is of relative Kloosterman integral type and is defined over $\text{GL}(n,F)$, where $E/F$ is a quadratic extension. Jacquet and Ye verified their conjecture for $\text{GL}(2)$ and $\text{GL}(3)$ over a local non-Archimedean field of characteristic zero. The aim of the paper under review is to prove the above conjecture of Jacquet and Ye for $\text{GL}(n)$ over local fields of positive characteristic. The main idea is to interpret the function-field analogies of the distributions of Kloosterman type defined by Jacquet and Ye as the trace of the Frobenius endomorphism over the $l$-adic cohomology of a certain algebraic variety, in the sense of Grothendieck. The conjectural identity of Jacquet and Ye in the function-field case becomes a certain quasi-invariant property of the trace with respect to a simply defined involution. It is a beautiful geometric argument. The original conjecture of Jacquet and Ye over local fields of characteristic zero is still open for $\text{GL}(n)$, $n\geq 4$. Ngô báo châu, \textit{Le lemme fondamental de Jacquet et Ye en caractéristique positive}, \textit{Duke Mathematical Journal}\textbf{96} (1999), 473-520. Representation-theoretic methods; automorphic representations over local and global fields, Étale and other Grothendieck topologies and (co)homologies The fundamental lemma of Jacquet and Ye in positive characteristic	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives with rational coefficients over a finite field when one assumes the Tate conjecture. The main result of the the paper under review is a similar result for motivic complexes. In a first section the statements of the Tate conjecture and of the generalised Tate conjecture are recalled and proven to be equivalent. The construction of ``the'' category of motivic complexes over a finite field $k$ proceeds then in essentially two steps. In a first step a semisimple Tannakian category of motives $\mathcal{M}_{\mathrm{num}}(k)$ over $k$ is constructed. This is essentially done by modifying the commutativity constraint in Grothendieck's category of pure motives (for numerical equivalence), using the Weil conjectures proven by Deligne. The group of Weil $q$-numbers can be identified with a subgroup of the fundamental group $\pi(\mathcal{M}_{\mathrm{num}}(k))$, and it is shown that equality holds if and only if the Tate conjecture holds over $k$. The second step of the construction consists of showing that the bounded derived category $\mathcal{D}^{\mathrm{b}}(\mathcal{M}_{\mathrm{num}}(k))$ is essentially the only candidate for a good triangulated category of motivic complexes. This category indeed has the hoped for properties if and only if the Tate conjecture holds over $k$ and numerical equivalence coincides with rational equivalence (always for rational coefficients). In a last section, the (conditional) results are used to produce an explicit description of Beilinson's algebra of correspondences at the generic point $A(X)$, for a smooth projective variety $X$ over $k$. Reviewer's remark: The reader might wish to consult the appendix (12p.) in the version of the paper provided on the first author's webpage, which fills in some details and proofs. Motives; Motivic complexes; Tate conjecture (Equivariant) Chow groups and rings; motives, Heights, Results involving abelian varieties, Complex multiplication and abelian varieties, Other abelian and metabelian extensions Motivic complexes over finite fields and the ring of correspondences at the generic point	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If $Y$ is a geometrically connected scheme of finite type over a field $k$, we have a surjective morphism (as profinite groups): $\text{ pr}_Y \pi _1(Y,) \rightarrow G_k $, where $$ is a suitable geometric point, $\pi _1$ is the étale fundamental group and $G_k$ is the absolute Galois group $\text{Gal} (k^{\text{sep}}/k)$. Grothendieck conjectured that when $k$ is finitely generated over ${\mathbb Q}$ and under suitable assumptions on $Y$ ($Y$ anabelian), the data $(\pi _1 (Y,),\text{ pr}_Y)$ determine (functorially) the isomorphism class of $Y$. One case where this conjecture is supposed to be true is for hyperbolic curves, i.e. when $Y$ is a smooth curve with $2-2g-n<0$, where $g$ is the genus of the smooth compactification $X$ of $Y$ and $n$ is the cardinality of $S(\overline k)$, with $S = X-Y$. In this paper the conjecture is shown to be true for affine (i.e. such that $n>0$) hyperbolic curves. More precisely what is shown is : Let $U_1$ and $U_2$ be two affine hyperbolic curves over a field $k$ finitely generated over ${\mathbb Q}$, then if there exists an isomorphism ${\mathcal F}: \pi _1 (U_1,) \rightarrow \pi_1(U_2,*)$ with $\text{ pr}_{U_1}=\text{ pr}_{U_2}\circ {\mathcal F}$, $U_1$ is isomorphic to $U_2$ (as schemes over $k$). The proof of the main theorem requires a lot of work and different techniques (study of the function fields and the Galois groups and their connections, study how to recover properties of the curve from data on the fundamental groups); this leads to results interesting per se, like the ones about affine (not necessarily hyperbolic) curves or about curves over discrete valuation rings. hyperbolic curves; fundamental group; Galois group; Grothendiek conjecture Tamagawa A., The Grothendieck conjecture for affine curves, Compos. Math. 109 (1997), no. 2, 135-194. Coverings of curves, fundamental group, Global ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry The Grothendieck conjecture for affine curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first result provides a generalization of Deligne formulae to the case where $n \geq 0:$ Theorem 1. Let $\pi : X \rightarrow B$ be a proper flat morphism of integral schemes of relative dimension $n \geq 0$ and let $L \rightarrow X$ be a line bundle which is very ample on the fibers. \noindent 1) There is a canonical functorial isomorphism \[ \lambda_{n+1}(L, X, B) = \langle L,..., L \rangle _{X/B}. \] 2) If $X$ and $B$ are smooth, and $K$ is the relative canonical line bundle of $X \rightarrow B,$ then there is a canonical functorial isomorphism \[ \lambda^{2}_{n}(L, X, B) = \langle L^{n}K^{-1}, L,..., L \rangle _{X/B}, \] where the right sides of 1) and 2) are Deligne pairings of $n + 1$ line bundles. The authors establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det(\pi_{*}L^{k})$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K.$ As a corollary, they showed that when $X$ is smooth, the line bundle $\eta$ associated to $X \rightarrow B,$ which was introduced by \textit{D.H. Phong} and \textit{J. Sturm} [Commun. Anal. Geom. 11, No. 3, 565--597 (2003; Zbl 1098.32012)], coincides with the CM bundle defined by \textit{S. Paul} and \textit{G. Tian} [CM stability and the generalized Futaki invariant, \url{arXiv: math.AG/0605278}]. line bundle; Deligne pairings; Knudsen-Mumford expansion D.\ H. Phong, J. Ross and J. Sturm, Deligne pairing and the Knudsen-Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475-496. Schemes and morphisms, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Analytic sheaves and cohomology groups, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Global differential geometry of Hermitian and Kählerian manifolds Deligne pairings and the Knudsen-Mumford expansion	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author introduces some complex of étale sheaves on a two-dimensional proper (not necessarily regular) $\mathbb Z$-scheme. This complex is a substitute of the complex $\mathbb Z(2, X)$ introduced by \textit{S. Lichtenbaum} [in: Number theory, Proc. Journ. arithm., Noodwijkerhut 1983, Lect. Notes Math. 1068, 127-138 (1984; Zbl 0591.14014)]. Modulo Gersten's conjecture (which is known for regular surfaces over a finite field) his complex is quasi-isomorphic to Lichtenbaum's one (up to 2-torsion). The main result is a duality theorem for constructible sheaves using the complex as a dualizing sheaf. The theorem extends the Artin-Verdier duality for regular one-dimensional schemes and Deninger's duality theorem for singular schemes to the case of two-dimensional schemes. arithmetic surfaces; Artin-Verdier duality; Deninger duality; two dimensional scheme M. Spieß, ''Artin-Verdier duality for arithmetic surfaces,'' Math. Ann., vol. 305, iss. 4, pp. 705-792, 1996. Étale and other Grothendieck topologies and (co)homologies, Arithmetic varieties and schemes; Arakelov theory; heights Artin-Verdier duality for arithmetic surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of the Jacobian on smooth plane curves are of key interest, mainly from Fermat and Klein curves. For degree 4, which are smooth plane quartics, the stratification by automorphism appears natural different families to study over the algebraic closure. This study is over an algebraic closed field of zero characteristic $K$. One family is the named Kuribayashi curves, with $a\in K$, defined by \[C_a: X^4+Y^4+Z^4+a(X^2Y^2+Y^2Z^2+Z^2X^2)=0\] which in order to be non-singular $a\notin\{-1,\pm 2\}$ and to have exact automorphism group equal to $S_4$ we impose $a\neq 0$ and $a\neq \frac{-1+\pm\sqrt{-7}}{6}$. We remind to the reader that Henn in 1976 [Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3, Inagural-Dissertation, Heidelberg] discovered such families by automorphism on quartics (and Kuribayashi with his coauthors are publishing results on this direction from 1978). Thus is interesting to study the Jacobian of such family as it is for Fermat or Klein quartic curves. \textit{L. Kulesz} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 503--506 (1999; Zbl 0954.11021)], proved that the Jacobian of $C_a$ is isogenous (isogeny defined in a maximal $\mathbb{Z}/2\times\mathbb{Z}/2$-Galois extension where the element $a$ is defined) to the product of $E_a^3$ with $E_a$ the elliptic curve defined by \[y^2=x(x-1)(x+a+1).\] The authors assume that a sextactic point $P$ appears in a fix smooth plane curve $C$ of degree $d\geq 4$ (we refer to the paper under review for the definition of sextactic point). Consider the natural map from $C$ to its Jacobian given by $\varphi:[Q]\mapsto [Q-P]$. We recall that there is a finite set of sextactic point on $C$. And there are sextactic points in $C_a$ if and only if $(a-14)(a^3+68 a^2-91 a+98)=0$ from [\textit{K. Alwaleed} and \textit{F. Sakai}, Saitama Math. J. 26, 67--82 (2009; Zbl 1202.14032)], and the case of the image of the group generated by the sextactic points for $C_{14}$ under $\varphi$ is obtained explicit in the preprint by \textit{K. Alwaleed} [``Group generated by total sextactic points of the Kuribayashi quartic curve'', Preprint]. Thus, in the paper under review, the authors deal the $C_a$'s such that $(a^3+68 a^2-91 a+98)=0$ obtaining first that any sextactic point on the Jacobian via $\varphi$ has order 8 or a divisor of 8, and more precisely such finite torsion group of the Jacobian of $C_a$ generated by the sextactic points via $\varphi$ (where we recall that we have fixed one of such three values of $a$ satisfying $a^3+68 a^2-91 a+98=0$) is isomorphic to $(\mathbb{Z}/8)^3$. Jacobian Kuribayashi curves; sextactic points Curves of arbitrary genus or genus $\ne 1$ over global fields, Special algebraic curves and curves of low genus On the Jacobian of Kuribayashi curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $k$ be a number field and $X$ a smooth complete $k$-variety. Based on the reciprocity law for the Brauer group of $k$, Manin isolated a condition which is necessary for the Hasse principle for $X$ to hold. A similar condition exists for weak approximation. Both have been widely studied under the name (Brauer-) Manin obstruction. These obstructions account for most of the known counterexamples to the Hasse principle [resp. to weak approximation]. Therefore people have been trying to find classes of varieties for which it could be proved that the Manin obstruction is the only one. The paper under review adds new examples of this type. To rephrase its main technical result, consider a dominant morphism $V\to \mathbb{A}^n_k$ of $k$-varieties whose generic fibre $V_\eta$ is geometrically integral, and such that a smooth projective model of $V_\eta$ has geometrically finite Brauer group and torsion-free Picard group. Assume that, for all $k$-points $x$ in a non-empty open subset of $\mathbb{A}^n_k$, the Manin obstruction to the Hasse principle [resp. to weak approximation] it the only one for the smooth projective models of the fibre $V_x$. Then, under additional technical conditions, which in particular guarantee the existence of a smooth $\overline k(D)$-point of $V \times_k\overline k(D)$ for almost all lines $D\subset\mathbb{A}^n_k$, it is proved that the Manin obstruction is the only one for the smooth projective models of $V$. Examples are given which are covered by this theorem, e.g., certain fibrations in conics or in Châtelet surfaces. This result generalizes a similar theorem about fibrations over the line, due to the same author [\textit{D. Harari}, Duke Math. J. 75, No. 1, 221-260 (1994; Zbl 0847.14001)]. A central technical tool is the study of the Brauer groups of the fibres of a smooth projective morphism $V\to U$, where $U \subset \mathbb{P}^1_k$ is open. If the generic fibre $V_\eta$ is geometrically integral and has geometrically finite Brauer group and torsion-free Picard group, it is shown that $\text{Br} (V_\eta)/ \text{Br} k(t) \cong\text{Br}(V_x)/ \text{Br} k$ for all $x$ in a Hilbertian subset of $\mathbb{P}^1(k)$. Again, this sharpens a result from the above-mentioned paper. Brauer-Manin obstruction; Hasse principle; weak approximation; Brauer group; Picard group Harari, D., \textit{flèches de spécialisation en cohomologie étale et applications arithmétiques}, Bull. Soc. Math. France, 125, 143-166, (1997) Global ground fields in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Brauer groups of schemes, Varieties over global fields Arrows of specializations in étale cohomology and arithmetical applications	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is an application of a new algebraic reformulation of the Picard group $\text{pic} (G)$ of a quasi-compact subset $G \subseteq X = \text{Spec} A$ for a commutative ring $A$. A torsion theoretic (algebraic) proof is given of $A$. Grothendieck's theorem that a complete intersection that is factorial in co-dimension 3 is factorial. Our proof is along the lines of Grothendieck's SGA2 proof, but eliminates the need for spectral sequences and (formal) sheaf theory. All torsion theoretic details are given in a lengthy appendix, including the proof of the long standing conjecture that $Q_G = \varinjlim Q_U$ for $G$ quasicompact and $U$ open, where $Q_G$ is defined in the following way: Denote by $F_G = \{I \subseteq A \|I \nsubseteq$ any $p \in G\}$ the torsion filter corresponding to $G$. $F_G$ is a directed set under reverse inclusion so we can form, for any $A$-module $M$, the $A$-module $P_G (M) : = \varinjlim_{I \in F_G} \Hom (I,M)$ and $Q_G (M) : = P_G (P_G(M))$. This yields the interesting result that restriction of an ${\mathcal O}_X$-Module to a quasi-compact, generically closed subset does not require the sheafification process. Picard group; complete intersection; factorial; torsion Call, F.: A theorem of Grothendieck using Picard groups for the algebraist. Math. scand. 74, 161-183 (1994) Torsion modules and ideals in commutative rings, Picard groups, Linkage, complete intersections and determinantal ideals A theorem of Grothendieck using Picard groups for the algebraist. -- Appendix	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $G$ be a split reductive group scheme over $\mathbb{Z}$ (recall that for any algebraically closed field $k$ there is a bijection $G\mapsto G\otimes k$ between isomorphism classes of such group schemes and isomorphism classes of connected reductive algebraic groups over $k$). Let $B$ be a Borel subgroup of $G$. Let $S$ be a scheme and $X$ a smooth proper scheme over $S$ with connected geometric fibers of pure dimension 1. Our goal is to prove the following theorems. Theorem 1. Any $G$-bundle on $X$ admits a $B$-structure after a suitable surjective étale base change $S'\to S$. Theorem 2. Any $G$-bundle on $X$ becomes Zariski-locally trivial after a suitable étale base change $S'\to S$. Theorem 3. Suppose that $G$ is semisimple. Let $D$ be a subscheme of $X$ such that the projection $D\to S$ is an isomorphism. Set $U:=X\setminus D$. Then for any $G$-bundle $F$ on $X$ its restriction to $U$ becomes trivial after a suitable faithfully flat base change $S'\to S$ with $S'$ being locally of finite presentation over $S$. If $S$ is a scheme over $\mathbb{Z}[n^{-1}]$ where $n$ is the order of $\pi_1(G(\mathbb{C}))$ then $S'$ can be chosen to be étale over $S$. Zariski-locally trivial bundle; reductive group scheme; $G$-bundle V. Drinfeld and C. Simpson, $B$-structures on $G$-bundles and local triviality, Math. Res. Lett. 2 (1995), 823--829. Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] $B$-structures on $G$-bundles and local triviality	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author is generalizing to the relative setting classical results as the theorem of Rojtman (the smooth case) and its generalization to singular varieties of Levine. His main theorem is: Theorem. Let $X$ be a smooth affine $k$-variety of dimension at least 2, $D$ an effective divisor on it. Then the Chow group of zero cycles on $X$ with modulus $D$, $\mathrm{CH}_0(X\|D)$ is torsion free except possibly for $p$-torsion if the characteristic of $k$ is $p>0$ The main idea of the proof is based on generalizing to the relative setting of the arguments of the paper [\textit{M. Levine}, Proc. Symp. Pure Math. 46, 451--462 (1987; Zbl 0635.14007)]. This paper is a continuation of the author's previous paper on torsion 0-cycles [\textit{F. Binda} et al., J. Algebra 469, 437--463 (2017; Zbl 1374.14007)]. Algebraic cycles, Algebraic cycles and motivic cohomology ($K$-theoretic aspects), Motivic cohomology; motivic homotopy theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Torsion 0-cycles with modulus on affine varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [J. $K$-Theory 12, No. 1, 183--201 (2013; Zbl 1291.19005)], \textit{G. Banaszak} and the author obtained the sufficient condition for the validity of the local to global principle for étale $K$-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale $K$-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by \textit{S. Barańczuk} [Expo. Math. 35, No. 2, 206--211 (2017; Zbl 1391.37089)]. We show that all our results remain valid for Quillen $K$-theory of $\mathcal{X}$ if the Bass and Quillen-Lichtenbaum conjectures hold true for $\mathcal{X}$. algebraic curve; étale $K$-theory; curve; Hasse principle $K$-theory of schemes, Galois cohomology, Relations of $K$-theory with cohomology theories, Curves of arbitrary genus or genus $\ne 1$ over global fields, Jacobians, Prym varieties, Hasse principle, weak and strong approximation, Brauer-Manin obstruction Note on linear relations in Galois cohomology and étale $K$-theory of curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $K$ be an algebraically closed field of characteristic zero and $C/K$ be a smooth projective curve over $K$ of genus $g\geq 2$. A point $P$ on $C$ is a Weierstrass point if there exists a holomorphic differential form on $C$ whose order at $P$ is $\geq g$. Let $J$ be the Jacobian of $C$. The Weierstrass subgroup of $C$ is defined as being the subgroup of $J(K)$ generated by the Weierstrass points. It is a geometric invariant of $C$. The authors are interested in the determination of the structure of this group. In some particular cases, for instance Fermat curves, these groups have been found to be torsion. M. Girard has previously obtained the first examples where this group has positive rank. Concerning the generic genus-$3$ curve, she has proved that this rank is at least $11$. In this paper, the authors complete this result by showing that there are no relations between the Weierstrass points on the generic genus-$3$ curve. They also obtain this fact for generic curves of any genus $g\geq 3$. They show that the Weierstrass subgroup of the generic curve of genus $g\geq 3$ has maximal rank, and is isomorphic to $\mathbb Z^{g(g^2-1)-1}$. Projective curves; Weierstrass points Curves of arbitrary genus or genus $\ne 1$ over global fields, Jacobians, Prym varieties, Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Computational aspects of algebraic curves The Weierstrass subgroup of a curve has maximal rank	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be an algebraic curve defined over $\overline{\mathbb Q}$ and contained in the algebraic torus $G_m^n$. We have the following result of Bombieri, Masser and Zanier: If $X$ is not contained in the translate of a proper algebraic subgroup, then only finitely many points of $X$ are contained in an algebraic subgroup of $G_m^n$ of dimension $n-2$. In this paper the previous result is generalized to show that there are at most finitely many points of $X$ that are ``close'' to an algebraic subgroup of dimension $n-2$, where the notion of close is given in terms of the Weil height. For any subset $H \subset G_m^n$ and any $\varepsilon \in \mathbb R$ the author defines the truncated cone: \[ C(H,\varepsilon)=\{ab; a \in H, b \in G_m^n, h(b) \leq \varepsilon(1+h(a)) \}, \] where $h(.)$ is denoting the Weil height on $G_m^n$. Also for $0 \leq m \leq n$, the special subset $H_m$ is defined as the points of $G_m^n$ that are contained in an algebraic subgroup of dimension at most $m$. With notation as above the result of Bombieri, Masser and Zanier states that $X \cap H_{n-2}$ is finite. Also Kronecker's theorem characterizing the points with Weil height zero as the torsion points, implies that $C(H_m,0)=H_m$. The main result of this paper is: Theorem: Let $X \subset G_m^n$ be an irreducible closed algebraic curve defined over $\bar{\mathbb Q}$, if $X$ is not contained in a proper coset there exist an $\varepsilon > 0$ effective and depending only on $h(X), \deg(X),$ and $n$ such that $X \cap C(H_{n-2},\varepsilon)$ is finite with cardinality bounded effectively in terms of $h(X)$, $\deg(X)$ and $n$. The theorem is proven in two steps: first showing that the height of points in $X \cap C(H_{n-2},\varepsilon)$ is bounded for $\varepsilon$ small and then showing that a subset of bounded height is finite for small $\varepsilon$. The second part proving the finiteness statement can be generalized to higher dimensional varieties in $G^n_m$. multiplicative group; algebraic curves; height functions; finiteness result; rational points P. Habegger, A Bogomolov property for curves modulo algebraic subgroups , Bull. Soc. Math. France 137 (2009), 93--125. Heights, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Arithmetic ground fields for curves, Counting solutions of Diophantine equations, Rational points, Global ground fields in algebraic geometry A Bogomolov property for curves modulo algebraic subgroups	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $C$ be a nonsingular projective curve of genus $g$ and let $J(C)$ denote its Jacobian variety. Let $A_\bullet(J(C))_\mathbb{Q}$ denote the group of rational algebraic cycles modulo algebraic equivalence, graded by dimension. The so-called tautological ring ${\mathcal R}(C)$ is defined to be the subgroup of $A_\bullet(J(C))_\mathbb{Q}$ containing the class of $C$ and stable under the Fourier transform, the intersection product, the Pontryagin product, pullbacks and pushforwards through multiplication maps by integers. In this paper the author constructs a set of generators of ${\mathcal R}(C)$, which enables us to control all the possible structures of the tautological ring. Let $[A_d(J(C))_{\mathbb{Q}}]_x= \{W\in A_d(J(C))_{\mathbb{Q}}; m^* W= m^{2g-2d-s} W$ for any $m\in\mathbb{Z}\}$. He takes the set of $s$-components $C_{(s)}\in[A_1(J(C))_{\mathbb{Q}}]_s$ of the curve $C$ as a starting point, and constructs inductively a certain cycle $\lambda_{\{i_1,\dots, i_d\}}\in [A_d(J(C))_{\mathbb{Q}}]_{i_1+\cdots+ i_d}$ which is annihilated by the principal polarization divisor and is congruent to $C_{(i_1)}\cdots C_{(i_d)}$ modulo the subspace spanned by elements of lower degree. As an application he gives an exhaustive description of ${\mathcal R}(C)$ for all the possibilities that may occur in the cases when $g\leq 9$. algebraic cycles; Chow group; Jacobian doi:10.1007/BF03191366 Algebraic cycles, (Equivariant) Chow groups and rings; motives, Jacobians, Prym varieties, Subvarieties of abelian varieties Tautological cycles on Jacobian varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $Y$ be a curve over an algebraically closed field $k$ of characteristic $p>0$. Let $W$ be a complete local ring with residue field $k$ and $G$ a finite flat group scheme over $W$ which acts faithfully on $Y$. Let $\text{Def}(Y,G)$ be the functor which associates to a local Artinian $W$-algebra $R$ with residue field $k$ the set of isomorphism classes of $G$--equivariant deformations of $Y$ to $R$. The functor $\text{Def}(Y,G)$ is studied using cohomological methods. The first 3 chapters of the paper give an exposition of certain cohomological methods for studying equivariant deformations of curves with group scheme actions. This theory is applied to the case of three point covers with bad reduction. There are three appendices including Picard stacks, the cohomology of affine group schemes and some spectral sequences. equivariant deformation; group schemes; cotangent complex [11] S. Wewers, `` Formal deformation of curves with group scheme action {'', $Ann. Inst. Fourier (Grenoble)$55 (2005), no. 4, p. 1105-1165. Cedram \| &MR 21 \| &Zbl 1079.} Local deformation theory, Artin approximation, etc., Coverings of curves, fundamental group, Deformations and infinitesimal methods in commutative ring theory Formal deformation of curves with group scheme action	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author establishes a Grothendieck-Riemann-Roch theorem for Deligne-Mumford stacks. More precisely: He defines ``cohomology with coefficients in representations'' of a Deligne-Mumford stack, constructs a Riemann-Roch transformation from $K$-theory to this new cohomology theory and proves a GRR theorem for this Chern character. The cohomology $H^\bullet_{\text{rep}}(F, \ast)$ with coefficients in representations of a stack $F$ is defined as the étale cohomology $H^\bullet((I_F)_{\text{et}}, \Gamma(\ast))$ of the ramification stack $I_F = F \times_{F\times F} F$ where $\Gamma$ is a cohomology theory in the sense of \textit{H. Gillet} [Adv. Math. 40, 203-289 (1981; Zbl 0478.14010)]. The main tool in the construction of the Riemann-Roch transformation is the so-called dévissage theorem for $G$-theory which may be viewed as a generalization of \textit{A. Vistoli}'s decomposition theorem for equivariant $K$-theory [Duke Math. J. 63, No. 2, 399-419 (1991; Zbl 0738.55002)] and which is proved by generalizing the étale descent theorem [see \textit{R.~W. Thomason} and \textit{Th. Trobaugh}, in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)] from schemes to stacks. The proof of the GRR theorem runs through the usual patterns (deformation to the normal cone, Chow envelopes, $\ldots$). Grothendieck-Riemann-Roch theorem; Deligne-Mumford stack; cohomology with coefficients in representations; equivariant $K$-theory; Vistoli's decomposition theorem; étale descent; homological descent; Chow envelope Bertrand Toën, ``Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford'', K-Theory18 (1999) no. 1, p. 33-76 Riemann-Roch theorems Riemann-Roch theorems for Deligne-Mumford stacks	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author states: ``The goal of this note is to describe the Grothendieck-Serre duality on an arithmetic surface after the fixing of a horizontal divisor on this surface. We apply this description to the generalization of $\theta$-invariants from the case of arithmetic curves to the case of arithmetic surfaces.'' Pontryagin dual group; ind-Euclidean lattice; pro-Euclidean lattice; Arakelov degree Arithmetic varieties and schemes; Arakelov theory; heights, Adèle rings and groups Grothendieck-Serre duality and theta-invariants on arithmetic surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a survey of the author's work with [\textit{A. J. De Jong, X. He} and \textit{J. M. Starr}, Publ. Math., Inst. Hautes Étud. Sci. 114, 1--85 (2011; Zbl 1285.14053)] on Serre's conjecture II in the case of the function field $K(S)$ of a surface $S$ over an algebraically closed field $k$: every $G$ torsor over $K(S)$ of a connected, simply connected, and semisimple algebraic group $G$ over $k$ is trivial, and more generally, a theorem about the existence of rational points for varieties defined over $K(S)$. The proof uses a geometric argument involving rational simple connectedness. The general principle is that a variety over the field $K(S)$ should have a rational point if it is ``rationally simply connected'' and if the elementary obstruction vanishes. However note that the definition of ``rational simple connectedness'' has been worked out only in the case of homogenous spaces ([Zbl 1285.14053] and [\textit{Y. Zhu}, ``Homogeneous fibrations over curves'', \url{arXiv:1111.2963}], and complete intersections [Zbl 1285.14053]. Roughly speaking, rational simple connectedness is the analogue of simple connectedness in topology, just as rational connectedness can be thought of as the analogue of connectedness. So part of the definition involves defining what it means for the moduli space of rational curves to be rationally connected. The actual definition is too technical to state in a review. The idea of producing the rational point is the following (again this is an over-simplified account of the proof). One first finds a model of the variety over $K(S)$ as an algebraic fibration over the surface $S$. Then using the Lefschetz fibration, one can write this as a family of varieties over a curve defined over $\kappa$, the function field of a curve (in this case, $\mathbb{P}^1$). It suffices to show that there is a section (over $\kappa$) of the family. To do this, one studies the space of sections and the Abel-Jacobi map to the Picard variety of the curve. If one can prove that this is surjective with rationally connected fibers, then one can conclude the existence of a section, which is the same as a rational point over $\kappa$ of the space of sections, by a theorem of \textit{T. Graber, J. Harris} and \textit{J. M. Starr} [J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)]. rationally connected varieties; semisimple algebraic groups; torsors Starr, J. M., Rational points of rationally simply connected varieties, Variétés rationnellement connexes: aspects géométriques et arithmétiques, 31, 155-221, (2010), Soc. Math. France: Soc. Math. France, Paris Rational and unirational varieties, Compactifications; symmetric and spherical varieties, Structure of families (Picard-Lefschetz, monodromy, etc.), Galois cohomology Rational points of rationally simply connected varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review deals with the problem of computing all integer solutions $(x, y)$ to the hyperelliptic equation $y^2 = f(x)$, where $f(x)$ is a separable polynomial of degree at least 3 with integer coefficients. \textit{R. F. Coleman}'s interpretation [Duke Math. J. 52, 765--770 (1985; Zbl 0588.14015)] of the method of Chabauty, allowing one to determine the rational points on a curve whose Jacobian has Mordell-Weil rank less than its genus. Over the last decade, \textit{M. Kim} [Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006)] has initiated a program aimed at removing this restricting on rank, allowing the study of rational points on hyperbolic curves through the use of nonabelian geometric objects generalizing the role of the Jacobian in the Chabauty-Coleman method. In this frame, the authors gave a method which they call \textit{quadratic Chabauty} [J. Reine Angew. Math. 720, 51--79 (2016; Zbl 1350.11067)] based on $p$-adic height pairings to $p$-adically approximate the set of integer solutions of equation $y^2 = f(x)$ in the case when the Jacobian of the corresponding algebraic curve has Mordell-Weil rank equal to its genus. In this paper, a full algorithm is presented for the computation in practice of all integer solutions of hyperelliptic equation $y^2 = f(x)$. It combines the quadratic Chabauty method with the Mordell-Weil sieve. All the necessary computations are described and an analysis of the $p$-adic precision which must be maintained throughout the computation is provided. A few examples are given. hyperelliptic equations; integer solution; Chabauty-Coleman method; quadratic Chabauty method; Mordell-Weil sieve J. S. Balakrishnan, A. Besser, and J. S. Müller, Computing integral points on hyperelliptic curves using quadratic Chabauty, Math. Comp. 86 (2017), 1403--1434. Curves of arbitrary genus or genus $\ne 1$ over global fields, Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.), Computer solution of Diophantine equations, Arithmetic varieties and schemes; Arakelov theory; heights Computing integral points on hyperelliptic curves using quadratic Chabauty	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study geometric properties of maximal curves over ${\mathbb F}_{q^2}$ having classical Weierstrass gaps. A maximal curve $X$ over ${\mathbb F}_{q^2}$ is a projective, geometrically irreducible, nonsingular algebraic curve, whose set of ${\mathbb F}_{q^2}$-rational points attains the Hasse-Weil bound: \[ \# X({\mathbb F}_{q^2})=q^2+1+2gq. \] For a maximal curve $X,$ let $P\in X({\mathbb F}_{q^2})$ and ${\mathcal D}=\| (q+1)P\| .$ The authors investigate the interplay between the canonical divisor and $\mathcal D$ via Stöhr-Voloch's approach to the Hasse-Weil bound. They prove that the support of the ${\mathbb F}_{q^2}$-Frobenius divisor associated with $\mathcal D$ is contained in the union of the Weierstrass points (for the canonical divisor) and the set $X({\mathbb F}_{q^2}).$ When the curves have classical Weierstrass gaps they show that this is an equality. The paper contains a number of examples of maximal curves with classical Weierstrass gaps. They conclude with an interesting remark and condition concerning maximal curves over ${\mathbb F}_{q^2}$ for which each rational point is not a Weierstrass point. This suggests closer investigation in the case when $g>1.$ maximal curves; Weierstrass points; Frobenius divisor A. Garcia, F. Torres, On maximal curves having classical Weierstrass gaps Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry, Finite ground fields in algebraic geometry On maximal curves having classical Weierstrass gaps.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main theorem proved in this article is the following: Theorem 1.1. Let $Y$ be the underlying variety of a connected affine algebraic group. Then two embeddings of the affine line $\mathbb{C}$ in $Y$ are equivalent (i.e. the same up to an automorphism of $Y$) provided that $Y$ is not isomorphic to a product of a torus $(\mathbb{C}^)^k$ and one of the three varieties $\mathbb{C}^3$, $\mathrm{SL}_2$, and $\mathrm{PSL}_2$. The case where $Y=\mathbb{C}^n$ with $n=2$ or $n \geq 4$ is classical; see the introduction of the article for an overview. Furthermore, the second author established in a former article a particular case of Theorem 1.1 (see the Main Theorem of [\textit{I. Stampfli}, Transform. Groups 22, No. 2, 525--535 (2017; Zbl 1454.14152)]): two embeddings of the affine line $\mathbb{C}$ into the underlying variety of $\mathrm{SL}_n$ are equivalent for all integers $n \geq 3$. Let us note that, as mentioned in the introduction of the article, the existence of non-equivalent embeddings of $\mathbb{C}$ into a product of a torus $(\mathbb{C}^)^k$ and one of the three varieties $\mathbb{C}^3$, $\mathrm{SL}_2$, and $\mathrm{PSL}_2$, is a well-known open problem. To prove Theorem 1.1, the authors use a lot of different results related to the theory of affine algebraic groups, their principal bundles, and their actions on algebraic varieties. For instance, one of the main tools they use is the shearing-tool: Let $X$ and $X'$ be affine lines embedded in an affine algebraic group G, and let $H \subseteq G$ be a closed subgroup such that $G/H$ is quasi-affine. If $\pi\colon G \to G/H$ restricts to an embedding on $X$ and $X'$ and if $\pi(X)=\pi(X')$, then there exist a $\pi$-fiber preserving automorphism of $G$ that maps $X$ to $X'$. automorphisms of algebraic groups; embedding of the affine line; principal bundles Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Automorphisms of surfaces and higher-dimensional varieties, Affine fibrations, Grassmannians, Schubert varieties, flag manifolds Uniqueness of embeddings of the affine line into algebraic groups	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $K$ be an algebraically closed field of characteristic zero and $C$ an integal, nodal curve of arithmetic genus $g$ over $K$. Assume $d\geq 2g-1$, $d \geq 0$ and $n \geq 2$ for certain integers $d$ and $n$. In this article, the author constructs a smooth compactification $\bar{M}$ of the moduli space $M$ of morphisms to projective space $\mathbb{P}^n$ of a fixed degree $d$ and computes the top intersection number $c_1(\mathcal{O}_{\bar{M}}(1))^{(n+1)(d+1-g)+g-1}[\bar{M}]$ of a certain Cartier divisor $X$ in $M$ to be $(n+1)^g$. where $g$ is the arithmetic genus of the curve $C$. The computation of the above result -- which was known in the case of a smooth curve (see \textit{A. Betram, G. Dasklopoulos} and \textit{R. Wentworth} [J. Am. Math. Soc. 9, No. 2, 529--571 (1996; Zbl 0865.14017)]) -- is made possible by the computation of the Chern character of the degree $d$ Picard bundle on the natural desingularization $\tilde{J}^d(C)$ of the compactified Jacobian $\bar{J}^d(C)$ (i.e. the moduli space of degree $d$ torsion-free sheaves of rank one on $C$). Further applications of this result to Brill-Noether loci are also given. nodal curves; torsion-free sheaves; generalized Jacobian; Picard bundle Bhosle Usha, N, Maps into projective spaces, Proc. Indian Acad. Sci. (Math. Sci.), 123, 331-344, (2013) Vector bundles on curves and their moduli Maps into projective spaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors investigate which equivalent definitions for Weierstrass points on smooth curves remain equivalent on Gorenstein curves. It is shown that a Wronskian definition using dualizing differentials is equivalent to a definition involving 1-special subschemes with support at a point. The appearance of 1-special subschemes, instead of just special divisors, is to be expected since the limit of a divisor may be a subscheme, but not a divisor, as smooth curves degenerate to a singular curve. Let X denote an integral, projective Gorenstein curve of arithmetic genus $g>1.$ An example is given to show that there may not be a locally principal 1-special subscheme (i.e. special Cartier divisor) of degree at most g at a (singular) Weierstrass point P on X. Another example is given to show that there may not exist a morphism $\phi: X\to {\mathbb{P}}^ 1$ of degree at most g such that $\phi^{-1}(\phi (P))=\{P\}$. Weierstrass points on smooth curves; Gorenstein curves Widland, C. and Lax, R. F.: Weierstraß points on Gorenstein curves,Pacific J. Math. 142 (1) (1990), 197-208. Riemann surfaces; Weierstrass points; gap sequences, Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Weierstrass points on Gorenstein curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This work comprises a summary of various approaches to the notion of infinitesimal, and a new ''intrinsic'' approach of the author's. The very readable introduction traces the antecedent ideas: a quick nod in the direction of Abraham Robinson, Weil's ''Théorie des points proches sur les variétés différentiables'', the use of nilpotents in algebraic geometry, the étale topology, the rise of topos theory, and synthetic differential geometry. Chapter 0 contains a quick survey of the interpretation of intuitionistic logic in a topos. Chapter 1 is entitled ''Use of Nilpotents''. It covers some of Kock's work on linear algebra in an intuitionistic setting, the Kock-Lawvere axiom for a ring of line type, models of the axiom in algebraic and differential geometry, infinitesimal linearity and the tangent bundle, 1- étale maps, and Reyes' notion of a Fermat ring. Chapter 2 is entitled ''Use of Infinitesimals''. The basic idea of this work is to exploit the observation that in the particular toposes discussed above as models for synthetic differential geometry it so happens that nilpotent elements in the ring of line type are also the elements x satisfying $NOT(NOT(x=0))$; recall that we are working in an intuitionistic setting. The author defines in general the relation of being infinitesimally close to be the double negation of equality. Of course, it is the fact that this notion is not geometric, i.e. not preserved by inverse image functors, which makes it difficult to work with, and also rather deeper. Infinitesimal invertibility and formal manifolds are examined from this point of view, and comparison is made with the nilpotent approach. Section 2 is particularly interesting; in it a class of toposes is introduced for which a ''Nullstellensatz'' holds, making the interpretation of infinitesimals work more smoothly. Chapter 3 is entitled ''Use of Intrinsic Neighbourhoods''. The idea here is that open neighbourhoods in a topology are interpreted as containing all points ''sufficiently close''. It follows that in an intuitionistic setting we have an intrinsic notion of ''open set'', namely a set closed under the relation of being infinitesimally close. This intrinsic topology is explored in this chapter. Chapter 4 investigates the relationship between local invertibility in the different senses now available, and what they mean in the different models under consideration. There is an appendix on the Dubuc topos. infinitesimal; nilpotents; synthetic differential geometry; intuitionistic logic; tangent bundle; ring of line type; formal manifolds; Nullstellensatz; intrinsic topology; local invertibility; Dubuc topos Penon, Jacques, De l'infinitésimal au local, Diagrammes, 13, suppl., iv+191~pages, (1985) Topoi, Synthetic differential geometry, Abstract manifolds and fiber bundles (category-theoretic aspects), Intuitionistic mathematics, Foundations of algebraic geometry De l'infinitésimal au local. (Thèse de Doctorat d'Etat, Université Paris VII)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article contains a theory of Weierstrass points for higher dimensional schemes. The theory is naturally flexible and well adopted to the many applications of Weierstrass points to arithmetical and geometrical questions. It gives the Weierstrass points as a closed subscheme of the original scheme, defined by the vanishing of sections of locally free sheaves. The scheme of Weierstrass points is naturally stratified by closed subschemes of the same type and they enjoy natural functorial properties. In order to illustrate the uses of our theory and to show how it is different from the families of sets suggested as Weierstrass sets of various types in the literature (Catanese-Schneider, Corti, Iitaka, Ogawa) we give some applications to rational normal scrolls and to hypersurfaces in projective space. Our approach is global and centered around the notion of a wronskian determinant. The construction of the wronskian determinants is the main innovation of the work. It is a section of a locally free sheaf and represents a far reaching generalization of the classical wronskian determinant. A novelty of our approach is that the point of departure is a system of maps of sheaves rather than the underlying scheme. This is important for applications. The main example of such a system is the sheaves of principal parts. The resulting theory is intimately connected with the properties of higher dimensional osculating spaces as defined by Pohl. An important part of the article consists in verifying that the resulting Weierstrass points have desirable properties. Weierstrass points for higher dimensional schemes; wronskian determinant; sheaves of principal parts; osculating spaces Laksov, D. andThorup, A., Weierstrass points on schemes,J. Reine Angew. Math. 460 (1995), 127--164. Riemann surfaces; Weierstrass points; gap sequences, Schemes and morphisms Weierstrass points on schemes	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a somewhat expanded version of the first edition that appeared in 1982 [Springer Graduate texts in mathematics 84, (1982; Zbl 0482.10001). The first edition of 1982 consisted of 18 chapters which have been included in the second edition without alterations. The second edition contains two new chapters, one on the Mordell-Weil theorem for elliptic curves and one on recent developments in arithmetic geometry. The authors wrote the first edition with the purpose to give insight into modern developments in number theory by showing their close relationship with classical, 19th century number theory. The authors wrote the new chapters 19 and 20 in the same spirit. In chapter 19, they give a proof of the Mordell-Weil theorem for elliptic curves over ${\mathbb{Q}}$ without using Kummer theory or algebraic geometry: they first give Cassels' proof of the weak Mordell-Weil theorem which uses only a weaker version of Dirichlet's unit theorem for number fields; and then they derive the Mordell-Weil theorem using the standard descent argument which is worked out by elementary arithmetic. Chapter 19 is meant as a preparation for chapter 20, in which the authors give a very interesting overview of the important developments in arithmetic geometry after the appearance of the first edition of their book. Among other things, they discuss the Mordell conjecture proved by Faltings, the Taniyama-Weil conjecture and the result of Frey, Serre and Ribet that this implies Fermat's last theorem, recent progress on the Birch-Swinnerton-Dyer conjecture by Coates-Wiles, Gross-Zagier, Rubin, and Kolyvagin, and the derivation of Gauss' class number conjecture from the results of Gross-Zagier. In chapter 20, the authors do not give proofs but they give sufficient background to understand and appreciate the results. Chapter 20 is an excellent introduction for those who want to study the subject more thoroughly. unique factorization; congruence; quadratic reciprocity; quadratic Gauss sums; Jacobi sums; cubic and biquadratic reciprocity; equations over finite fields; zeta functions; quadratic and cyclotomic fields; Stickelberger relation; Eisenstein reciprocity law; Bernoulli numbers; Dirichlet L-functions; Mordell-Weil theorem for elliptic curves; Mordell conjecture; Taniyama-Weil conjecture; Fermat's last theorem; Birch- Swinnerton-Dyer conjecture; Gauss' class number conjecture K. Ireland and M. Rosen, \textit{A Classical Introduction to Modern Number Theory}, (2nd ed.), Springer, 1990. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Elementary number theory, Arithmetic algebraic geometry (Diophantine geometry), Multiplicative number theory, Algebraic number theory: global fields, Finite fields and commutative rings (number-theoretic aspects), Diophantine equations, Elliptic curves over global fields, $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Higher degree equations; Fermat's equation, Elliptic curves A classical introduction to modern number theory.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper gives a survey on the recent studies on fundamental groups of curves in positive characteristic. Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field of characteristic $p>0$, $U$ an affine open subscheme of $X$ with complement $S=X \backslash U$, and let $\pi_A(U)$ be the collection of finite groups appearing in quotients of the fundamental group $\pi_1(U)$. (A strong form of) Abhyankar's conjecture states that a finite group $G$ is in $\pi_A(U)$ if and only if every prime-to-$p$ quotient of $G$ has $2g+\|S\|-1$ generators, and that such $G\in \pi_A(U)$ can be realized by a Galois étale cover of $U$ tamely ramified over $X$ with at most one exceptional wild-ramification point in $S$. In the paper under review, the author describes contributions to the solution of this conjecture, mainly focusing on those by \textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 311, No. 6, 341-346 (1990; Zbl 0726.14021)], \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)], and the author himself [\textit{D. Harbater}, Invent. Math. 117, No. 1, 1-25 (1994; Zbl 0805.14014)]. Expository explanations on the main ingredients of formal/rigid-analytic patching methods are also included. In the last section, there are described several related topics and stimulating open problems on structures of $\pi_1(U)$, $\pi_1(X)$ and embedding of fields. Among others, questions raised here on Grothendieck's anabelian geometry over finite fields have recently been settled by a remarkable paper by \textit{A. Tamagawa} [Compos. Math. J. (to appear)]. fundamental groups of curves; positive characteristic; quotients of the fundamental group; Abhyankar's conjecture; formal/rigid-analytic patching -, Fundamental groups of curves in characteristic $p$, in Proceedings of the International Congress of Mathematicians, 1, 2 (Zürich, 1994), Birkhäuser, 1995, pp. 656-666. Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Inverse Galois theory, Finite ground fields in algebraic geometry, Local ground fields in algebraic geometry Fundamental groups of curves in characteristic $p$	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For part I see J. Math. Kyoto Univ. 28, No.1, 91-110 (1988; Zbl 0654.14021).] Kodaira introduced the notion of logarithmic transformations and showed that a non-Kähler elliptic surface and a Kähler one can be transformed to each other via logarithmic transformations. The main purpose of this paper is to prove the counterpart of Kodaira's theorem for elliptic bundles. Here by an elliptic bundle f: $Y\to X$ over X, we mean that Y is a principal fiber bundle over a complex manifold X whose typical fiber and structure groups are non-singular elliptic curves. For an elliptic bundle f: $Y\to X$ over X, we shall define generalized logarithmic transformations along an arbitrary $Cartier\quad divisor\quad D$ on X, which is not necessarily assumed to be reduced, irreducible or effective. We state our main theorem: Theorem (A). Let f: $Y\to X$ be an elliptic bundle over a projective manifold X, which satisfies either of the following conditions: (1) Y is Kähler; (2) $h^{2,0}(X):=\dim (H^ 0(X,\Omega^ 2_ X))=0.$ Then Y can be obtained from the trivial elliptic bundle over X by a succession of generalized logarithmic transformations. However, theorem (A) does not necessarily hold if we drop the assumption that Y is Kähler. We can give the characterization of elliptic bundles which are obtained from the trivial bundle by generalized logarithmic transformations. logarithmic transformations; elliptic bundle Fujimoto, Y.: Logarithmic transformations on elliptic fibre spaces II, Math. Ann.288, 589--570 (1990) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Elliptic surfaces, elliptic or Calabi-Yau fibrations, Rational and birational maps Logarithmic transformations on elliptic fiber spaces. II: Elliptic bundle case	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It has been proven by \textit{J.-P. Serre} [``Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981'', in: Œuvres. Collected papers. Vol. IV: 1985--1998. Berlin: Springer. 1--12 (2000)], \textit{M. Larsen} and \textit{R. Pink} [Math. Ann. 302, No. 3, 561--579 (1995; Zbl 0867.14019)] and \textit{C. Chin} [J. Am. Math. Soc. 17, No. 3, 723--747 (2004; Zbl 1079.14029)], that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have ``the same'' $\pi_0$ and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent $F$-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent $F$-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent $F$-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects. $p$-adic cohomology, crystalline cohomology, Galois representations The monodromy groups of lisse sheaves and overconvergent $F$-isocrystals	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this book the author studies properties of germs of plane curves, i.e., power series in ${\mathbb C}[[x, y]]$, the ring of formal power series in two indeterminates over the complex numbers, or power series in ${\mathbb C}\{x, y\}$, the ring of convergent power series. After some introductory remarks, the first chapter deals with Puiseux's theorem and the Newton-Puiseux algorithm, and the second chapter introduces branches of a germ and the notion of intersection multiplicity of two curves. Blowing up and infinitely near points are the subject of the third chapter; Enrique's definition of infinitely near points is also mentioned. This chapter deals also with the notion of proximate points, free and satellite points, and the resolution of singularities for germs of curves. There are also algebraic descriptions of the rings in the successive neighborhoods of the ring of a plane germ (cf. section 3.10 and 3.11). The notion of virtual multiplicity of clusters is defined in chapter 4; this chapter contains also a proof of Noether's $Af + B\varphi$ theorem. Chapter 5 deals with characteristic exponents, the position of infinitely near points and the semigroup of values of an irreducible germ. In particular, it is shown that two irreducible germs are equisingular (i.e., have the same characteristic exponents) iff they have the same semigroup. (The well-known necessary and sufficient condition for a numerical semigroup to be the semigroup of a plane irreducible germ is given as exercises 5.10 and 5.11.) Polar germs are dealt with in chapter 6. Here the author gives Pham's example (showing that equisingular curves may have non-equisingular polars), proves Merle's result on the polar quotients, introduces the Milnor number, and shows [using result of his two papers: Math. Ann. 287, 429-454 (1990; Zbl 0675.14009); Comp. Math. 89, 339-359 (1993; Zbl 0806.14021)] that in the case of a single characteristic exponent, the polars of a germ may be used for unveiling properties of a germ that depend on its isomorphism class and not only on its equisingularity class. In section 6.10 and 6.11 the author treats the polar invariants of a reduced germ $\xi$ and shows how to compute them from an Enrique diagram of $\xi$. Linear families of germs are the subject of chapter 7. In section 7.2 the author introduces the weighted clusters of base points of pencils and proves their main properties. Section 7.5 and 7.6 deals with the notion of $E$-sufficiency (or $C^0$-sufficiency): A positive integer $n$ is said to $E$-sufficient for a reduced $f \in {\mathbb C} [[x,y]]$ iff all $h \in f + m^n$ are non-zero, reduced and equisingular to $f$ ($m$ is the maximal ideal of ${\mathbb C} [[x, y]]$). ({}The letter $E$ stands for equisingularity: there is a similar notion with regard to being analytically isomorphic which is dealt with in section 7.7.) The last chapter presents -- in the context of this book -- a classification of valuations of ${\mathbb C}\{x,y\}$ and a proof of Zariski's factorization theorem for complete ideals in ${\mathbb C}\{x, y\}$. This book covers a lot of classical material in a modern fashion; in particular, it clarifies some obscure definitions, remarks and statements in older papers from the end of the 19th century and the beginning of the 20th century. As prerequisites for reading this book a course in algebra should be enough. The clear presentation of the proofs makes life for a reader very easy; exercises at the end of each chapter are either examples to give the reader the possibility of proving for himself that he has mastered the text or provide new theorems not dealt with in the main text. Taken all together: A book making nice reading; it belongs to the shelf of every mathematician interested in curves and their singularities. Puiseux's theorem; Newton-Puiseux algorithm; infinitely near points; proximate points; satellite and free points; resolution of singularities; equisingularity; semigroup of a curve; characteristic exponents; linear families of germs; clusters; polar germ; formal power series; convergent power series E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser. 276, Cambridge University Press, Cambridge 2000. Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Equisingularity (topological and analytic), Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Germs of analytic sets, local parametrization Singularities of plane curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with Grothendieck's standard conjectures on algebraic cycles, introduced independently by Grothendieck and Bombieri to explain the Weil conjectures on the $\zeta$-function of algebraic varieties. We prove that the semisimplicity of the algebra of algebraic correspondences ${\mathcal A}^({\mathcal X}\times {\mathcal X})$ of a projective irreducible smooth variety ${\mathcal X}$ implies the standard conjecture of Lefschetz type for ${\mathcal X}$. It was proved by U. Jannsen that the algebra ${\mathcal A}^({\mathcal X}\times {\mathcal X})$ is semisimple when the numerical and homological equivalences of algebraic cycles on ${\mathcal X}$ are the same. Thus, with Jannsen's theorem our result asserts that the standard conjecture of Lefschetz type follows from Grothendieck's conjecture about the equality of the numerical and homological equivalences. This was known before only in the presence of the standard conjecture of Hodge type. algebraic cycles; Weil conjectures; algebraic correspondences; standard conjecture of Lefschetz type doi:10.1007/s002220050140 Algebraic cycles, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Graded rings and modules (associative rings and algebras) Graded associative algebras and Grothendieck standard conjectures	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This booklet contains the author's doctoral dissertation accomplished at the University of Köln, Germany under the guidance by U. Jannsen as academic adviser. Thematically, the research conducted here is devoted to a number of problems linking algebraic geometry over local fields, and that in its motivic-cohomological aspects, and the accompanying theory of local Galois representations. More precisely, the author investigates more closely two related conjectures due to \textit{S. Bloch} and \textit{K. Kato} [in: The Grothendieck Festschrift, Collect. Artic. Honor A. Grothendieck, Vol. I, Prog. Math. 86, 333--400 (1990; Zbl 0768.14001)], on the one hand, and to \textit{U. Jannsen} on the other, which predict certain good properties of the so-called motivic pairs arising from the étale cohomology of smooth and proper arithmetic schemes over $\mathbb{Q}$. In order to tackle these conjectures, at least so in manageable special cases, the author introduces what is called adelic pairs, describes then the underlying algebraic framework of monodromy representations and Galois representations, and applies the general results obtained here to the geometric case of arithmetic abelian varieties and arithmetic surfaces. The main results of this subtle analysis enable the author to verify the above-mentioned conjectures in those particular cases, which certainly represents a remarkable progress in this current field of research within arithmetic algebraic geometry. Diophantine geometry; local ground fields; étale cohomology; Galois representations; monodromy representations; Abelian varieties; arithmetic surfaces; algebraic geometry over local fields; motivic-cohomological aspects; adelic pairs; arithmetic algebraic geometry Local ground fields in algebraic geometry, Varieties over finite and local fields, Arithmetic ground fields for surfaces or higher-dimensional varieties, Arithmetic ground fields for abelian varieties, Brauer groups of schemes, Arithmetic varieties and schemes; Arakelov theory; heights Finiteness theorems for local Galois representations	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{P. Berthelot} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185--272 (1996; Zbl 0886.14004)] generalized the notion of PD structure in crystalline cohomology theory to that of ``$m$-PD structure (PD structure of level $m$)'' for each natural number $m$. From this, one can generalize the usual crystalline cohomology theory to $m$-crystalline cohomology using a level-$m$ version of the crystalline site. In this paper, the author proves the boundedness, the base change, and the finiteness of the crystalline cohomology of level $m$. \textit{B. Le Stum} and \textit{A. Quirós} [J. Algebra 240, 559--588 (2001; Zbl 1064.14015)] proved the exact crystalline Poicaré lemma, namely the ``jet complex of order $p^m$'' calculates the crystalline cohomology of level $m$. Unfortunately, this complex is not bounded and the proof to its local freeness contains a gap. So the finiteness of crystalline cohomology of level $m$ does not follow from the exact crystalline Poincaré lemma. The main contribution of this paper is to introduce an auxiliary ``de Rham-like'' complex which locally resolves a \textit{direct sum} of finitely many copies of the structure sheaf (as opposed to the structure sheaf itself). This allows the author to prove cohomological boundedness and the base change theorem. Together with the exact crystalline Poincaré lemma of Le Stum and Quirós, the author concludes the finiteness of crystalline cohomology of level $m$. crystalline cohomology of higher level; Poincaré lemma; PD structure of higher level Miyatani, K.: Finiteness of crystalline cohomology of higher level. Ann. inst. Fourier 65, No. 3, 975-1004 (2015) $p$-adic cohomology, crystalline cohomology Finiteness of crystalline cohomology of higher level	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author establishes an analogy between a modern proof of the classical theorem of N. H. Abel giving the rational equivalence classes of divisors $P$ and $Q$ of degree $d$ on a Riemann surface from equality of their images in the Jacobian variety $J(X)$, on the one hand, and a construction for certain cohomologous curves $P$ and $Q$ on a threefold $Y$ for which the image $Q- P$ in the intermediate Jacobian $IJ(Y)$ is zero, on the other hand. After recasting the classical proof of Abel's theorem in the language of differential forms with values in distributions (or currents), the author turns to smooth projective threefolds and constructs, for two suitable effective algebraic one-cycles $P$ and $Q$ therein, a rank-2 vector bundle $E_\infty$ whose first Chern class is trivial and whose second Chern class is represented by $P$ or by $Q$. Then, using Chern-Simons theory and following the distribution-theoretic proof of Abel's theorem for divisors on curves, he produces a certain three-form on the underlying threefold, which depends on a fixed quaternionic structure on the bundle $E_\infty$. If $P$ and $Q$ are algebraically equivalent curves and $P- Q$ is zero in the intermediate Jacobian of the threefold, then this differential three-form ``almost'' gives the rational equivalence of $P$ and $Q$. As the author points out, this paper was motivated by some work of R. Thomas [cf.: \textit{S. K. Donaldson} and \textit{R. Thomas}, in: The geometric universe: Science, geometry, and the work of Roger Penrose, 31--47 (1998; Zbl 0926.58003)]. Further details can be found in the author's recent paper [``Cohomology and obstructions II: Curves on $K$-trivial threefolds'', Preprint, \texttt{http://arxiv.org/abs/math.AG/0206219}]. intermediate Jacobians; rational equivalence of cycles; vector bundles; connections; currents H. Clemens, An analogue of Abel's theorem (English summary), The legacy of Niels Henrik Abel, Springer-Verlag, Berlin, 2004, pp. 511-530. $3$-folds, Picard schemes, higher Jacobians, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Transcendental methods, Hodge theory (algebro-geometric aspects), Currents in global analysis An analogue of Abel's theorem	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with some aspects of the theory of algebraic systems of effective divisors on a non singular curve and of one dimensional systems of curves on a smooth surface. In the case of curves, defined over an algebraically closed field, a short proof is given of a remarkable result due to Allen, which extends to non-linear systems of effective divisors the well known formula of De Jonquières. As an application of Allen's theorem a criterion of linear equivalence by Castelnuovo and Torelli is derived; this, in turn, implies a well known inequality by Castelnuovo and Severi concerning correspondences on a curve. Turning to surfaces and assuming the base field of characteristic zero, a formula is proved, giving a bound for the number of curves in a one dimensional system of generically smooth curves having a singular point. This bound was already given, but with incomplete proof, by Torelli. From this result a criterion of linear equivalence, analogous to the aforementioned one of Castelnuovo and Torelli for curves, is deduced. algebraic systems of effective divisors; correspondences on a curve C. Ciliberto--F. Ghione,Serie algebriche di divisori su una curva e su una superficie, Ann. Mat. Pura Appl.,136 (1984), pp. 329--353. Divisors, linear systems, invertible sheaves, Rational and birational maps Algebraic series of divisors on a curve and on a surface	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ denote a smooth integral complex algebraic curve of genus $g \geq 2$. Fix points $x_1, \dots, x_m$ of $X$ and integers $n_i$ attached to each $x_i$. For a semisimple simply connected algebraic group $G$, parahoric bundles or parahoric $G$-torsors on $X$ are pairs $(E, \theta)$ where $E$ is a torsor (i.e. a principal homogeneous space) on $X$ under a parahoric Bruhat-Tits group scheme $\mathcal{G}$ and weights $\theta \in Y(T)\otimes {\mathbb Q}$. Here $Y(T)$ denotes the group of $1$-parameter subgroups for a (fixed) maximal torus $T$ of $G$. To a set $\tau$ of conjugacy classes, one associates weights $\theta_{\tau} = \{ \theta_i \} \in (Y(T)\otimes \mathbb{Q})^m$ and a parahoric group scheme $G_{\theta_{\tau},X}$ with only ramification points $x_i$. The authors introduce notions of stability and semistability for parahoric $G$-torsors, construct their moduli spaces $M_X(G_{\theta_{\tau},X})$ and show that they are irreducible, normal, projective varieties. Let $K$ be a maximal compact subgroup of $G$. The authors show that there is a Fuchsian group $\pi$ and a homeomorphism of the space of conjugacy classes of representations of $\pi$ in $K$ of local type $\tau$ onto the moduli space $M_X(G_{\theta_{\tau},X})$ which identifies the subset corresponding to irreducible representations with the subset of $M_X(G_{\theta_{\tau},X})$ corresponding to isomorphism classes of stable $G_{\theta_{\tau},X}$-torsors. This generalises the work of \textit{V. B. Mehta} and \textit{C. S. Seshadri} on parabolic vector bundles [Math. Ann. 248, 205--239 (1980; Zbl 0454.14006)] to parahoric $G$-torsors. Parahoric torsors (without weights) were studied by \textit{G. Pappas} and \textit{M. Rapoport} [Adv. Math. 219, No. 1, 118--198 (2008; Zbl 1159.22010); Adv. Stud. Pure Math. 58, 159--171 (2010; Zbl 1213.14028)], many results on moduli stacks of these torsors were proved by \textit{J. Heinloth} [Math. Ann. 347, No. 3, 499--528 (2010; Zbl 1193.14014)] over arbitrary ground fields. $G$-torsors; parahoric bundles; representations of Fuchsian groups; Riemann surface Balaji, V. and Seshadri, C. S. Moduli of parahoric G-torsors on a compact Riemann surface J.~Algebraic Geom.24 (2015) 1--49 Math Reviews MR3275653 Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moduli of parahoric $\mathcal G$-torsors on a compact Riemann surface	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first volume of \textit{G. Harder's} ``Lectures on algebraic geometry I. Sheaves, cohomology of sheaves, and applications to Riemann surfaces'' appeared three years ago [Aspects of Mathematics E 35. Wiesbaden: Vieweg (2008; Zbl 1129.14001)] and was predominantly devoted to those fundamental concepts, methods, and techniques which are absolutely indispensable for formulating and treating algebraic geometry in its scheme-theoretic and cohomological language à la A. Grothendieck and his successors. Actually, two thirds of this first volume dealt with the basic prerequisites from general category theory, abstract homological algebra, sheaf theory, and cohomology theory of sheaves, nevertheless so with a steady prevailing view toward their later applications to algebraic geometry, algebraic topology, and complex-analytic geometry. The book under review is the long-announced second volume within the author's respective publication program, which still is intended to culminate in a future third volume devoted to the current active research topic of the cohomology of arithmetic groups. While the first volume presented the first five chapters of the whole treatise, this second volume completes the introduction to modern algebraic geometry by the subsequent five chapters. Each chapter is divided into several sections and subsections, which certainly enhances the overall lucidity of the entire exposition. As for the more precise contents, Chapter 6, the first chapter of the current second volume, is titled ``Basic Concepts of the Theory of Schemes''. The author develops here the language of schemes in its general and abstract setting. This includes affine schemes and general schemes, quasi-coherent sheaves of modules, algebraic vector bundles on schemes, affine and flat morphisms, relative schemes and base change, the various concepts of points in a scheme, representable functors in algebraic geometry, and an introduction to the theory of descend for later use. Chapter 7 provides some basic commutative algebra and its use in algebraic geometry. Among the selected topics discussed here are those which are especially related to geometry and algebraic number theory: finitely generated algebras, low-dimensional rings and basic results from arithmetic, flat morphisms, formal schemes, infinitesimal schemes, regular rings and smoothness, relative differentials, vector fields, derivations, infinitesimal automorphisms, group schemes and basic examples, and actions of group schemes on relative schemes. Chapter 8 gives an introduction to projective geometry, that is, to projective schemes, their special morphisms, their sheaf cohomology, and their specific geometric properties. Among other topics, the reader gets here acquainted with locally free sheaves on $\mathbb{P}^n_A$, valuative criteria for morphisms, ample and very ample sheaves, the cohomology of quasi-coherent and coherent sheaves, the fundamental finiteness theorems for the latter, the constructions of blowing-up and contracting subschemes, base change properties, basic intersection theory, and Bertini's theorem. Chapter 9 is devoted to algebraic curves and the Riemann-Roch theorem. The main objects of study are here smooth projective curves over an arbitrary ground field, together with their function fields, divisors, and line bundles. The main theme in the first part of this chapter is the classical Riemann-Roch-theorem, together with the Serre duality theorem, which are treated in a very interesting, highly enlightening, and somewhat non-standard manner. In fact, the author's approach uses classical ideas of \textit{R. Dedekind} and \textit{H. Weber} [cf.: ``Theorie der algebraischen Funktionen einer Veränderlichen'', Kronecker J. XCII. 181--291 (1882; JFM 14.0352.01)] and combines them effectively with the modern sheaf-theoretic and cohomological framework. The second part of this chapter discusses modern generalizations and applications of the Riemann-Roch theorem, ranging from the coarse moduli space of elliptic curves to the general idea of moduli spaces for arbitrary smooth projective curves. At the end, the general Riemann-Roch-Grothendieck theorem is explained in a more informal way, with the emphasis on some important special cases. However, the final highlight is the application to curves over finite fields, where the relationship between the Riemann-Roch theorem and the Zeta-function of such a curve is analyzed, on the one hand, and an analogue of the Riemann hypothesis (à la Mattuck-Tate and Grothendieck, 1958) is expounded on the other hand. Chapter 10, the most advanced part of the book, describes another moduli problem in algebraic geometry, namely the representability of the Picard functor for algebraic curves and their Jacobians. Referring to the construction of the Jacobian of a compact Riemann surface as described in Chapter 5 of Volume I, the author first defines the Picard functor on the category of schemes of finite type over a base scheme B, establishes its local representability and the construction of the Picard scheme of a curve, and obtains the Jacobian of a curve as a particular Picard scheme, which turns out to be a connected projective variety equipped with the structure of a commutative group scheme, that is, an abelian variety. In the sequel, the theory of abelian varieties and their Picard schemes is developed, with Jacobians of curves as prototype, and endomorphism rings of Jacobians, the Weil pairing, Néron-Severi groups, and the ring of correspondences are treated just as well as the $\ell$-adic Tate modules. As the author points out in the preface to the present volume, the final goal of this book is to bring the reader to the foothills of the mountain range of étale cohomology. This is done in the last section of this concluding chapter, where an outlook to this subject is given. After explaining the basic concepts and some of the fundamental theorems, this section touches upon the famous Weil conjectures for projective varieties over a finite field, Deligne's confirming theorem, and the special case of abelian varieties and of curves. Finally, the study of a degenerating family of elliptic curves serves as an illustrating, highly instructive example for the nature of the Weil conjectures and for compactifications of moduli spaces, respectively. All together, this second volume of [loc. cit.] stands out by the same features that already distinguished the foregoing Volume 1. Again, the very special and individual disposition of the subject matter, the purposeful arrangement of the material, the broad spectrum of topics and their interrelations, the great clarity of exposition, the topicality, the mathematical depth, and the reader-friendly style of writing make this text a particularly valuable enrichment of the existing course book literature in algebraic geometry and its arithmetic aspects. algebraic geometry (textbook); schemes and morphisms; projective schemes; algebraic curves; Jacobians; Picard functor; abelian varieties; étale cohomology Harder, G.: Algebraic Geometry. Vieweg. Manuscript (to appear) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Relevant commutative algebra, Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies, Jacobians, Prym varieties, Algebraic theory of abelian varieties Lectures on algebraic geometry II. Basic concepts, coherent cohomology, curves and their Jacobians	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a smooth algebraic curve of genus $g$ with Brill-Noether number $\varrho(g,r,d) = 0$, a finite number of linear series of degree $d$ and dimension $r$ over $C$ exists. A classical result by Castelnuovo counts these series and a generalization by \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 90, 359--387 (1987; Zbl 0631.14023)] provides a formula for the number of linear series with a prescribed vanishing sequence $a: 0 \leq a_0 \leq \cdot \leq a_r \leq d$ at a fixed general point~$p$. In this article, the authors consider curves with adjusted Brill-Noether number $\varrho(g,r,d,a) = -1$ and count the number of pairs of points $p$ and linear series with prescribed vanishing sequence $a$ at $p$. Pointed curves with $\varrho(g,r,d,a) = -1$ at the marked point form a divisor in the moduli stack ${\mathcal M}_{g,1}$ and the authors compute classes of closures in $\overline{\mathcal M}_{g,1}$ of such divisors. Moreover, they show that classes of Brill-Noether loci of codimension two are generally not proportional. For Brill-Noether divisors, this is known to be true [\textit{D. Eisenbud} and \textit{J. Harris}, Invent. Math. 90, 359--387 (1987; Zbl 0631.14023)]. Brill-Noether theory; linear series; vanishing sequence; enumerative geometry; Castelnuovo numbers Farkas, G; Tarasca, N, Pointed Castelnuovo numbers, Math. Res. Lett., 23, 389-404, (2016) Special divisors on curves (gonality, Brill-Noether theory), Computational aspects of algebraic curves Pointed Castelnuovo numbers	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $K$ be a field of characteristic different from 2 and $K_a$ its algebraic closure. The main result proved in the paper is the following: Let $f(x)\in K[x]$ be an irreducible and separable polynomial of degree $n\geq 5$ whose Galois group is $S_n$ or $A_n$ and let $J(C_f)$ be the Jacobian of the hyperelliptic curve of equation $y^2=f(x)$ whose $K_a$-endomorphism ring is denoted by $\text{End} (J(C_f))$. Then either $\text{End} (J(C_f))= \mathbb{Z}$ or $\text{char}(K)>0$ and $J(C_f)$ is a supersingular abelian variety. The proof is based on the study of representation of $\text{End} (J(C_f))$ in the $\mathbb{F}_2$-vector space of 2-torsion points of $J(C_f)$. hyperelliptic Jacobians; complex multiplication; representations; endomorphism ring Yu. G. Zarhin, Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett. 7 (2000), 123--132. Jacobians, Prym varieties, Complex multiplication and abelian varieties, Abelian varieties of dimension $> 1$ Hyperelliptic Jacobians without complex multiplication	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this work we consider the relation between the residues of a Weil differential $\omega$ on a non-singular projective algebraic curve $C$ over a perfect constant field. Let $P$ is a plase of curve $C$ of degree $n, F_P$ is the residue class field, which is a finite Galois extension of constant field $K, F=K(C)$ is the field of rational functions on $C, F'=F \cdot F_P$ is the constant field $F$ by means of $F_P$. Then in $F'$ there exist exactly $n$ places $Q_1, \dots, Q_n$ lying over $P$. The degree of $Q_i$ is one, for any $i:1 \leq i \leq n$. Moreover $\text{Re}s_{P} (\omega)=\sum \limits_{i=1}^{n} \text{Re}s_{Q_{i}}(\omega)$, and a local component $\omega_P$ of the Weil differential $\omega$ in $P$ can be viewed as $\omega_{P}(u)=\text{Re}s_{P}(u \cdot \omega)$ for any $u \in F$, that gives a simple proof of the Residue Theorem. Algebraic functions and function fields in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials About residues of Weil differentials in a constant field extension of an algebraic curve.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a survey article devoted to recent developments in constructing moduli spaces of geometric objects such as vector bundles or principal $G$-bundles on a fixed smooth projective variety defined over an algebraically closed field $k$ of arbitrary chracteristic. In view of the tremendous recent progress achieved in this field of contemporary research, the author had to focus on a few selected aspects of moduli theory. Thus, according to both his taste and expertise, the author has chosen the following guiding principles for the current survey article: 1. Analytic techniques are throughout avoided, and only a purely algebraic part of the theory is discussed, with special emphasis on the case of positive characteristics. 2. The main objective is to present some recent constructions of certain moduli spaces, that is, to establish their very existence, whereas any comments concerning their (mostly still unknown) geometry have been largely neglected. 3. Methodologically, the framework of Geometric Invariant Theory (GIT) is generally adopted, and therefore the classes of geometric objects to be classified are chosen with respect to their feasibility within this framework. With these principles in mind, the author has divided this survey into two major parts. The first part (Sections 2 and 3) explains the theory underlying the construction of moduli spaces of semistable (torsion-free) sheaves on a smooth projective variety $X$. This includes the concept of Gieseker semistability, slope semistability in characteristic zero, and strong slope semistability in positive characteristic likewise, together with some old and new results on the basic properties of semistable sheaves such as Bogomolov's inequality, restriction theorems, and dimension bounds for their cohomology groups. Along the way, the author explains his own recent contributions in the case of positive characteristic in greater detail [cf. \textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)]. These basic results allow to apply the standard GIT methods to construct the corresponding moduli spaces as projective schemes right away. The second part (Sections 4 and 5) turns to the construction of (semistable) principal $G$-bundles. Historically, this theory was initiated by \textit{A. Ramanathan} in his Ph.D.thesis at the Tata Institute, Mumbai, mainly in the case of compact Riemann surfaces, Math. Ann. 213, 129--152 (1975; Zbl 0284.32019). In the last few years, this theory (in characteristic zero) was extended to higher-dimensional projective varieties by \textit{T. Gómez} and \textit{I. Sols} [Moduli space of principal-sheaves over projective varieties, Ann. Math. (2) 161, No. 2, 1037--1092 (2005; Zbl 1079.14018)], on the one hand, and by \textit{A. H. W. Schmitt} [Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.14001)] on the other. But only very recently, the construction of compactified moduli spaces of semistable principal $G$-bundles on smooth projective varieties in arbitrary characteristic has been accomplished by \textit{T. L. Gómez}, \textit{A. Langer}, \textit{A. H. W. Schmitt} and \textit{I. Sols} [Moduli spaces for principal bundles in large characteristic, Adv. Math. 219, No. 4, 1177--1245 (2008; Zbl 1163.14008)], and a fairly detailed sketch of their novel approach is the main topic of the second part of the paper. Also, it is carefully explained how this final result is based upon the author's foregoing work in this direction [cf. \textit{A. Langer}, Semistable principal $G$-bundles in positive characteristic, Duke Math. J. 128, No. 3, 511-540 (2005; Zbl 1081.14018)]. Written in a very lucid and panoramic style, the survey under review effectively combines classical and utmost topical aspects of algebraic moduli theory, thereby providing an excellent source for further reading in a field at the forefront of current research in algebraic geometry. survey articles (algebraic geometry); algebraic moduli problems; moduli of semistable sheaves; principal $G$-bundles; geometric invariant theory; moduli of vector bundles A. Langer, Moduli spaces of sheaves and principal \textit{G}-bundles, Algebraic geometry. Part 1 (Seattle 2005), Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 273-308. Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vanishing theorems in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Geometric invariant theory Moduli spaces of sheaves and principal $G$-bundles	0

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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X_1\) be a smooth geometrically connected projective curve over the finite field of \(q\)-elements \(\mathbb{F}_q\). Let \(S_1\subseteq X_1\) be a reduced divisor consisting of \(N_1\) closed points. Set \(\mathbb{F}\) to be the algebraic closure of \(\mathbb{F}_q\). Let \((X,S):=(X_1, S_1)\otimes_{\mathbb{F}_q}\mathbb{F}\), \(l\) a prime number not dividing \(q\). There is an equivalence of categories between the category of \(\bar{\mathbb{Q}}_l\)-lisse sheaves and the category of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), where \(x\in (X-S)(\mathbb{F})\) is a geometric point on which the equivalence functor depends. Since \(X-S\) is obtained from \(X_1-S_1\) by base change, the Galois group \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the scheme \(X-S\) and hence on the set of isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, or equivalently on the set of isomorphism classes of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), via transport of structures. There are several equivalent ways to describe this action. One way is to look at the fundamental exact sequence of the étale fundamental groups \[ 1\to \pi_1^{\text{ét}}(X-S,x)\to \pi_1^{\text{ét}}(X_1-S_1,x) -^{{\phi}}\rightarrow {\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\to 1. \] For any \(\sigma\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) we choose an element \(\tau\in \phi^{-1}(\sigma)\), then for any continous finite dimensional representation \(\rho: \pi_1^{\text{ét}}(X-S,x)\to \text{GL}(V)\) we have \(\sigma(\rho)=( g\mapsto \rho(\tau^{-1}g\tau))\) for all \(g\in\pi_1^{\text{ét}}(X-S,x)\). As an action of \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) on the \textit{isomorphism classes} of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, the so defined action does not depend on the choice of the element \({\tau\in \phi^{-1}(\sigma)}\). Let \(F_1\) be the fraction field of \(X_1\), \(F=F_1\otimes_{\mathbb{F}_q}\mathbb{F}\) be the fraction field of \(X\), \(s\in S\). Then the choice of a place \(\bar{s}\) of \(\bar{F}\) above \(s\) defines an inertia group \(I_s\subset {\text{Gal}}(\bar{F}/F)\). A \(\bar{\mathbb{Q}}_l\)-lisse sheaf is said to have ``\textit{Principal unipotent local monodromy at \(s\)}'' if the composition \(I_s\subset {\text{Gal}}(\bar{F}/F)\twoheadrightarrow \pi_1^{\text{ét}}(X-S,x)-^{\rho}\rightarrow {\text{GL}}(V)\) factors through the largest pro-\(l\) quotient (\(\cong \mathbb{Z}_l\)) of \(I_s\) with an element of \(I_s\) with image \(a\) in \( \mathbb{Z}_l\) acting on \(V\) as \text{exp}\((aN)\), where \(N\) is nilpotent with one Jordan block. Let \(\mathcal{T}^{(n)}(X,S)\) be the set of isomorphism classes of rank \(n\) irreducible \(\bar{\mathbb{Q}}_l\)-smooth sheaves on \(X-S\), with principal unipotent local monodromy at each \(s\in S\). \(\mathcal{T}^{(n)}(X,S)\) as a subset of the isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves is stable under the \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\)-action. Let \(T(X_1,S_1,n)\) denote the number of fixed points of \(\mathcal{T}^{(n)}(X,S)\) by the geometric Frobenius \(\text{Frob}\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\). For each \(m\geq 1\), let \((X_m,S_m):=(X_1,S_1)\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}\), then \(T(X_1,S_1,n,m):= T(X_m,S_m,n)\), where \((X_m,S_m)\) is viewed as a pair over \(\mathbb{F}_{q^m}\). The aim of the article under review is to give a computation of the number \(T(X_1,S_1,n,m)\). It starts with a formula for \(T(X_1,S_1,n)\), under the assumption that \(n\) and \(N_1\) are \(\geq 2\), in terms of \(N_1\), \(n\), \(q\), the degrees \(\deg(s)\) for \(s\in S_1\) and the coefficients of the polynomial \(f(t):= \det(1-\text{Frob}\cdot t, H^1(X))\), where \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the \(l\)-adic cohomology group \(H^1(X)\) by transport of structures. However, the first formula is not helpful to understand how the number of fixed points varies with \(m\). One problem is that when one replaces \((X_1,S_1)/\mathbb{F}_q\) by \((X_m,S_m)/\mathbb{F}_{q^m}\), the number \((n/S_1):=\{\text{the largest divisor of }n\) that is prime to all \(\deg(s)\) for \(s\in S_1\) local systems; \(\ell\)-adic smooth sheaf; Lefschetz fixed point formula; automorphic representations; function fields; \(\mathrm{GL}(n)\); principal unipotent local monodromy; trace formula P. Deligne and Y. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921-982. Étale and other Grothendieck topologies and (co)homologies, Cohomology of arithmetic groups, Finite ground fields in algebraic geometry, Modular and Shimura varieties, Structure of families (Picard-Lefschetz, monodromy, etc.) Counting local systems with principal unipotent local monodromy	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let A be an abelian variety over \({\mathbb{C}}\) and let \(X\subset A\) be an algebraic curve; denote by T the sub-group of torsion points of A(\({\mathbb{C}})\); if X is not an elliptic curve (hence \(genus(X)\geq 2\) in this case), then \(T\cap X({\mathbb{C}})\) is a finite set. This is the main theorem proved in this paper. Note that this was called ''the Manin- Mumford conjecture''. It is interesting what Manin wrote (in 1966) when talking about such problems: ''As Lang remarked, Mordell's conjecture is equivalent to the assertion that a curve in an abelian variety has only a finite number of points in common with any subgroup of finite type of variety. This formulation contains no explicit references to the ground field. Nevertheless, it seems probable that the proof of Mordell's conjecture cannot be achieved without using the arithmetic (in the wide sense of the word) of the ground field. If that is so, then our result can be regarded as a reduction of the general problem to the case when the ground field is an algebraic number field: a case which has so far remained inviolate to known methods'' [cf. \textit{Yu. I. Manin}, Am. Math. Soc., Translat., II. Ser. 50, 189-234 (1966), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)]. Note that Bogomolov proved that the n-primary part \(T(n^{\infty})\cap X({\mathbb{C}})\) is finite for every integer \(n>1\) (\textit{F. A. Bogomolov}, C. R. Acad. Sci. Paris, Sér. A 290, 701-703 (1980; Zbl 0457.14020)]. Note that together with the proof of the Mordell conjecture given by Faltings [\textit{G. Faltings}, ''Endlichkeitssätze für abelsche Varietäten über Zahlkörpern'', Invent. Math. 73, 349-366 (1983); Erratum, Invent. Math. 75, 381 (1984)], this result by the author proves a conjecture by S. Lang [cf. \textit{S. Lang}, Ann. Mat. Pura Appl., IV. Ser. 70, 229-234 (1965; Zbl 0151.274)]. For the proof the author first remarks that it suffices to prove the theorem in case \(X\subset A\) is defined over a number field (under specialization \(X\cap T\) ''cannot become smaller''). Note that an analogous statement over the algebraic closure of a finite field is not correct. Thus it is understandable that in the proof of the theorem a careful study is made what happens under reduction modulo \(p^ 2\), where p is a carefully chosen prime number. This study of the torsion points in the local case is the main (and technically difficult) point of the proof. - We like to stress that the problem studied can be formulated purely over \({\mathbb{C}}\), but at the moment we have no proof which does not use arithmetic properties. Does there exist a proof using analytic tools? - For a related theorem, see VI.4 in the book edited by \textit{G. Faltings} and \textit{G. Wüstholz}, ''Rational points'' (Seminar Bonn/Wuppertal, 1983/84). This is a nice paper, the methods of proof are ingeneous and convincing. algebraic curve on abelian variety; Mordell conjecture; rational points on algebraic curves; Manin-Mumford conjecture; torsion points M. Raynaud, Courbes sur une variété abélienne et points de torsion. \textit{Inventiones} \textit{Mathematica 71}(1983), 207--233.Zbl 0564.14020 MR 0688265 Arithmetic ground fields for abelian varieties, Rational points, Global ground fields in algebraic geometry, Special algebraic curves and curves of low genus Courbes sur une variété abélienne et points de torsion	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a nonsingular projective curve, and \(J(C)\) be its jacobian. To a divisor \(D\) of degree zero on \(C\) is associated a point \(\gamma (D)\) of the jacobian by integration, and so-called Abel's theorem states that the image of \(D\) in the jacobian vanishes if and only if \(D\) is rationally equivalent to zero, i.e., \(D\) is a divisor of a rational function of \(C\), in other words, it gives an algebraic condition for \(\gamma (D)=0\), whereas the jacobian is defined complex-analytically. The following conjecture of \textit{S. Bloch} [``Lectures on algebraic cycles'', Duke Univ. Math. Ser. 4 (1980; Zbl 0436.14003)] can be regarded as the weight 2 counterpart of the above equivalence: For any smooth projective variety \(V\) over \(\mathbb{C}\), there exists a filtration on the Chow group of 0-cycles \(\text{CH}_ 0(V)\). Let \(S\) be a surface and let \(z\) be a cycle on \(V \times S\) with \(\dim z=m=\dim V\). Then \(z\) induces a map \(z:\text{CH}_ 0 (V) \to \text{CH}_ 0 (S)\). - - We get also \([z]:gr^ \bullet \text{CH}_ 0(V) \to gr^ \bullet \text{CH}_ 0(S)\). Conjecture (\textit{S. Bloch}). The map \([z]\) depends only upon the cohomology class \(\{c\} \in H^ 4(V \times S)\). Metaconjecture. There is an equivalence of category between a suitable category of polarized Hodge structures of weight 2 and a category built up from \(gr^ 2\text{CH}_ 0 (S)\). The aim of this article is two-fold: to give a condition for the vanishing of cycles in the intermediate jacobian, and to construct filtrations on the Chow groups which satisfy the above conjectures. polarized Hodge structures of weight 2; vanishing of cycles in the intermediate jacobian; filtration on the Chow group Saito, H.: Generalization of Abel's theorem and some finiteness properties of 0-cycles on surfaces. Compositio math. 84, 289-332 (1992) Parametrization (Chow and Hilbert schemes), Algebraic cycles, Picard schemes, higher Jacobians, Jacobians, Prym varieties, Surfaces and higher-dimensional varieties Generalization of Abel's theorem and some finiteness property of zero- cycles on surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives a method of computing topological fundamental groups of complex algebraic varieties by expressing them as fibres of good fibrations and relating their fundamental groups to those of the total space of the fibration. Let \(Y\) and \(B\) be nonsingular algebraic varieties over \(\mathbb{C}\). Let \(p: Y\to B\) be a dominant morphism with general fibre \(Y_0\) connected. Assume that \(p\) has a section \(s\) over a classical open set \(U\subset B\). Under certain conditions on \(s\) and \(U\), the author determines the kernel of the morphism \(\pi_1 (Y_0, s(0))\to \pi_1 (p^{-1} (U), s(0))\). He also gives a set of sufficient conditions for this kernel to be trivial. This is used to prove the main theorem: Let \(F\) be a nonsingular connected quasi-projective variety of dimension \(\geq 2\) and \(\overline {F}\) its nonsingular projective compactification such that \(\overline {F}- F\) is of codimension \(\geq 2\) in \(\overline {F}\). Let \(W\subset F\times \mathbb{A}^n\) be a reduced (possibly reducible) effective divisor, \(W_a=\) fibre over \(a\in \mathbb{A}^n\). Assume that the locus of \(a\in \mathbb{A}^n\) such that \(W_a\) is not a reduced divisor in \(F\) is of codimension \(\geq 2\) in \(\mathbb{A}^n\). Then the natural homomorphism \(\pi_1 (F-W_a)\to \pi_1 ((F\times \mathbb{A}^n- W)\) is an isomorphism for a general \(a\in \mathbb{A}^n\). Two applications of the theorem are given. For the first, consider general enough hyperplanes \(V_i\subset \mathbb{P}^n\), degree \(V_i= d_i\), \(i=1, \dots, m\). Let \(U\) be the subset of the Grassmannian of lines in \(\mathbb{P}^n\) consisting of all lines which intersect \(V_1\cup \dots \cup V_m\) in distinct \(d_1+ \cdots+ d_m\) points and \(U_p\) the subset of \(U\) consisting of lines passing through a fixed point \(p\in \mathbb{P}^n\). The author shows that for \(n\geq 3\) and \(\sum d_i\geq 3\), the natural monodromy representation of \(\pi_1 (U)\) in the generalized mapping class group is surjective with kernel an abelian group generated by at most \(m\) elements contained in the centre of \(\pi_1 (U)\). Further, if \(p\) is also general, then this representation is an isomorphism. The second application is the determination of the fundamental group of the complement of a projective plane curve \(f^q+ g^p =0\), \(p\geq 2\) and \(q\geq 2\) being degrees of the two general homogeneous polynomials \(f\) and \(g\). Such results are due to \textit{O. Zariski} [Am. J. Math. 51, 305-328 (1929)], \textit{M. Oka} [Math. Ann. 218, 55-65 (1975; Zbl 0335.14005) and J. Math. Soc. Japan 30, 579-597 (1978; Zbl 0387.14004)]\ and \textit{A. Nemethi} [Math. Proc. Camb. Philos. Soc. 102, 453-457 (1987; Zbl 0679.14004)]. Zariski problem; topological fundamental groups of complex algebraic varieties; good fibrations; mapping class group; fundamental group of the complement of a projective plane curve DOI: 10.1016/0040-9383(94)00045-M Homotopy theory and fundamental groups in algebraic geometry, Fundamental group, presentations, free differential calculus Fundamental groups of open algebraic varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this very clearly written paper is to provide a tool for construction of explicit examples of hyperelliptic curves of genus two such that its Jacobian variety admits real multiplication. Let \(K\) be a field of characteristic \(p\neq 2\). Let \(f(x)\in K[x]\) be an irreducible separable polynomial of degree six. Let \(C_f\) denote the hyperelliptic curve defined by \(y^2=f(x)\) and \(J(C_f)\) its Jacobian. Assuming that the Galois group of \(f\) is isomorphic to the alternating group of degree five and that \(\text{End}^0_K(J(C_f))\) contains a real quadratic field \(D\), the main theorem proves that \(\text{End}_K(J(C_f))\) is isomorphic to an order of \(D\) with discriminant \(\equiv 5 \pmod 8\). For the proof, the notion of very smallness of a group representation, which is introduced in [\textit{Yu. G. Zarhin}, Prog. Math. 195, 473--490 (2001; Zbl 1047.14015)] plays a crucial role. Armed with this notion, the author proves the theorem by employing the representation theory of finite groups and algebras. Furthermore, in the last two sections he gives quite a few concrete examples of hyperelliptic curves satisfying the hypotheses of the theorem. hyperelliptic curve; real multiplication; Jacobian variety Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for abelian varieties Hyperelliptic Jacobians with real multiplication	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, the author gives a generalization of a result of \textit{B. van Geemen} [Invent. Math. 78, 329--349 (1984; Zbl 0568.14015)] for any field \(k\) of characteristic different from \(2\). Namely, let \(\mathcal{A}_g\) be the moduli space of principally polarized abelian varieties of dimension \(g \geq 2\). Let \(\mathfrak{M}_g\) be the moduli space of smooth curves of genus \(g\). Let \(\mathfrak{I}_{g,k}\) be the closed subscheme of \(\mathcal{A}_g \otimes k\) defined as the scheme theoretic image of \(\mathfrak{M}_g \otimes k\) via the Torelli map. Let \(\mathfrak{S}_{g,k}^{\text{small}}\) be the closed subscheme of \(\mathcal{A}_g \otimes k\) defined by relations analogous to the classical Schottky-Jung relations. Theorem. The irreducible component of \(\mathfrak{S}_{g,k}^{\text{small}}\) containing \(\mathfrak{I}_{g,k}\) is equal to \(\mathfrak{I}_{g,k}\). The main difficulty and the main achievement of the paper is to carry out a purely algebraically proof of the result. In order to define the Schottky relations for instance, the author uses the formalism of Mumford's theta function. The second novelty is the use of results of [\textit{G. Faltings} and \textit{C.-L. Chai}, ``Degeneration of abelian varieties'', Ergebnisse der Mathematik und ihrer Grenzbiete, 3. Folge, Band 22, Springer-Verlag (1990 ; Zbl 0744.14031)] on toroidal compactifications and the use of Fourier-Jacobi expansion of theta functions that enables to go deeper in the boundary. Schottky-Jung relations; theta functions; Mumford's uniformization Theta functions and curves; Schottky problem The small Schottky-Jung locus in positive characteristics different from two.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper contains two parts. In the first one, the following result is proved: ``Let M be a Riemann surface of genus \(g,\) G(n,m) the Grassmann manifold of n-dimensional quotients of \({\mathbb{C}}^ m\) and k a positive integer with \(k\geq n-2g\). Then the inclusion \(Hol_ d(M,G(n,m))\to Map_ d(M,G(n,m))\) induces cohomology isomorphisms up to dimension \(k- 2m^ 2g\), provided that \(d\geq d_ 0(k,n,g)\). - Here \(Hol_ d\) (resp. \(Map_ d)\) stands for holomorphic (resp. continuous) maps of degree \(d.\) The bound \(d_ 0\) is done explicitly. For maps in the projective space, the result was shown by \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] who conjectured more general results, as the above ones. The proof is done by induction, starting with Segal's result, and uses extensively holomorphic bundles on M. One uses all this, and previous technique of the author [``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020)], to rederive in the second part of the paper the formula given by \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond., A 308, 523-615 (1982; Zbl 0509.14014)] for the cohomology of the moduli spaces of holomorphic bundles on curves. holomorphic maps; Riemann surface; Grassmann manifold; quotients; cohomology of the moduli spaces of holomorphic bundles on curves Frances Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. 24 (1986), no. 2, 221-275. Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Geometric invariant theory, Homogeneous spaces and generalizations On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f:X\rightarrow C\) be a family of curves of genus \(g\) with smooth general fiber. When \(g=1\), \textit{K. Kodaira} proves a fundamental formula in [Ann. Math. (2) 78, 1--40 (1963; Zbl 0171.19601)] which represents the global invariants of the surface \(X\) by the local invariants of the singular fibers together with the modular invariant. The purpose of this paper is to generalize Kadaira's formula to the higher genus cases. In the author's previous paper [Math. Z. 222, No. 4, 655--676 (1996; Zbl 0864.14016)], he defines the Chern numbers \(c_{1}^{2}(F),c_{2}(F),\chi_{F}\) of a singular fiber \(F\), and shows that, if \(g=1\), then \(c_{1}^{2}(F)=0\) and \(c_{2}(F)\) is exactly the coefficient in Kodaira's formula according to the type of the singular fiber. His generalization is formulated with the help of these local invariants together with some natural modular invariants constructed from a holomorphic map from \(C\) to the moduli space of semistable curves of genus \(g\). Chern numbers; family of curves; singular fiber Chen, Z., Lu, J. and Tan, S.-L., On the modular invariants of a family of non-hyperelliptic curves of genus 3, http://math.ecnu.edu.cn/preprint/2010-006.pdf Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Chern numbers of a singular fiber, modular invariants and isotrivial families of curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Grothendieck's anabelian conjecture for smooth, hyperbolic curves over finite fields states that, given any two hyperbolic curves \(U\) and \(V\) over finite fields, any isomorphism between the arithmetic fundamental groups \(\pi_1(U)\) and \(\pi_1(V)\) arises from a unique isomorphism \(U \to V\). In particular, if \(U\) is a smooth hyperbolic curve over a finite field, the fundamental group of \(U\) determines the isomorphism class of \(U\). This was proven in \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)] in the affine case and in \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47 (2007; Zbl 1143.14305)] in the proper case. The current paper proves a stronger, ``prime-to-\(p\)'' version of this result: For any smooth curve \(X\) over a finite field \(k_X\), there is a natural inclusion \(\pi_1(\overline{X} := X \times_{\text{Spec } k_X} \text{Spec } \overline{k_X}) \hookrightarrow \pi_1(X)\). Let \(\Delta_X\) be the maximal prime-to-\(p\) quotient of \(\pi_1(\overline{X})\), and let \(S\) be the kernel of the canonical map \(\pi_1(\overline{X}) \to \Delta_X\). Write \(\Pi_X := \pi_1(X)/S\). Then, if \(U\) and \(V\) are smooth hyperbolic curves defined over finite fields, any isomorphism \(\Pi_U \to \Pi_V\) arises from a unique isomorphism \(U \to V\). In particular, \(\Pi_U\) determines the isomorphism class of \(U\). A similar birational result is also proved for absolute Galois groups of function fields, strengthening a result of \textit{K. Uchida} [Ann. Math. 106 (1977; Zbl 0372.12017)]. The outline of the proof is similar to that of \textit{A. Tamagawa} [Compositio Math. 109, 135--194 (1997; Zbl 0899.14007)]. Given \(\Pi_X\), one first addresses the ``local theory'', attempting to recover the closed points of \(X\) by recovering the conjugacy classes of decomposition groups of closed points of \(X\) as subgroups of \(\Pi_X\). In the case treated by the current paper, the points of \(X\) can only be recovered up to a finite set \(E_X\), the ``exceptional set''. Then, one uses Kummer theory to recover the group \(\mathcal{O}_{E_X}^{\times}\) of rational functions on \(X\) with support away from \(E_X\). Lastly, one recovers the additive structure on the set of rational functions \(k_X(X)\). The problem of the exceptional set \(E_X\) is dealt with by a trick allowing one to pass to an infinite extension of \(k_X\) over which \(X\) has infinitely many points, followed by a descent argument. \bibSaidi-Tamagawa1article label=Saïdi-Tamagawa1, author=Saïdi, Mohamed, author=Tamagawa, Akio, title=A prime-to-\(p\) version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic \(p>0\), journal=Publ. Res. Inst. Math. Sci., volume=45, number=1, pages=135--186, date=2009, doi=10.2977/prims/1234361157, issn=0034-5318, review=\MR2512780, Finite ground fields in algebraic geometry, Coverings of curves, fundamental group, Arithmetic ground fields for curves, Curves over finite and local fields A prime-to-\(p\) version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic \(p>0\)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the equivalence of three ``points of view'' on the notion of a \(G\)-torsor when the base scheme is a Dedekind scheme. The proof is based on author's generalization of Chevalley's theorem on semi-invariants in [\textit{A. Borel}, Linear algebraic groups. 2nd enlarged ed. Graduate Texts in Mathematics, 126. New York etc.: Springer-Verlag. (1991; Zbl 0726.20030)] and a Tannakian description of \(G\)-torsors given [\textit{N. Saavedra Rivano}, in: Catégories Tannakiennes. Lecture Notes in Mathematics. 265. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0241.14008)] and by \textit{M. V. Nori} [Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)]. This interesting paper includes as an application that the fibered category of \(G\)-torsors on a regular proper curve over a field \(k\) is an Artin stack locally of finite presentation over \(k\). In the introduction, the author fixes notation and presents main results of the paper under review. Let \(X\) be a Dedekind scheme, \(T\) the \(X\)-scheme, \(V\) a vector bundle on \(X\), \(L \subset V\) a locally split line bundle. In this notation let \(\underline{Aut}(V,T)\) be the representable functor whose \(T\)-points are automorphisms \(f\) of \(V\otimes{\mathcal O}_T\) such that \(f(L\otimes {\mathcal O}_T) = L\otimes {\mathcal O}_T \). Let \(V\) be a representation of a flat, affine group scheme \(G\) of finite type over \(X\). Let \(\{X_1, \ldots , X_r \}\) be the (nonempty) connected components of \(X\) and \({\mathbf i}=(i_1, \ldots , i_r)\) a sequence of natural numbers. Let \(\bigwedge^{\mathbf i} V\) be the vector bundle such that \(\bigwedge^i V\| X_k = \bigwedge^{i_k} V\| X_k \) for \(k = 1, \ldots , r\) and let \( t(V)\) be some finite iteration of the operations \(\otimes, \bigwedge^i, \mathrm{Sym}^j, \bigoplus, (\cdot)^* \). The author calls such an iteration a tensorial construction. Theorem 1.1. Let \(G\) be a flat algebraic group over a Dedekind scheme \(X\). There is a representation \(V\) of \(G\), a tensorian construction \( t(V)\), and a locally split line bundle \(L \subset t(V)\) such that \(G \simeq \underline{Aut}(V,T)\). Theorem 1.2. Let \(G\) and \(X\) be as above. Let \(Y\) be a scheme faithfully flat over \(X\). There is a natural equivalence that is functorial in \(Y\) of the following groupoids: (i) The groupoid of \(G_Y\)-torsors; (ii) The groupoid of tensor functors \(F: {\mathbf{Rep}}\; G \to {\mathbf{Bun}}_Y\), that on each fiber over \(X_{\mathrm{Zar}}\) are faithful and exact. The second section is devoted to the application of Theorem 1.1 to the stack of \(G\)-torsors over a curve. Let \(k\) be a field, let \(GTor_X\) denotes the fibered category that assigns to a \(k\)-scheme \(T\) the groupoid of \(G_{X_T}\)-torsors. Theorem 2.1. The fibered category \(GTor_X\) is an Artin stack, locally of finite presentation over \(k\). In section 3 algebraic groups over Dedekind schemes are dealt with. Here, the author proves Theorem 1.1. The last section treats the Tannakian viewpoint. It begins with a treatment of the category of representations \({\mathbf{Rep}} G \) of a flat algebraic group \(G\) over a Dedekind scheme and for \(X\)-scheme \(Y\) the fibered category \({\mathbf{Bun}}_Y\), where for an object \(U\) in \(X_{\mathrm{Zar}}\), \({\mathbf{Bun}}_Y(U) = \mathbf{Bun}_{Y_U}\) is the category of all finite rank vector bundles on \( Y(U)\). Let \(P \to Y\) be a \(G_Y\)-torsor, \(F_P\) denotes the tensor functor from \({\mathbf{Rep}} G\) to \({\mathbf{Bun}}_Y\), that on each fiber over \(X_{\mathrm{Zar}}\) is faithful and exact (Lemma 4.5). The converse is true. Let \(F: {\mathbf{Rep}} G \to {\mathbf{Bun}}_Y\) be a tensor functor that on each fiber over \(X_{\mathrm{Zar}}\) is faithful and exact. The author shows that there is a natural equivalence \(F \simeq F_P\) for a uniquely defined \(G_Y\)-torsor \(P\). Following Nori and generalizing Nori`s approach to the investigated case the author of the paper under review extends (uniquely) the functor \(F\) to a tensor functor from the fibered category \({\mathbf{Rep}}^` G\) over \(X_{\mathrm{Zar}}\), where for each \(U\) in \(X_{\mathrm{Zar}}\) , \({\mathbf{Rep}}^` G(U) = {\mathbf{Rep}}_{U}^` G\) is the category of flat quasicoherent \({\mathcal O}_U \)-modules to the fibered category \({\mathbf{QCoh}}_Y\), where for each \(U\) in \(X_{\mathrm{Zar}}\), \({\mathbf{QCoh}}_Y(U) = \mathrm{QCoh}_{Y_U}\) is the category of quasicoherent \({\mathcal O}_{Y_U} \)-modules. The main theorem of the section is Theorem 4.8. It`s prove gives Theorem 1.2. Torsor; Dedekind scheme; fibered category; Artin stack; tensorial construction; groupoid; tensor functor; category of representations Broshi, M., \(G\)-torsors over a Dedekind scheme, J. Pure Appl. Algebra, 217, 11-19, (2013) Group schemes, Linear algebraic groups over finite fields \(G\)-torsors over a Dedekind scheme	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In {\S}1 the author develops a general theory of divisors and linear systems on an integral projective curve with Gorenstein singularities. A (generalized) divisor is synonymous with a fractional ideal, i.e. a subsheaf of the constant sheaf of the function field of the curve C which is coherent as \({\mathcal O}_ C\)-module. The sum of two divisors is defined only if one of them is a Cartier divisor. The notion of complete linear systems, Riemann-Roch theorem and Serre duality theorem remain valid. As an example the author gives a new proof of a theorem of Fujita which says that the canonical system \(\| K\|\) has no base points if \(p_ a(C)>0\), and is very ample if \(p_ a(C)>1\). - In {\S}2 he applies the results of {\S}1 to prove a theorem about an irreducible plane curve of degree \(k.\) If Z is a subscheme of finite length d of C, it gives a precise upper bound for \(h^ 0({\mathcal O}_ C(Z))\) in terms of k and d and tells when the bound is attained. The proof is by induction on k, and even when C is non-singular (the case Max Noether considered) one has to deal with singular curves in the course of induction, and this is the point ignored by Noether. subscheme of irreducible plane curve; linear systems; integral projective curve with Gorenstein singularities; Cartier divisor R. Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), 375--386. Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special algebraic curves and curves of low genus Generalized divisors on Gorenstein curves and a theorem of Noether	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this thesis we prove cohomological descent between the (small) étale site and the logarithmic sites for the sheaf of logarithmic differential forms. The existing techniques in the literature allow us to prove cohomological descent for a broader class of sheaves. Specifically, we compare the Kummer log étale cohomology with the étale cohomology for quasi-coherent sheaves defined on the Kummer log étale site. By using this, we prove that morphisms of log schemes where the underlying morphism of schemes is affine have trivial higher direct images for ket locally classical quasi-coherent sheaves on the Kummer log étale site. In the same spirit, we compare the log étale cohomology with the étale cohomology for classical vector bundles on the log etale site. We also give a proof for the fact that log regular log schemes are rationally singular, something which is considered known. The case of Zariski log schemes has been already treated by \textit{K. Kato} [Am. J. Math. 116, No. 5, 1073--1099 (1994; Zbl 0832.14002)]. We provide a proof for the case of étale log schemes. As a final application we define algebraic log de Rham cohomology for fs log schemes using the log étale site. Along these lines, we prove with a different method the theorem of A. Ogus, [\textit{A. Ogus}, Lectures on logarithmic algebraic geometry. Cambridge: Cambridge University Press (2018; Zbl 1437.14003), Theorem V.4.2.5(1)], regarding the generalization of algebraic de Rham cohomology to schemes with toric singularities. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Generalizations (algebraic spaces, stacks) Cohomological descent for logarithmic differential forms in the log etale topology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1993 \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] gave a ``GIT-free'' construction of the moduli space of semistable vector bundles on smooth projective curve \(X\) over an algebraically clased field \(k\). This construction was presented by him at German-Spanish Workshop on Moduli spaces of Vector Bundles, Essen, February 2007. The present notes give a self-contained and clear introduction to the moduli spaces of vector bundles on a curve and the generalized \(\Theta\)-divisor. For the sake of simplicity, all the exposition is devoted to the rank 2 case with determinant isomorphic to \(\omega_X\). Also there is a brief discussion of generalizations the arbitrary rank and degree, supplied with necessary references. Notes contain following sections. -- Outline of the construction. -- Background and notation. -- A nice over-parameterizing family. -- The generalized \(\Theta\)-divisor. -- Raynaud's vanishing result for rank two bundles. -- Semistable limits. -- Positivity. -- The construction. -- Prospect to higher dimension. moduli space; vector bundles on a curve; generalized Theta divisor Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Faltings' construction of the moduli space of vector bundles on a smooth projective curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve of genus \(g\ge 2\). For all integers \(d\) and \(r\ge 1\) let \(U(r,d)\) be the moduli space of rank \(r\) and degree \(d\) stable vector bundles on \(X\). For each positive integer \(k\) set \(B_{r,d}^k:= \{E\in U(r,d)\mid h^0(E)\ge k\}\) (the Brill-Noether loci for rank \(r\)) [\textit{I. Grzegorczyk} and \textit{M. Teixidor i Bigas}, Lond. Math. Soc. Lect. Note Ser. 359, 29--50 (2009; Zbl 1187.14038)]. The Brill-Noether loci \(B_{r,d}^k\) and their twisted version [\textit{S. Casalaina-Martin} and \textit{M. Teixidor i Biga}, Math. Nachr. 284, No. 14--15, 1846--1871 (2011; Zbl 1233.14025); \textit{G. H. Hitching} et al., ``Nonemptyness and smoothness of twisted Brill-Noether loci'', Ann. Mat. Pura ed Appl., (to appear)] are very interesting and give Torelli-type theorems. He study the tangent cones to \(B_{r,d}^k\) and give a rank \(r\) generalization of the Riemann-Kempf singularity theorem. To prove it fr any rank \(r\) vector bundle \(V\) over \(X\) the author defines a surjective rational map from the Hilbert scheme \(\mathrm{Hilb}^d(\mathbb {P}V)\) to the Quot-scheme \(\mathrm{Quot}^{0,d}(V^\ast)\). As a corollary if \(k=r\) and \(h^0(X,E) =r+n\) he proves that the \(n\)-th secant variety of the rank \(1\) locus of \(\mathbb{P}\mathrm{End}E\) is contained in the tangent cone of \(B_{r,d}^r\). vector bundles; Brill-Noether theory for vector bundles; singularities; stable vector bundles; Riemmann-Kempf singularity theorem Vector bundles on curves and their moduli, Special divisors on curves (gonality, Brill-Noether theory) A Riemann-Kempf singularity theorem for higher rank Brill-Noether loci	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the present paper, we discuss the homotopy sequences for varieties over curves. We prove that, for instance, for a morphism from a normal variety to a certain smooth curve over a field of characteristic zero, it holds that the induced outer homomorphism between the étale fundamental groups satisfies some group-theoretic conditions if and only if the homotopy sequence for the morphism of varieties is exact. Moreover, we also give a refinement of a known result concerning the Grothendieck conjecture for hyperbolic polycurves of dimension two by means of our study of homotopy sequences. Homotopy theory and fundamental groups in algebraic geometry Homotopy sequences for varieties over curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In an earlier paper [\textit{A. Moriwaki}, J. Am. Math. Soc. 11, No. 3, 569--600 (1998; Zbl 0893.14004)] the second author proved the relative Bogomolov's inequality for algebraic varieties over an algebraically closed field of characteristic zero. This capacious paper under review consisting of an introduction, 10 sections and an appendix is devoted to prove the arithmetic analogues of the mentioned result and of Cornalba-Harris-Bost's inequality [\textit{M. Cornalba} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455--475 (1988; Zbl 0674.14006)] in the case of semistable families of arithmetic varieties. Since the usual push-forward for arithmetical cycles is insufficient for the authors' purposes, they need to introduce the notion of arithmetic \(L^1\)-cycle and to extend the usual Chow groups defined by Gillet-Soulé [\textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. In section 1 of the paper the investigation of locally integrable forms is done. Section 2 is devoted to the study of the properties of three variants of arithmetic Chow groups. In section 3 the authors introduce the notion of weak positivity of arithmetic divisors. In sections 4 and 5 the authors prove the Riemann-Roch theorems for generically finite morphisms and for stable curves. Sections 6 and 7 provide respectively an asymptotic upper bound of analytic torsion and formulae for arithmetic Chern classes. The first main result of the paper consists of a relative Bogomolov inequality in the arithmetic case and appears in section 8. The second main result of the paper is a relative Cornalba-Harris-Bost inequality (theorem 10.1.4). Finally, a comparison of theorems 8.1 and 10.1.4 is given (each of them has its advantage). This is a substantial paper which contains many deep results on the subject. vector bundles; relative Bogomolov's inequality; semistable families of arithmetic varieties; arithmetic Chow group; Riemann-Roch theorems; locally integrable forms [23] Kawaguchi (S.) and Moriwaki (A.).-- Inequalities for semistable families of arithmetic varieties, J. Math. Kyoto Univ. 41 (2001), no.~1, 97-182. &MR~18 \| &Zbl~1041. Arithmetic varieties and schemes; Arakelov theory; heights, (Equivariant) Chow groups and rings; motives, Topological properties in algebraic geometry Inequalities for semistable families of arithmetic varieties.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Generalizing a construction by G. Humbert, the author constructs a family of curves of genus \(g>0\) whose Jacobian admit real multiplication by \(2\cdot\cos(2\pi/(2g+1))\). This generalized construction is used to establish the existence of a regular extension of the rational function field \({\mathbb{Q}}(T)\) having the Galois group \(PSL_ 2({\mathbb{F}}_{p^ 2})\) provided that \(p\equiv \pm 2\) mod 5. Examples of 2-parameter families of hyperelliptic curves over \({\mathbb{Q}}\) of genus \(g=7\) (resp. 19) are given whose Jacobian is isogenous over \({\mathbb{Q}}\) (resp. an extension of \({\mathbb{Q}})\) to a product of 7 (resp. 19) elliptic curves. Proofs are outlined only or omitted. complex multipliction; isogenous Jacobian; real multiplication; rational function field; hyperelliptic curves Mestre, J. F.: Courbes hyperelliptiques à multiplications réelles. C. R. Acad. sci. Paris, ser. I math. 307, 721-724 (1988) Elliptic curves, Algebraic functions and function fields in algebraic geometry, Complex multiplication and abelian varieties, Galois theory Courbes hyperelliptiques à multiplications réelles. (Hyperelliptic curves with real multiplications)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an integral projective curve defined over the complex numbers and \(D_ 1,\dots,D_ n\) mutually disjoint effective Cartier divisors on \(X\). A parabolic structure on a torsion free coherent sheaf \(E\) on \(X\) consists on a filtration of the vector space \(H^ 0(E\mid D_ j)\) for each \(j=1,\dots,n\), and on some weights associated to these filtrations allowing one to define a parabolic sheaf and the notion of (semi-)stable generalized parabolic sheaf. This technical notion occurs naturally when one studies vector bundles on singular curves by pulling them back to the desingularization of the curve. The author proves existence and studies the moduli spaces of semi-stable generalized parabolic sheaves of a certain type (for which the filtrations have only 2 steps). As an application, she obtains partial desingularizations of moduli spaces of torsion free sheaves on a nodal curve. effective Cartier divisors on integral projective curve; vector bundles on singular curves; desingularization; moduli spaces of semi-stale generalized parabolic sheaves Vector bundles on curves and their moduli, Singularities of curves, local rings, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) Generalized parabolic sheaves on an integral projective curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a translation of the book ``Le problème des modules pour les branches planes'' of \textit{O. Zariski} [Hermann, Paris (1973; Zbl 0317.14004)] by Ben Lichtin. It is based on notes from a course of Zariski at Centre de Mathématiques de l'École Polytechnique 1973 and contains an appendix by B. Teissier considering the moduli problem from the point of view of deformation theory. Zariski's aim is to study the space of isormorphism classes of plane curve singularities (analytically irreducible curve germs) of given equsingularity type, i.e. the moduli space \(M(\Gamma)\) of plane curve singularities with fixed semigroup \(\Gamma\). His ideas and results were the basis for further research in this direction [cf. for example \textit{O. A. Laudal} and the reviewer, ``Local moduli and singularities'', Lect. Notes Math. 1310 (1988; Zbl 0657.14005)]. The first three chapters of the book introduce the basic notions, especially invariants as the semigroup, the conductor, the characteristic of a branch, short parametrizations, etc. Then the moduli space \(M(\Gamma)\) is studied. It is proved that \(M(\Gamma)\) is not seperated in general. The structure of \(M(\Gamma)\) is analyzed for special examples. The dimension of the generic component of the moduli space for \(\Gamma=\langle n, n+1\rangle\) is computed. This was generalized later on by C. Delorme. The appendix of Teissier is based on the idea that each plane curve singularity with semigroup \(\Gamma\) appears as a deformation of the associated monomial curve, more precise in its negatively weighted part. It turns out that every plane branch has a miniversal equisingular deformation, over a smooth base, with an equisingular section. Among several results a ``natural'' compactification of \(M[\Gamma)\) is given, as well as an interpretation of the generic component. moduli problem; plane curve singularity; semi group; deformation Zariski, O., \textit{The Moduli Problem for Plane Branches (with an Appendix by Bernard Teissier)}, 39, (2006), AMS, Providence RI Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry The moduli problem for plane branches. With an appendix by Bernard Teissier. Transl. from the French by Ben Lichtin	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book gives a full account of the theory of moduli for elliptic curves order \({\mathbb{Z}}\) from a scheme-theoretical point of view. In order to give representable moduli problems elliptic curves need to be rigidified by introducing supplementary structures not preserved by the automorphism group. The \(\Gamma\) (N)-problem, or full level-N problem, is determined by a choice of generators of the group of division points with order dividing N. For elliptic curves over an arbitrary base ring this has to be stated in terms of subgroup schemes exactly generated by two divisors. There are similar statements for the \(\Gamma_ 0(N)-\), \(\Gamma_ 1(N)\)-problems and their variants. The authors introduce functorial machinery to treat these problems in a uniform way and avoid algebraic stacks by the device of relative representability. The first four chapters give a self-contained account of the elementary theory of elliptic curves over a scheme, their finite subgroup and moduli problems. The reader is referred to other sources for deeper results such as the Serre-Tate theorem on deformations and the construction of Tate curves. The subsequent chapters deal with the intricate details involved in completing the moduli scheme at infinity and describing its structure at points corresponding to super-singular curves. In particular two chapters cover problems specific to characteristic p and Igusa curves. The culminating result is a general good-reduction theorem for Jacobians. Altogether this book is an essential complement to the many works on the transcendental theory of elliptic curves. full level-N problem; elliptic curves; moduli scheme at infinity; characteristic p; Igusa curves; good-reduction theorem for Jacobians 26. Katz, Nicholas M. and Mazur, Barry \textit{Arithmetic moduli of elliptic curves}Annals of Math. Studies, vol. 108, Princeton Univ. Press, Princeton, NJ, 1985 Math Reviews MR772569 Families, moduli of curves (algebraic), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fine and coarse moduli spaces, Elliptic curves, Algebraic moduli problems, moduli of vector bundles, Modular and automorphic functions, Special algebraic curves and curves of low genus Arithmetic moduli of elliptic curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this third lecture of these proceedings the author introduces the Picard functor of a morphism \(f:X\to Y\) of schemes and discusses the usual representability criteria due to Grothendieck, Serre, Murre, Oort and others. The case of the Picard functor of a proper flat curve \(f:X\to S\), locally of finite presentation, is discussed in details: its neutral component, its unipotent part and its torical and abelian component. Picard functor; Picard scheme; relative effective divisors; flat curve Picard groups, Arithmetic ground fields for curves, Local ground fields in algebraic geometry The Picard functor	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $\mathcal{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points. The Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the geometric fundamental groups of the ``Teichmüller tower'' of $\{ \mathcal{M}_{g,n} \}$ (i.e. the collection of all the stacks and the natural maps between them). Ihara showed that this action extends to a faithful action of the profinite Grothendieck-Teichmüller group $\widehat{\mathrm{GT}}$, which contains $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. \par The authors of the paper under review prove that $\widehat{\mathrm{GT}}$ is actually equal to the group of homotopy automorphisms of the Teichmüller tower in genus $g = 0$. They use an operadic model $\mathcal{M}$ of the genus $0$ Teichmüller tower: they replace marked point by boundary circles, which allows to glue curves along boundary components and to fill boundary components. These operations yield maps between the corresponding moduli spaces, and the operad obtained this way is equivalent to the classical framed little disks operad. The authors' first main theorem states more precisely that $\widehat{\mathrm{GT}}$ is the group of homotopy automorphisms of the profinite completion of $\mathcal{M}$. The difficult part of this theorem is showing that the obvious faithful action of $\widehat{\mathrm{GT}}$ on each $\mathcal{M}(n)$ is compatible with the operad structure (which is related to [\textit{A. Hatcher} et al., J. Reine Angew. Math. 521, 1--24 (2000; Zbl 0953.20030)]) and that all homotopy automorphisms are obtained this way. \par The authors' second main theorem deals with the compactification $\overline{\mathcal{M}}_{g,n}$ of $\mathcal{M}_{g,n}$, obtained by allowing nodal singularities in the curves. These moduli spaces admit a natural operadic structure by gluing curves along marked points. The operad $\mathcal{M}$ maps to $\overline{\mathcal{M}}_{0,\bullet+1}$ (the latter being a homotopy quotient of the former under an $\mathrm{SO}(2)$-action [\textit{G. C. Drummond-Cole}, J. Topol. 7, No. 3, 641--676 (2014; Zbl 1301.55005)]). The authors prove that the action of $\widehat{\mathrm{GT}}$ on the profinite completion of $\mathcal{M}$ extends to a nontrivial action on the profinite completion of $\overline{\mathcal{M}}_{0,\bullet+1}$. Note that actions of $\widehat{\mathrm{GT}}$ are usually constructed on schemes whose associated complex analytic spaces are $K(\pi,1)$, whereas $\overline{\mathcal{M}}_{0,n+1}$ is simply connected. infinity operads; Grothendieck-Teichmüller group; absolute Galois group; moduli space of curves , Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Loop space machines and operads in algebraic topology, Abstract and axiomatic homotopy theory in algebraic topology Operads of genus zero curves and the Grothendieck-Teichmüller group	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Grothendieck-Riemann-Roch theorem plays an essential role in the study of algebraic geometry. Roughly speaking, it describes how far the Chern character map between the algebraic \(K\)-theory and the algebraic Chow-theory to be commutative with the push-forward operations w.r.t. proper morphisms. This theorem vastly generalizes the classical Riemann-Roch theorem on compact Riemann surfaces, and hence it can be viewed as an Atiyah-Singer type index theorem. In previous work, the Grothendieck-Riemann-Roch theorem was extended by several mathematicians to an arithmetic setting in the sense of Arakelov. In this situation, the generators of algebraic \(K\)-groups and algebraic Chow groups are equipped with additional Hermitian structures so that the fibers of an arithmetic variety over the infinite places can be treated as the same footing as the finite ones. Historically, \textit{H. Gillet} and \textit{C. Soulé} [Invent. Math. 110, No. 3, 473--543 (1992; Zbl 0777.14008)] proved a degree one version of the arithmetic Grothendieck-Riemann-Roch theorem, they used the theory of Ray-Singer analytic torsion to define a reasonable push-forward on the level of arithmetic \(K\)-groups. Whence \textit{J.-M. Bismut} and \textit{K. Köhler} [J. Algebr. Geom. 1, No. 4, 647--684 (1992; Zbl 0784.32023)] developed the analytic torsion theory to higher degree case, a complete proof of the arithmetic Grothendieck-Riemann-Roch was given by Gillet, Rössler and Soulé [\textit{H. Gillet} et al., Ann. Inst. Fourier 58, No. 6, 2169--2189 (2008; Zbl 1152.14023)]. But, due to technical reason, these two works had to restrict to push-forward by morphisms which are smooth over generic fibers. The aim of the article under review is to break this limit, namely is to generalize the works of Gillet-Soulé and Gillet-Rössler-Soulé to arbitrary projective morphisms, not necessarily smooth over the generic fibers. This forces the authors to enlarge the arithmetic \(K\)-groups and the arithmetic Chow groups to afford corresponding push-forward functorialities. The strategy the authors followed was replacing smooth differential forms in the theory of Gillet-Soulé by currents with possibly non-empty wave front sets. As a consequence, by explaining the relationship between the topological correction term and the generalized analytic torsion classes, the authors obtained all the possible forms of the arithmetic Grothendieck-Riemann-Roch theorem. Arakelov theory; Grothendieck-Riemann-Roch theorem; projective morphism Burgos, J. I.; Freixas, G.; Litcanu, R., The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms, Ann. Fac. Sci. Toulouse Math. (6), 23, 3, 513-559, (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Riemann-Roch theorems The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A classical theorem of \textit{K. Brauner} [Abh. Hamburg 6, 1--55 (1928; JFM 54.0373.01)] gives a formula for the generators and relations of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular curve in \(\mathbb C^2\). These elegant formulas depend only on the characteristic pairs of a Puiseux series expansion of the curve. Zariski states this theorem as part of his discussion on Puiseux series in the chapter on resolution of singularities in his book ``Algebraic surfaces''. Berlin: Springer (1935; Zbl 0010.37103)]. The arithmetic analogue of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular plane curve is thus the algebraic fundamental group \(\pi_1(\text{Spec}(R)-V(f))\), where \(R=k[[x,y]]\) is a power series ring over an algebraically closed field \(k\), and \(f\in R\) is irreducible. In positive characteristic, Puiseux series expansions do not always exist. In this paper we prove an arithmetic analogue of Brauner's theorem, valid in arbitrary characteristic. The generators and relations in our theorem for the prime to \(p\) part of the algebraic fundamental group coincide with those of Brauner's theorem. Main theorem. Let \(R=k[[x,y]]\) be a power series ring over an algebraically closed field \(k\) of characteristic \(p\geq 0\). Suppose that \(f\in R\) is irreducible. Let \(U=\text{Spec}(R_f)= \text{Spec}(R)-V(f)\). Let \((m_i,n_i)\), \(1\leq i\leq g\), be the characteristic pairs of \(f\). Then \(\pi_1^{(p)}(U)\) is isomorphic to the prime to \(p\) part of the pro-finite completion of the free group on the symbols \(Q_0,\dots,Q_g,P_1, \dots,P_g\) with the relations \[ Q_i^{m_i}= P_i^{\overline n_i} Q_{i-1}^{m_{i-1}m_i},\;1\leq i\leq g,\quad Q_0=1, \] \[ P_{i+1} P_i^{y_i}Q_{i-1}^{m_{i-1} x_i}=Q_i^{x_i},\;1\leq i\leq g-1, \] \(x_i\) and \(y_i\) are integers such that \(x_i\overline n_i=y_im_i+1\), \(\overline n_i=n_i-n_{i-1}m_i\). The prime to \(p\) part of a fundamental group \(\pi_1\) is the quotient \(\pi_1^{(p)}\) of \(\pi_1\) by the closed normal subgroup generated by its \(p\)-Sylow subgroups. Let \(C=V(f)\), \(m\) be the maximal ideal of \(R\). The basic strategy of the proof is to construct an embedded resolution of singularities \(\tau:X\to\text{Spec}(R)\) of \(C\) such that the preimage of \(C\) is a simple normal crossing divisor. Then Abhyankar's lemma [\textit{S. Abhyankar}, Am. J. Math. 81, 46--94 (1959; Zbl 0100.16401)] shows that \(\pi_1^{(p)} (\text{Spec} (\widehat{\mathcal O}_{X,x})- \tau^{-1}(C))\) has an extremely simple form for any \(x\in\tau^{-1}(m)\). We further compute the prime to \(p\) part of the algebraic fundamental group of the complement of \(\tau^{-1}(C)\) in the formal completion of \(X\) along each of the exceptional curves of \(X\to\text{Spec}(R)\). curve singularity; algebraic fundamental group; arbitrary characteristic; power series ring Cutkosky, S. D.; Srinivasan, H.: The algebraic fundamental group of a curve singularity. J. algebra 230, 101-126 (2000) Coverings of curves, fundamental group, Singularities of curves, local rings, Formal power series rings, Algebraic functions and function fields in algebraic geometry, Germs of analytic sets, local parametrization, Homotopy theory and fundamental groups in algebraic geometry The algebraic fundamental group of a curve singularity	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The starting point for the paper under review is a series of (mostly unpublished) investigations by Faber and Zagier into relations between tautological classes on the moduli space of curves. Faber and Zagier empirically discovered an infinite sequence of polynomial expressions in the tautological classes on \(M_g\), governed by two curious power series, which as far as they could check on computer would all vanish in the tautological ring. They conjectured that this would indeed always be the case, which was later proved by \textit{R. Pandharipande} and \textit{A. Pixton} [``Relations in the tautological ring'', \url{arXiv:1101.2236}]. Later, Pixton ingeniously wrote down an extension of their formulas to the spaces \(M_{g,n}\), and then to the Deligne-Mumford boundary \(\overline{M}_{g,n}\). Pixton conjectured that not only should these extended formulas also always produce relations; moreover, all tautological relations on all spaces \(\overline{M}_{g,n}\) should arise in this way. We remark that this conjecture contradicts the Faber conjecture on \(M_g\), so Faber and Zagier did not expect that all relations would arise from this construction. The main result of this paper is that Pixton's extended Faber-Zagier relations are indeed true in cohomology for all \(g\) and \(n\). After this paper was written, \textit{F. Janda} [``Tautological relations in moduli spaces of weighted pointed curves'', \url{arXiv:1306.6580}] proved that these relations are in fact true also in the Chow rings. The main tool used here is the Givental-Teleman classification of semisimple CohFTs. This is the reason their proof only works in cohomology -- the Givental-Teleman classification is only known in cohomology, due to the fact that it uses the Madsen-Weiss theorem. We remark that the Madsen-Weiss theorem, hence also the Givental-Teleman classfication, is expected to remain valid also in Chow. In any case, the authors consider the behavior of Witten's 3-spin CohFT as they move towards a non-semisimple point of the Frobenius manifold, in terms of the action of the R-matrix. Applying the R-matrix gives what appears to be a divergent expression. Knowing that this expression must in fact converge, it follows that all coefficients before negative exponents are actually zero, which produces relations between tautological classes. The relations they obtain in this way are exactly Pixton's relations. \textit{F. Janda} [``Relations in the Tautological Ring and Frobenius Manifolds near the Discriminant'', \url{arXiv:1505.03419}] has recently proved a striking universality result: all relations obtained by this method, for \textit{any} generically semisimple CohFT, follow from the 3-spin relations (i.e. Pixton's). moduli of curves; tautological rings; cohomological field theories R. Pandharipande, A. Pixton, and D. Zvonkine, Relations of \(\overline{M}_{g,n}\) via \(3\)-spin structures, J. Amer. Math. Soc. 28 (2015), 279--309. Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Relations on \(\overline{\mathcal{M}}_{g,n}\) via \(3\)-spin structures	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Grothendieck abelian categories play the role of models of possibly noncommutative schemes [\textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002); \textit{J. T. Stafford} and \textit{M. Van den Bergh}, Bull. Am. Math. Soc., New Ser. 38, No. 2, 171--216 (2001; Zbl 1042.16016; \textit{M. Kontsevich} and \textit{A. L. Rosenberg}, in: The Gelfand Mathematical Seminars, 1996--1999. Dedicated to the memory of Chih-Han Sah. Boston, MA: Birkhäuser. 85--108 (2000; Zbl 1003.14001)], which is motivated by the Gabriel-Rosenberg reconstruction theorem claiming that a quasi-separated scheme can be reconstructed, up to isomorphism of schemes, solely from the abelian category of quasi-coherent sheaves on the scheme, which is a Grothendieck category [\url{https://stacks.math.columbia.edu/}]. The theorem was initially established for noetherian schemes by \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323--448 (1962; Zbl 0201.35602)] and generalized to quasi-separated schemes by \textit{A. Rosenberg} [\url{https://archive.mpim-bonn.mpg.de/id/eprint/1516/}]. The Gabriel-Popescu theorem [\textit{N. Popesco} and \textit{P. Gabriel}, C. R. Acad. Sci., Paris 258, 4188--4190 (1964; Zbl 0126.03304)] allows of interpreting Grothendieck categories as a linear version of Grothendieck topoi [\textit{W. Lowen}, J. Pure Appl. Algebra 190, No. 1--3, 197--211 (2004; Zbl 1051.18007)]. Different \(2\)-categories obtain, depending on the choice of morphisms. \begin{itemize} \item[\(\mathsf{Grt}\)] the \(2\)-category of Grothendieck categories and left adjoints as morphisms \item[\(\mathsf{Grt}_{\flat}\)] the \(2\)-category of Grothendieck categories and left exact left adjoints as morphisms \end{itemize} The \(2\)-category \(\mathsf{Grt}_{\flat}\)\ is the main object of study in this paper. It is shown that \(\mathsf{Grt}_{\flat}\)\ can be endowed with a monoidal structure, where the exponentiable objects are characterized. From an algebro-geometric standpoint, this can be seen as a contribution to the understanding of exponentiable schemes or Hom-schemes when restricted to the flat case. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] aims to \(\mathsf{Grt}_{\flat}\)\ can simulate flat algebraic geometry via a collection of examples. \item[\S 3] shows that the monoidal structure \(\boxtimes\)\ on \(\mathsf{Grt} \)\ [\textit{W. Lowen} et al., Int. Math. Res. Not. 2018, No. 21, 6698--6736 (2018; Zbl 1408.18024)] nicely restricts to \(\mathsf{Grt}_{\flat}\), which is easy on the level of objects, but a highly non-trivial task on the level of morphisms. The problem of exponentiability is introduced. It is shown that the category of linear presheaves \(\mathrm{Mod}\left( \mathfrak{a}\right) \)\ are exponentiable (Proposition 3.15). \item[\S 4] investigates the properties of the forgetful functor \[ :\mathsf{Grt}_{\flat}^{\circ}\rightarrow\mathsf{Cat}_{k} \] showing that it is representable (Prposition 4.2). \item[\S 5] introduces and investigates quasi-injective Grothendieck categories (\S 5.1), continuous linear categories (\S 5.2) and then connect the two concepts (\S 5.3). These are technical tools for the main theorem. \item[\S 6] gives the main theorem: Theorem 6.1. A Grothendieck category is exponentiable in \(\mathsf{Grt}_{\flat}\) iff it is continuous. In particular, every finitely presentable Grothendieck category is exponentiable. \item[\S 7] is a collecction of examples and instances of the main theorem. The most relevant is the following: Proposition 7.2. Let \(X\) be a quasi-compact quasi-separated scheme over \(k\), then \(\mathsf{Qcoh}\left( X\right) \)\ is exponentiable. \end{itemize} Grothendieck categories; noncommutative algebraic geometry; flatness; monoidal structures; exponentiability; continuous categories; quasi-coherent sheaves Noncommutative algebraic geometry, Sheaves in algebraic geometry, Topoi, Accessible and locally presentable categories, Grothendieck topologies and Grothendieck topoi, Monoidal categories, symmetric monoidal categories, Abelian categories, Grothendieck categories Exponentiable Grothendieck categories in flat algebraic geometry	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y\) be an integral Noetherian scheme. Let \(f : X \rightarrow Y\) be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension \(d\) and with geometrically connected fibres. Assume that \(X\) is reduced. Let \(\omega ^d_{X/Y}\) be the sheaf of relative regular differential forms. The aim of the paper is to study when the direct images of this sheaf are torsion-free. The question being local, it is assumed that \(Y = \text{Spec } D\) for some complete discrete valuation ring \(D\) with residue class field \(k\) algebraically closed and \(X = \text{Proj } S\) for some graded reduced \(D\)-algebra \(S\). The authors give a set of equivalent conditions for \(R^if_\omega^d_{X/Y}\) to be torsion-free. Using them the following results are proved. (1) Suppose that \(X/Y\) is arithmetically \(S_{k+2}\) (Serre's condition). Then \(R^{d-i}f_\omega^d_{X/Y}\) is torsion-free and commutes with base change for all \(i\leq k\). (2) If \(R^{d-i}f_\omega^d_{X/Y}\) is torsion-free for all \(i\leq k\) then \(R^if_{\mathcal O}_X\) is torsion-free for all \(i\leq k\). (3) Suppose that \(X/Y\) is globally a complete intersection. Then the sheaf \(R^if_\omega^d_{X/Y}\) is torsion-free for all \(i\). (4) If \(X\) is a surface (i.e. \(d=1)\) then \(R^1f_\omega ^1_{X/Y}\) is torsion-free if and only if there exists a graded Cohen-Macaulay algebra \(S/D\) such that \(X=\text{Proj }S\). Specialise now to the case of arithmetic surfaces. Thus \(D\) is further assumed to be a discrete valuation ring with quotient field of characteristic zero and residue field of positive characteristic, \(X\) is a regular scheme of dimension \(2\) and \(f\) is proper, flat and with connected fibres. Let \(X_y\) denote the special fibre of \(f\). Write \(X_y = \sum n_i F_i\), \(F_i\) being reduced and irreducible curves, \(n = \text{g.c.d.}(n_1,\cdots ,n_m)\). Theorem. With the above notations assume that \(p \nmid n\). (1) Then \(h^0(X,{\mathcal O}_{X_y}) = 1,\) and \(R^1f_\omega ^1_{X/Y}\), \(R^1f_{\mathcal O}_X\) are torsion-free. (2) Assume that \(X/Y\) is projective. Then there exists a graded Cohen- Macaulay algebra \(S/D\) such that \(X =\text{Proj} (S)\). Finally the authors examine the effects of base extension on \(R^1f_{\mathcal O}_X\) and \(R^1f_\omega ^1_{X/Y}\). \textit{J. Kollár} [Ann. Math., II. Ser. 123, 11-42 (1986; Zbl 0598.14015)] had shown that if \(X\) is a complex projective manifold and \(Y\) is a reduced variety then \(R^if_*\omega _X\) is torsion-free for all \(i\). relative regular differential forms; torsion freeness of direct image of sheaf; dualizing sheaves; arithmetic surfaces DOI: 10.1090/S0002-9939-98-04499-2 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Modules of differentials On the cohomology of regular differential forms and dualizing sheaves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In these notes we give a reasonably self contained proof of three of the main theorems of the Diophantine geometry of curves over function fields of characteristic zero. Let \(F\) be a function field of dimension one over the field of the complex numbers \(\mathbb{C}\) i.e. a field of transcendence degree one over \(\mathbb{C}\). Let \(X_F\) be a smooth projective curve over \(F\). We prove that: \begin{itemize} \item[--] If the genus of \(X_F\) is zero then it is isomorphic, over \(F\), to the projective line \(\mathbb{P}^1\). \item [--] If the genus of \(X_F\) is one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil). \item[--] If the genus of \(X_F\) is strictly bigger than one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) is finite (former Mordell Conjecture). \end{itemize} The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [\textit{R. Hartshorne}, ``Algebraic geometry'', Graduate Texts in Mathematics, 52, 496 p. (1977; Zbl 0367.14001)] or \textit{P. Griffiths} and \textit{J. Harris} [Principles of algebraic geometry. New York, NY: John Wiley \& Sons Ltd. (1994; Zbl 0836.14001)]. arithmetic over function fields; arithmetic of algebraic curves; Mordell Weil theorem; Mordell conjecture Rational points, Global ground fields in algebraic geometry, Arithmetic ground fields for curves, Heights, Elliptic curves over global fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic theory of algebraic function fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Diophantine geometry on curves over function fields	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A result of André Weil allows one to describe rank \(n\) vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set \(\mathrm{GL}_{n}(\mathbb{A})\) of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to \(G\)-torsors for a reductive algebraic group \(G\)). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson's co-simplicial ring of adèles \(\mathbb{A}_X^\bullet \), we have an equivalence \(\mathsf{Perf}(X)\simeq\|\mathsf{Perf}(\mathbb{A}_X^\bullet)\|\) between perfect complexes on \(X\) and cartesian perfect complexes for \(\mathbb{A}_X^\bullet\). Using the Tannakian formalism for symmetric monoidal \(\infty\)-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand-Naimark's reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest. adèles; vector bundles; perfect complexes; moduli stacks Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Stacks and moduli problems Adelic descent theory	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to present an algebro-geometric approach to the study of the geometry of the moduli space of stable bundles on a smooth projective curve defined over an algebraic closed field \(k\), of arbitrary characteristic. One of the basic ideas is to consider a notion of divisor of higher rank and a suitable Abel-Jacobi map generalizing the classical notions in rank one. Let \({\mathcal O}_C\) be the structure sheaf of the curve \(C\) and let \(K\) be its field of rational functions, considered as a constant \({\mathcal O}_C\)-module. We define a divisor of rank \(r\) and degree \(n\), an \((r,n)\)-divisor for short, to be any coherent sub-\({\mathcal O}_C\)-module of \(K^r= K^{\oplus r}\) having rank \(r\) and degree \(n\). Since \(C\) is smooth, these submodules are locally free and coincide with the matrix divisors defined by \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)]. Denote by \(\text{Div}^{r,n}_{C/k}\) the set of all \((r,n)\)-divisors. This set can be identified with the set of rational points of an algebraic ind-variety \({\mathcal D} iv^{r,n}_{C/k}\) that may be described as follows. For any effective ordinary divisor \(D\) set \(\text{Div}^{r,n}_{C/k} (D)= \{E\in \text{Div}^{r,n}_{C/k}\mid E\subseteq {\mathcal O}_C (D)^r\}\) where \({\mathcal O}_C (D)^r\) is considered as a sub \({\mathcal O}_C\)-module of \(K^r\). The elements of the set \(\text{Div}^{r,n}_{C/k} (D)\) can be identified with the rational points of the scheme \(\text{Quot}^m_{{\mathcal O}_C (D)^r/X/k}\), \(m=r\cdot \deg D-n\), parametrizing torsion quotients of \({\mathcal O}_C (D)^r\) having degree \(m\). It is natural to stratify the ind-variety \({\mathcal D} iv^{r,n}_{C/k}\) according to Harder-Narasimhan type \({\mathcal D} iv^{r,n}_{C/k}= ({\mathcal D} iv^{r,n}_{C/k} )^{ss} \cup \bigcup_{P\neq ss} {\mathcal S}_P\) where \(({\mathcal D} iv^{r,n}_{C/k} )^{ss}\) is the open ind-subvariety of semistable divisors. The cohomology of each stratum stabilizes and this stratification is perfect. (Here cohomology means \(\ell\)-adic cohomology for a suitable prime \(\ell\).) In particular, there is an identity of Poincaré series \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= P\bigl( ({\mathcal D} iv^{r,n}_{C/k} )^{ss}; t\bigr)+ \sum_{P\neq ss} P({\mathcal S}_P, t)\cdot t^{2d_r}, \] where \(d_P\) is the codimension of \({\mathcal S}_P\). Let \(r\) and \(n\) be coprime. Then the notations of stable and semistable bundle over \(C\) coincide, and the moduli space \(N(r,n)\) of stable vector bundles having rank \(r\) and degree \(n\) is in this case a smooth projective algebraic variety. It is natural to define, by analogy with the classical case, Abel-Jacobi maps \(\vartheta: ({\mathcal D} iv^{r,n}_{C/k} )^{ss}\to N(r,n)\), by assigning to a divisor \(E\) its isomorphism class as a vector bundle. In order to find the Betti numbers of \(N(r,n)\) it suffices to know those of \(({\mathcal D} iv^{r,n}_{C/k} )^{ss}\). This computation reduces, for arbitrary \(r\) and \(n\), to the calculation of those of \({\mathcal D} iv^{r,n}_{C/k}\). The varieties \(\text{Div}^{r,n}_{C/k} (D)\) are analogous to Grassmannians and share with them the property of having a decomposition into Schubert ``strata''. One obtains \[ P({\mathcal D} iv^{r,n}_{C/k}; t)= {{\prod^r_{j=1} (1+t^{2j-1} )^{2g}} \over {(1-t^{2r}) \prod^{r-1}_{j=1} (1- t^{2j})^2}}. \] moduli space of stable bundles; divisor of rank \(r\) and degree \(n\); algebraic ind-variety; semistable divisors; stratification; Poincaré series; Abel-Jacobi maps \beginbarticle \bauthor\binitsE. \bsnmBifet, \bauthor\binitsF. \bsnmGhione and \bauthor\binitsM. \bsnmLetizia, \batitleOn the Abel-Jacobi map for divisors of higher rank on a curve, \bjtitleMath. Ann. \bvolume299 (\byear1994), page 641-\blpage672. \endbarticle \OrigBibText \beginbarticle \bauthor\binitsE. \bsnmBifet, \bauthor\binitsF. \bsnmGhione and \bauthor\binitsM. \bsnmLetizia, \batitleOn the Abel-Jacobi map for divisors of higher rank on a curve, \bjtitleMath. Ann. \bvolume299 (\byear1994), page 641-\blpage672. \endbarticle \DOI10.1007/BF01459804 \endOrigBibText \bptokaddids, structpyb \endbibitem Mathematical Reviews (MathSciNet): Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Abel-Jacobi map for divisors of higher rank on a curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0747.00028.] The purpose of this paper is to determine explicitly the local structure of the Hilbert scheme of curves in \(\mathbb{P}^ 3\) at certain points. Actually, the main body of this work is devoted to a quite detailed exposition of the various deformation theories of a closed subscheme \(Y\) of a projective scheme \(X = \text{Proj}(S)\), where \(K\) is a field and \(S\) is a graded Noetherian \(K\)-algebra, with \(S_ 0 = K\). The types of deformations are: the deformations of \(Y\) as a subscheme of \(X\), the deformations of the ideal sheaf \({\mathcal I}_ Y\) as a sheaf of \({\mathcal O}_ X\)-modules, the deformations of the ideal \(I(Y)\) as a homogeneous \(S\)-module, and ``conical deformations'' of \(Y\). This exposition builds on previous work by \textit{O. A. Laudal} [in Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 218-240 (1986; Zbl 0597.14010)] and \textit{J. O. Kleppe} [Math. Scand. 45, 205-231 (1979; Zbl 0436.14004)], but the material is presented here with a view towards explicit calculations. Various conditions are worked out under which these deformation theories are isomorphic. The theory is then applied to the calculation of the deformation theory of particular space curves, yielding explicit local equations for some singular points of the Hilbert scheme and explicit examples of obstructed curves (i.e. corresponding to singular points of the Hilbert scheme) of maximal rank. The same examples were found independently by \textit{Bolondi}, \textit{Kleppe} and \textit{Miró-Roig}. The question remains whether curves with seminatural cohomology (i.e. curves \(C\) such that for each \(n\), the sheaf \({\mathcal J}_ C(n)\) has at most one nonzero cohomology group) are unobstructed. isomorphic deformation theories; deformation theory of space curves; Hilbert scheme of curves; deformation theories of a closed subscheme Charles H. Walter, Some examples of obstructed curves in \?³, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324 -- 340. Plane and space curves, Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory Some examples of obstructed curves in \(\mathbb{P}^ 3\)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first goal of this paper is to generalize in local complex geometry the main idea in \textit{W. L. Chow} and \textit{B. L. van der Waerden}' construction [Math. Ann. 113, 692-704 (1937; Zbl 0016.04004)]: associating the ``incidence divisor'' of a cycle with a given analytic family of cycles (the linear projective varieties of suitable dimension in the case of \(\mathbb{P}_N (\mathbb{C}))\). Under very general hypotheses we associate a Cartier divisor on the parameter space in which arbitrary singularities are allowed to a local complete intersection. The second goal is to study the singularities of functions obtained by integration of meromorphic cohomology classes on an analytic family of cycles. Our result enables us to control the pole order of the meromorphic functions on the parameter space from the pole order of the given cohomology class. local complex geometry; Cartier divisor; local complete intersection; singularities of functions; integration of meromorphic cohomology classes D. Barlet - J. Magnusson , Integration de classes de cohomologie méromorphes et diviseur d'incidence , Ann. Sci. École Norm. Sup. ( 4 ) 31 ( 1998 ), 811 - 842 . Numdam \| MR 1664218 \| Zbl 0963.32019 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Complete intersections Integration of meromorphic cohomology classes and incidence divisors	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a Weil cohomology theory for smooth proper varieties over a field. Grothendieck's ``conjecture C-plus'' asserts the Künneth projectors are algebraic. His ``conjecture D'' asserts that in the group of algebraic cycles numerical equivalence agrees with homological equivalence. Differential graded categories are regarded as ``noncommutative schemes'' by the specialists; the differential graded category associated with an ordinary scheme being the category of perfect complexes. In the noncommutative world, noncommutative analogues of Grothendieck's conjectures have been formulated. Let \(k\) be a perfect field of characteristic \(p\). In the article under review, the author proves that these noncommutative conjectures (for ``smooth proper \(k\)-linear dg categories'') are equivalent to the corresponding commutative conjectures (for smooth proper \(k\)-varieties). Here, one uses the rational crystalline cohomology in the conjectures C-plus and D, and the topological periodic cyclic homology in the noncommutative versions of these conjectures. The author gives some applications. From the abstract: ``As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of `low-dimensional' orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot's cohomological interpretation of the classical zeta function and of Grothendieck's conditional approach to 'half' of the Riemann hypothesis.'' Reviewer's remark: There is a citation with broken hyperlink ``[?book]'' on p. 5052 in the published version of the article. It should point to the author's book [Noncommutative motives. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1333.14002)]. noncommutative algebraic geometry; motive; crystalline cohomology Noncommutative algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, \(K\)-theory and homology; cyclic homology and cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Chain complexes (category-theoretic aspects), dg categories A note on Grothendieck's standard conjectures of type \(\mathrm{C}^+\) and \(\mathrm{D}\) in positive characteristic	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The notion of a ``Galois point'' of an algebraic curve was introduced by Yoshihara in 1996 and can be described as follows. Let \(C\) be an irreducible smooth projective curve defined over an algebraically closed field \(K\) of characteristic \(p \geq 0\). Let \(K(C)\) be the corresponding function field. Let \(P\) be a point in \(\mathbb{P}^2\) and \(\varphi\) a rational map for \(C\) to \(\mathbb{P}^2\) being birational onto its image. A point \(P \in \mathbb{P}^2\) is a Galois point for \(\varphi(C)\) if the function field extension \(K(\varphi(C))/{\pi_P}^*K(\mathbb{P}^1)\) induced by the projection \(\pi_P\) is Galois. If \(P\) is a smooth point of \(\varphi(C)\) (resp. contained in \(\mathbb{P}^2 \setminus \varphi(C)\)) then it is said to be inner (resp. outer). If \(P\) is a Galois point, then the corresponding Galois group will be denoted with \(G_P\). One of the most interesting problems in studying Galois points is to determine the number of Galois points that a plane curve can have. For smooth plane curves, this number was determined by \textit{H. Yoshihara} [J. Algebra 239, No. 1, 340--355 (2001; Zbl 1064.14023)] and \textit{S. Fukasawa} [Rend. Semin. Mat. Univ. Padova 129, 93--113 (2013; Zbl 1273.14066)]. Not so many examples of singular plane curves \(C\) having two Galois points are known, but following the works of Yoshihara, Takahashi and Terasoma characterizations are known for \(C\) to have \(2\) Galois points. In this paper applications of these results for rational and elliptic curves are presented. More precisely it is shown that if \(p \neq 2\), then there exist a morphism \(\varphi: \mathbb{P}^1 \rightarrow \mathbb{P}^2\) which is biratonal onto its image, such that {\parindent=0.7cm \begin{itemize}\item[--] \(\deg(\varphi(C))=5\), \(\varphi(C)\) admits two Galois points \(\varphi(P_1)\) and \(\varphi(P_2)\) such that \(G_{\varphi(P_i)}\) is cyclic of order \(4\), for \(i=1,2\) and \(p \neq 3\); \item[--] \(\deg(\varphi(C))=5\), \(\varphi(C)\) admits two Galois points \(\varphi(P_1)\) and \(\varphi(P_2)\) such that \(G_{\varphi(P_i)} \cong (\mathbb{Z}/4\mathbb{Z})^{\oplus 2}\), for \(i=1,2\); \item[--]\(\deg(\varphi(C))=5\), \(\varphi(C)\) admits two Galois points \(\varphi(P_1)\) and \(\varphi(P_2)\) such that \(G_{\varphi(P_1)}\) is cyclic of order \(4\) and \(G_{\varphi(P_2)} \cong (\mathbb{Z}/4\mathbb{Z})^{\oplus 2}\); \item[--] \(\deg(\varphi(C))=6\), \(\varphi(C)\) admits two Galois points \(\varphi(P_1)\) and \(\varphi(P_2)\) such that \(G_{\varphi(P_i)}\) is cyclic of order \(5\), for \(i=1,2\). \end{itemize}} As an application, it is shown in Theorem 3 that if \(p \neq 3\) and \(E \subset \mathbb{P}^2\) is the curve defined by \(X^3+Y^3+Z^3=0\), then there exists a morphism \(\varphi: E \rightarrow \mathbb{P}^2\) which is birational onto its image with \(\deg(\varphi(C))=4\), such that \(\varphi(E)\) admits two inner Galois points. Galois points; plane curves; Galois groups; automorphism groups Fukasawa, S., A birational embedding of an algebraic curve into a projective plane with two Galois points, preprint Plane and space curves, Algebraic functions and function fields in algebraic geometry, Automorphisms of curves A birational embedding of an algebraic curve into a projective plane with two Galois points	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the main theorem of this paper: if \(f: P\to X\) is a flat morphism of algebraic varieties over an algebraically closed field \(k\) of arbitrary characteristic, with \(X\) a non-singular projective curve and the geometric generic fiber of \(f\) is a normal, separably rationally connected variety, then \(f\) has a section. Here, a normal variety is said to be separably rationally connected if there is a morphism \(C\to V\), with \(C\) a smooth rational curve, whose image does not contain any singular point of \(V\) and such that the pull-back via \(f\) of the tangent bundle of \(V\) is ample on \(C\). This result easily implies the statement of the title of the present article. The main theorem is closely related to a question posed by Kollar, Miyaoka and Mori, who asked whether the analagous statement, but using a different notion of rationally connected variety, is true. Both notions agree if \(V\) admits a resolution of singularities. When \(k\) is the field of complex numbers, the main theorem was proved by using a not purely algebraic argument in [\textit{T. Graber}, \textit{J. Harris} and \textit{J. Starr}, J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)]. The proof given in the present paper, partially based on that of the mentioned article, is purely algebraic and rather convoluted. By means of a normalized base change via a suitable finite, generally étale morphism \(Y\to X\) there is a reduction to the case where all the fibers of \(f: P\to X\) are reduced. In this situation, a key ingredient in the proof is a theorem saying that given a finite morphism \(Y\to X\) of irreducible smooth projective curves over \(k\) there is a family \(\{Y_t\}_{t\in X}\) of curves such that (among other properties) for suitable points \(0\), \(\infty\) in \(X\) we have that \(Y_0\) is nodal, with \(Y\) as normalization, and \(Y_\infty\) has an irreducible component which is naturally isomorphic to \(X\). Techniques from the theory of Hilbert schemes also play an important role in the proof. These and other auxiliary results and methods discussed in the paper may be of independent interest. rational point; section; rationally connected variety; family of curves; Hilbert scheme de Jong, A.J., Starr, J.: Every rationally connected variety over the function field of a curve has a rational point. Am. J. Math. 125(3), 567--580 (2003) Rational points, Schemes and morphisms, Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry Every rationally connected variety over the function field of a curve has a rational point	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an arbitrary smooth complex projective model of the fibre product \(X_1\times_C X_2\), where \(X_1\rightarrow C\) is an elliptic surface over a smooth projective curve \(C\) and \(X_2\rightarrow C\) is a family of \(K3\) surfaces. Such a family is said to be with semistable degenerations of rational type, if, for every singular fibre, the Picard-Lefschetz transformation \(\gamma\) satisfies the condition \((\log(\gamma))^2\neq 0\). In the present paper the author proves that the Grothendieck standard conjecture \(B(X)\) of Lefschetz type holds for \(X\) under the assumption that \(X_2\rightarrow C\) is a family of \(K3\) surfaces with semistable degenerations of rational type such that \text{rank NS}\((X_{2s})\neq 18\) for a generic geometric fibre \(X_{2s}\). Furthermore he proves that \(B(X)\) holds for every smooth projective compactification \(X\) of the Néron minimal model of an abelian scheme of relative dimension 3 over an affine curve if the generic fibre is an absolutely simple abelian variety with reductions of multiplicative type at all infinite places. algebraic cycle; elliptic variety; standard conjecture of Lefschetz type; \(K3\) surface; abelian scheme Tankeev, S G, On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models, Izv. Ross. Akad. Nauk Ser. Mat., 78, 181-214, (2014) Algebraic cycles, Variation of Hodge structures (algebro-geometric aspects), Classical real and complex (co)homology in algebraic geometry, \(4\)-folds On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an algebraic curve of genus \(g\geq 2\), defined over an algebraically closed field \(k\) of characteristic zero, let \(S_{n,d}(X)\) be the variety of all stable vector bundles \(E\) on \(X\) of rank \(n\) such that \(\wedge^nE\) is isomorphic to a fixed line bundle of degree \(d\), \(n\) and \(d\) are relatively prime. The analogue of Torelli's theorem for \(S_{n,d}\) is the assertion: \[ \text{If } S_{n,d}(X_1) \cong S_{n,d}(X_2), \text{ then } X_1\cong X_2. \tag{} \] For \(n=1\) this is Torelli's classical theorem. The case \(k=\mathbb C\), \(g=2\), \(n=2\) was studied by \textit{P. E. Newstead} [Topology 7, 205--215 (1968; Zbl 0174.52901)], who described the complete structure of \(S_{2,d}\), showed a projective embedding of \(S_{2,d}\) in \(\mathbb P_5\) and proved that the variety of lines of \(S_{2,d} \subset \mathbb P_5\) is isomorphic to the Jacobian \(J(X)\) of \(X\). In the case \(k=\mathbb C\), \(g\geq 2\), \(n=2\) \textit{D. Mumford} and \textit{P. E. Newstead} [Am. J. Math. 90, 1200--1208 (1968; Zbl 0174.52902)] proved () and showed an isomorphism of the intermediate Jacobian \(J^2(S_{2,d}(X)\) to \(J(X)\). The author [Math. USSR, Izv. 3 (1969), 1081--1101 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1149--1170 (1969; Zbl 0225.14009)] gave the following geometric reconstruction of \(X\) from \(S_{2,d}(X)\): if \(V\) denotes the base variety of the family of lines of some type on \(S_{n,d}\), \(\varphi: V \to\mathrm{Alb}(V)\) is the Albanese's application and \(\gamma: J(X)\to J(X)\) is the endomorphism of doubling, then \(\mathrm{Alb}(V)\cong \bar J(X)\) and \(X\cong \gamma(\varphi(V))\). In the present paper it is proved (*) in the general case and pointed out the geometric way of reconstructing of \(X\) from \(\prod S_{n,d}(X)\). The procedure is the following. Let \(M\) be the set of varieties of form \(\prod S_{n,d}(X)\) \(((n,d)\) runs through a finite set of pairs \((n,d)\), \(n\) and \(d\) are coprime). For \(V\in M\) let \(\bar B_n(V)\) denote a connected component of the base variety of projective spaces \(\mathbb P_n\subset V\) such that \((-K_V)\cdot \mathbb P_n= 2 \mathbb P_{n-1}\), where \(K_V\) is the canonical class on \(V\) and \(\mathbb P_{n-1}\) is a hyperplane in \(\mathbb P_n\) (all connected components of the base variety are actually isomorphic to each other). Let \(B_n(V)\) be a fibre of Albanese's application \(\bar B_n(V)\to\mathrm{Alb}\bar B_n(V))\) (for \(V\in M\) the fibres are isomorphic to each other). The pair \((n,d)\) defines a sequence of natural numbers \(\{m_1,m_2,\dots,m_N\}\) such that \[ S_{2,1}(X) =B_{m_N}(B_{m_{N-1}}(\ldots B_{m_1}(S_{n,d}(X))\ldots)). \] For example the pair \((n,1)\) gives the sequence \(\{(n-1)(g-1), (n-2) (g-1),\dots, 2(g-1)\}\). By examination of projective subspaces of \(S_{n,d}\) were found very useful the new notions of a rigid pair of bundles and rigid bundle: a pair \((E_1,E_2)\) is rigid if \(\text{rk}\,E_1 \deg E_2 - \text{rk}\,E_2 \deg E_1=1\) and a bundle \(E\) is rigid if it is representable as an extension \(0\to E_1\to E\to E_2\to 0\) where \((E_1,E_2)\) is a rigid pair. Tjurin, A.: Analogs of Torelli's theorem for multi-dimensional vector bundles over an arbitrary algebraic curve. Izv. Akad. Nauk SSSR, Ser. Mat. Tom34 (2), 338--365 (1970), Math. USSR Izvestija Vol.4 (2), 343--370 (1970). Torelli problem, Vector bundles on curves and their moduli Analogs of Torelli's theorem for multidimensional vector bundles over an arbitrary algebraic curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review gives a survey of recent results on rational points of algebraic varieties over number fields, mainly obtained by the fibration method, and thus can be viewed as a natural continuation of an earlier author's publication [in: Number theory. Proc. int. cong. discr. math. number theory, India 1996, Contemp. Math. 210, 19--39 (1998; Zbl 0910.14008)]. Note however that a substantial part of mentioned results is given with complete proofs (especially in chapter 1); moreover, chapter 3 contains original results due to the author. Here are some details on the content of the paper. The introduction contains a brief account of basic notions necessary to introduce the reader into the context of problems under consideration: the Brauer-Manin pairing and its use for describing obstructions to the Hasse principle and weak approximation, particularly in the special case when the variety under investigation can be fibred over the projective line so that all geometric fibres contain a component of multiplicity one. Chapter 1 is focused on the descent method for non-complete varieties (``open descent''). First, the author presents a simplified proof of the ``formal lemma'' of \textit{D.~Harari} [Duke Math. J. 75, No. 1, 221--260 (1994; Zbl 0847.14001)] which is indispensable while working with the Brauer pairing on non-complete varieties. Second, he gives a summary of the descent method explaining the relationship between torsors and the Brauer group. Finally, he discusses a beautiful example of \textit{D.~ R. Heath-Brown} and \textit{A.~N.~Skorobogatov} [Acta Math. 189, 161--177 (2002; Zbl 1023.11033)] when this method (combined with the circle method) allows one to prove that for certain hypersurfaces of norm type the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. The author gives a slightly simplified version of the proof of this result. (It is worth mentioning that this result has been generalized, using the same method of open descent, by \textit{J.-L.~Colliot-Thélène, D.~Harari, and A.~N.~Skorobogatov} [Lond. Math. Soc. Lecture Notes Ser. 303, 69--89 (2003; Zbl 1087.14016)]. Chapter 2 is devoted to a new method invented by \textit{P. Swinnerton-Dyer} [in: Abelian Varieties, Walter de Gruyter, Berlin, 273--292 (1995; Zbl 0849.14009)] and improved by \textit{J.-L.~Colliot-Thélène, A.~N.~Skorobogatov} and \textit{P.~Swinnerton-Dyer} [Invent. Math. 134, 579--650 (1998; Zbl 0924.14011)] which ensures the existence of many rational points on a variety fibred over the projective line into principal homogeneous spaces under an abelian variety. The author describes in some detail the main features and gives an overview of the main results obtained by the method. The emphasis is made on a recent result by \textit{P.~Swinnerton-Dyer} [in: Ann. Sci. Éc. Norm. Sup. 34, 891--912 (2001; Zbl 1003.11028)] establishing the Hasse principle for all diagonal cubic hypersurfaces of dimension at least 3 (as well as for some classes of diagonal cubic surfaces) defined over the field of rational numbers (under the assumption of finiteness of the 3-primary part of the Tate-Shafarevich group of certain elliptic curves). In chapter 3 the author proves an (unconditional) analogue of the above mentioned result of Swinnerton-Dyer in the case where the ground field is a function field in one variable over a finite field \({\mathbb F}_q\) (\(q\) odd, \(q\equiv 2\pmod 3)\). Hasse principle; torsor; descent; Brauer group Jean-Louis Colliot-Thélène, Points rationnels sur les fibrations, Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Society Mathematical Studies 12, Springer, 2003, p. 171-221 Rational points, Arithmetic ground fields (finite, local, global) and families or fibrations, Global ground fields in algebraic geometry Rational points on fibrations	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Logarithmic geometry is a general tool to work with singular varieties like toric varieties or varieties with ''semi-stable reduction'' as well as if they were smooth (over some base). In this theory, the analog of a smooth morphism is what is called a log smooth morphism (eg. the structural morphism of a toric variety). A general setting for this theory was developed by \textit{K. Kato} [in: Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore/MD 1988, 191-224 (1989; Zbl 0776.14004)], including the main aspects of an extension of crystalline cohomology [developed in the classical case by \textit{P. Berthelot} ``Cohomologie cristalline des schemas de caractéristique \(p>0\)''. Lect. Notes Math. 407 (1974; Zbl 0298.14012)] to log schemes \((X, M)\) over a base with a log structure. Among all the generalizations of the classical results or constructions one can ask in this new setting, the paper under review treats the case of Poincaré duality (this kind of result can potentially be used to prove that the Frobenius is an isomorphism in crystalline cohomology\(\otimes \mathbb{Q}\)), including the case of coefficients locally free of finite type. The base scheme has to be the finite length Witt vectors \(W_{m}\) of a perfect field of positive characteristic with trivial log structure or with the log structure associated to \(\mathbb{N} \rightarrow W_{m}\) sending \(1\) to \(0\) which play a key role in the proof, by \textit{O. Hyodo} and \textit{K. Kato} [in: Périodes \(p\)-adiques, Sém. 1988, Astérisque 223, 221-268 (1994; Zbl 0852.14004)], of a conjecture of Fontaine, the underlying scheme \(X\) of the log scheme \((X,M)\) has to be proper and there is a slight technical assumption about the kind of log structure needed. The Poincaré duality takes therefore, mutatis mutandis, the standard form. The proof follows step by step some constructions and proofs by Hartshorne and Berthelot done in the classical case showing clearly what has to be modified. Poincaré duality; logarithmic crystalline cohomology; logarithmic geometry; Witt vectors Tsuji, T, Poincaré duality for logarithmic crystalline cohomology, Compositio Math., 118, 11-41, (1999) \(p\)-adic cohomology, crystalline cohomology Poincaré duality for logarithmic crystalline cohomology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a one-hour report of ICM 1998, \textit{C. Deninger} [Doc. Math., J. DMV , 23--46 (1998; Zbl 0899.14001)] conjectured a cohomological formalism for algebraic varieties over rings of integers of number fields which relates to arithmetic zeta functions, particularly to Riemann zeta function; as well-known, here for these zeta functions there exists the Hasse-Weil conjecture. Such a conjectural arithmetic cohomology can be taken as a counterpart of \(\ell\)-adic cohomology, developed by Grothendieck's school for the construction of a Weil cohomology, for algebraic varieties over finite fields which relates to local zeta functions; also here for these local zeta functions one has the Weil conjecture proved by Deligne. Deninger's conjectural cohomology arises from J.-P. Serre's conjectural approach of local factors of zeta functions attached to algebraic varieties and Deninger has tried to realise it by leafwise cohomology of certain foliated dynamical systems. This is on one hand. On the other hand, \textit{S. Lichtenbaum} [Compos. Math. 141, No. 3, 689--702 (2005; Zbl 1073.14024)] defined a type of Grothendieck topology, called Weil-étale topology, for an arithmetic variety. He conjectured that the Euler characteristic of the Weil-étale cohomology gives the special values of the arithmetic zeta function for a given arithmetic variety. In this paper, the author discusses Deninger's dynamical systems and Weil-étale topology in terms of topoi over finite fields and over rings of integers of number fields, respectively. Then several properties for the interplay between the dynamical systems and the Weil-étale topoi are also obtained. In particular, the author defines a topos morphism which describes a relationship between the Weil-étale topos and the topos associated to a Deninger's dynamical system. Deninger's dynamical system; topos; Weil-étale topology Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta functions and \(L\)-functions of number fields On the analogy between Deninger dynamical systems and the Weil-étale topos	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let S be a geometrically irreducible smooth scheme over a finitely general \({\mathbb{Z}}\)-algebra \(R\subset {\mathbb{C}}\). Let \(X\to S\) be an abelian scheme. Fix some general point \(\eta: Spec {\mathbb{C}}\to S\) and let L be the function field of \(S\otimes_ R\bar R\), where \(\bar R\) denotes the algebraic closure. - The author gives relations between the Lie algebra \(G_{Hod}\) of the special Mumford-Tate group, the Lie algebra \(G_{\ell}\) of the Zariski closure of the image of \(\pi_ 1(S,\eta)\) in \(GL(H^ 1_{et}(X,{\mathbb{Q}}_{\ell}))\) and the Lie algebra of the differential Galois group of the Gauss-Manin connection \((H^ 1_{DR}(X_ L),\nabla_ L)\). He also gives the interrelations between the Hodge and the Tate conjecture for all powers of X and some stronger conditions on X which imply these conjectures. The author announces for certain classes of abelian schemes a proof of the conjectures of Hodge, Tate, and Mumford and Tate (i.e. \(G_{Hod}\otimes {\mathbb{Q}}_{\ell}\oplus {\mathbb{Q}}_{\ell}=G_{\ell})\), which is inspired by ideas of Ribet and Tankeev. These abelian varieties also provide interesting examples for other questions. Assume for example that X is a product of simple, polarized abelian varieties A such that the involution on \(E=End^ 0A\) is of the first kind and such that E contains a commutative subfield of degree dim(A). The above conjectures are true for X and, moreover, so is a conjecture of Katz, which asserts that the Lie algebra of the differential Galois group is generated by the p-curvature of \(\nabla_ L\). Finally, the author indicates a new approach to the Chowla-Selberg formula in a special case. Hodge conjecture; abelian scheme; Mumford-Tate group; differential Galois group of the Gauss-Manin connection; Tate conjecture André, Y. , Sur certaines algèbres de Lie associées aux schémas abéliens , Note C.R.A.S.t. 299 I n^\circ 5 (1984), 137-140. Analytic theory of abelian varieties; abelian integrals and differentials, Connections (general theory), Lie algebras of linear algebraic groups, Arithmetic problems in algebraic geometry; Diophantine geometry Sur certaines algèbres de Lie associées aux schémas abéliens. (On certain Lie algebras associated with abelian schemes)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians From publisher's description: ``This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.'' The topic of this book is the unramified Brauer group and its application to rationality problems and Hasse principles. The theory is developed not as in most other textbooks, but mainly through exercises with many intermediate steps and hints making them very accessible e.g. for graduate students or researchers looking for a quick introduction to the field. Difficult theorems like Merkurjev-Suslin are cited. I think the strategy of this concise book allows one to quickly learn the subject if you like the approach through exercises in the style of Serre's books. The book has a good balance between modern geometric language and explicit equations. You should not take it as a reference containing full proofs. At the end of most chapters, further reading recommendations are given. The Russian original, extended notes of a 2011 reading seminar at the Steklov Mathematical Institute, can be found at [\url{arXiv:1512.00874}]. Other books covering the theory of the Brauer group and the Brauer-Manin obstruction are [\textit{A. Skorobogatov}, Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015); \textit{B. Poonen}, Rational points on varieties. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1387.14004)]. These books contain more proofs. [Zbl 0972.14015] covers more applications of Brauer groups and torsors, but does not contain the basics as the book under review and [Zbl 1387.14004]. The applications to rationality problems (Part III) are unique to this book. Part I ``Preliminaries on Galois cohomology'' is an introduction to group and Galois cohomology. In Part II (``Brauer group'') treats the special case of the Brauer group of a field and its relation to Severi-Brauer varieties in Chapter 3 and then in Chapter 4 the (Azumaya algebra) Brauer group of a scheme first defined by \textit{A. Grothendieck} [in: Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 46--66 (1968; Zbl 0193.21503); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 67--87 (1968); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 88--188 (1968; Zbl 0198.25901)] and the unramified Brauer group \(\mathrm{Br}^{nr} (K)\) (cf. the first part of the title of the book) [\textit{J. L. Colliot-Thélène}, Proc. Symp. Pure Math. 58, 1--64 (1995; Zbl 0834.14009)] of a function field \(K\) finitely generated over a constant field \(k\) of characteristic 0 defined as the intersection of the kernels of the residue maps \(\mathrm{res}_v:\mathrm{Br}(K)\to\mathrm{Br}(K_v)\to\mathrm{Hom}(G_{\kappa_v},\mathbb Q/\mathbb Z)\) for all discrete valuations \(v\) of \(K\) trivial on \(k^\ast\) with residue field \(\kappa_v\), also with their geometric meaning and the unramified Brauer group of a normal irreducible variety over \(k\). The unramified Brauer group of a smooth proper variety is a stably birational invariant. The book restricts to finitely generated function fields of characteristic 0 because they have perfect residue fields. Part III ``Applications to rationality problems'' is the second part of the title and the heart of the book: Chapter 5 produces an example of a quotient of a variety over an al-gebraically closed field of characteristic 0 by a finite group and a proof that is not stably rational using the non-triviality of the unramified Brauer group. Chapter 6 ``Arithmetic of Two-dimensional Quadrics'' is on the discriminant and the Clifford invariant of quadrics and their geometric meaning. Chapter 7 ``Non-rational Double Covers of \(\mathbb P^3\)'' proves using the unramified Brauer group that certain unirational threefolds are not rational. In Chapter 8 the Weil restriction of scalars and algebraic tori are introduced and applied to rationality problems. Chapter 9 is on an example [\textit{A. Beauville} et al., Ann. Math. (2) 121, 283--318 (1985; Zbl 0589.14042)] of a variety over a perfect field of characteristic \(\neq 2\) which is non-rational, but stably rational, hence unirational. In Part IV (``The Hasse principle and its failure'') the Hasse-Minkowski theorem on quadratic forms over global fields is proved by a geometric reduction to quadrics of dimension 1 and the Hasse principle for the Brauer group (Chapter 11). In Chapter 12 the Brauer-Manin obstruction introduced by \textit{Yu. I. Manin} [in: Actes Congr. internat. Math. 1970, No. 1, 401--411 (1971; Zbl 0239.14010)] is defined and applied to explaining the failure of the Hasse principle for the Lind-Reichardt equation \(2y^2=x^4-17\). Appendix A contains a short introduction to etale cohomology without proofs or exercises, but references to the standard source [\textit{J. S. Milne}, Étale cohomology. Princeton Mathematical Series. 33. Princeton, New Jersey: Princeton University Press. (1980; Zbl 0433.14012)]. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Brauer groups of schemes, Rationality questions in algebraic geometry, Rational points, Rational and unirational varieties, Galois cohomology Unramified Brauer group and its applications	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By analyzing the Galois action on the étale fundamental group, the author studies continuous \(\ell\)-adic representations of the geometric fundamental group of a variety \(X\) over a field \(k\) \[\rho: \pi^{\text{ét}}_1(X_{\bar{k}},\bar{x}) \to \mathrm{GL}(\mathbb{Z}_{\ell}) \] which arise from geometry. Let \(X\) be a geometrically connected variety over a field \(k\), \(\bar{x}\) a geometric point, and \(\ell\) a prime. A representation \(\rho\) is arithmetic if it is a subquotient of a representation \[ \tilde{\rho}: \pi^{\text{ét}}_1(X_{k'},\bar{x}) \to \mathrm{GL}(\mathbb{Z}_{\ell}) \] where \(k'\) is a finite extension of \(k\). It is geometric if it is a subquotient of the monodromy representation \[ \pi^{\text{ét}}_1(X_{\bar{k}},\bar{x}) \to \mathrm{GL}((R^i\pi_*\underline{\mathbb{Z}_{\ell}})_{\bar{x}}) \] for some smooth proper morphism \(\pi: Y \to X\) and some integer \(i\geq 0\). Moreover, if \(X\) is normal and \(k\) is finitely generated of characteristic zero, the main theorems claim that there exists a positive integer \(N=N(X,\ell)\) such that any arithmetic (resp. geometric) representation which is trivial mod \(\ell^N\) is unipotent (resp. trivial). The strategy to study the action of \(\mathrm{Gal}(\bar{k}/k)\) on \(\pi^{\text{ét}}_1(X_{\bar{k}},\bar{x})\) is anabelian methods, and these technical results are the main ingredients of the proof. For Part II, see [the author, Duke Math. J. 170, No. 8, 1851-1897 (2021; Zbl 07369863)]. arithmetic representation; fundamental group; weight; unipotent Homotopy theory and fundamental groups in algebraic geometry, Galois representations, Étale and other Grothendieck topologies and (co)homologies Arithmetic representations of fundamental groups. I.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For part I see \textit{D. S. Nagaraj} and \textit{C. S. Seshadri}, ibid. 107, No. 2, 101-137 (1997; Zbl 0922.14023). Let \(X_0\) be an irreducible projective curve of arithmetic genus \(g\geq 2\) whose singularity is one ordinary point. The authors give a generalisation of Gieseker's construction for arbitrary rank. They construct a birational model \(G(n,d)\) of the moduli space \(U(n,d)\) of stable torsion free sheaves in the case \((n,d)=1,\) such that \(G(n,d)\) has normal crossing singularities and behaves well under specialization, i.e. if a smooth projective curve specializes to \(X_0,\) then the moduli space of stable vector bundles of rank \(n\) and degree \(d\) on \(X\) specializes to \(G(n,d)\). This generalizes an earlier work of Gieseker in the rank two case. Theorem 1. There exists a canonical structure of a quasi-projective variety on \(G(n,d)\) and a canonical proper birational morphism \(\pi _{*}:G(n,d)\rightarrow U(n,d)_s\) onto the moduli space of stable torsion free sheaves on \(X_0.\) The singularities of \(G(n,d)\) are (analytic) normal crossings. If \((n,d)=1\), \(G(n,d)\) is projective, since \(U(n,d)_s=U(n,d)\) is projective. projective curves with singularities; moduli space of stable vector bundles Nagaraj, D. S.; Seshadri, C. S., \textit{degenerations of the moduli spaces of vector bundles on curves, II: generalized gieseker moduli spaces}, Proc. Indian Acad. Sci. Math. Sci., 109, 165-201, (1999) Vector bundles on curves and their moduli, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Singularities of curves, local rings Degenerations of the moduli spaces of vector bundles on curves. II. (Generalized Gieseker moduli spaces)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is a global survey of approximation theorems; more precisely, it contains approximation theorems for sections of sheaves \({\mathcal F}\otimes{\mathcal E}_X\), where \({\mathcal F}\) is an analytic (or algebraic) coherent sheaf defined over a coherent real analytic space \(X\) (or a real affine variety \(X)\) and \({\mathcal E}_X\) is the sheaf of germs of \(C^\infty\) functions on \(X\). The paper is divided into three parts: in the first one, the analytic case, we prove that the set of sections of \({\mathcal F}\) is dense in the strong (or Whitney) topology in the space of sections of \({\mathcal F}\otimes {\mathcal E}_X\). The second part deals with the algebraic case. In this context we use the classical Weierstrass theorem and obtain approximation in the weak (or compact open) topology. In the real algebraic context, Theorems A and B are not true, so we study two subcategories of coherent sheaves: \(A\)-coherent and \(B\)-coherent sheaves. They are studied in \S 6 and \S 7; we prove approximation theorems for sections of \({\mathcal F}\otimes {\mathcal E}_X\) by means of sections of \({\mathcal F}\). The third part of this article is devoted to fiber bundles. Approximation theorems were known also for sections of fiber bundles. In \S 8, via a duality theory, we find new approximation results for sections of fiber bundles of a more general class, which are not, in particular, locally trivial. Locally trivial algebraic vector bundles do not have the usual good behaviour. For example, the fiber is not in general generated by global sections. Bundles having the latter property are called strongly algebraic and are studied in \S 9. We prove that an algebraic vector bundle is strongly algebraic if, and only if, its total space is an affine variety. By applying the approximation results to the sheaf \(\Hom({\mathcal F}, {\mathcal G})\), we can extend to general vector bundles the results about the equivalence of the analytic and \(C^\infty\) classification. For strongly algebraic vector bundles, if \(X\) is compact, we have similar results about \(C^\infty\) and algebraic classification. survey; approximation theorems; fiber bundles Tognoli, A.: Approximation theorems in real analytic and algebraic geometry, Lectures in real geometry (Madrid, 1996), de Gruyter Exp. Math., 23, pp. 113-166. Berlin, (1996) Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs, Real-analytic manifolds, real-analytic spaces, Real-analytic and semi-analytic sets Approximation theorems in real analytic and algebraic geometry	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper arises from the extensions of the Manin-Mumford conjecture, where we shall focus on the case of (complex connected) commutative algebraic groups \(G\) of dimension 2. This context predicts finiteness for the set of torsion points in an algebraic curve inside \(G\), unless the curve is `special', i.e. a translate of an algebraic subgroup of \(G\). Here we shall consider not merely the set of torsion points, but its topological closure in \(G\) (which equals the maximal compact subgroup). In the case of abelian varieties this closure is the whole space, but this is not so for other groups \(G\); actually, we shall prove that in certain cases (where a natural dimensional condition is fulfilled) the intersection of this larger set with a non-special curve must still be a finite set. Beyond this, in the paper we shall briefly review some of the basic algebraic theory of group extensions of an elliptic curve by the additive group \(G_a\), which are especially relevant in the said result. We shall conclude by stating some general questions in the same direction and discussing some simple examples. The paper concludes with the reproduction of a letter of Serre (whom we thank for his permission) to the second author, explaining how to obtain explicit projective embeddings of the said group extensions. Pietro Corvaja, David Masser, and Umberto Zannier, Sharpening 'Manin-Mumford' for certain algebraic groups of dimension 2, Enseign. Math. 59 (2013), no. 3-4, 225 -- 269. Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties, Global ground fields in algebraic geometry Sharpening `Manin-Mumford' for certain algebraic groups of dimension 2	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a number field, \(X\) a smooth projective curve of genus \(g\ge 2\) over \(K\), \(J\) the Jacobian of \(X\) and \(r\) the rank of the Mordell-Weil group of \(J(K)\). Chabauty's method, made effective by Coleman, produces a way to compute, in many cases, the finite set \(X(K)\) of rational points on \(X\), provided that \(r<g\) [\textit{C. Chabauty}, C. R. Acad. Sci., Paris 212, 882--885 (1941; JFM 67.0105.01); \textit{R. F. Coleman}, Ann. Math. (2) 121, 111--168 (1985; Zbl 0578.14038); \textit{W. McCallum} and \textit{B. Poonen}, Panor. Synth. 36, 99--117 (2012; Zbl 1377.11077)]. Replacing the Jacobian with non-abelian Selmer varieties and the abelian integrals with iterated Coleman integrals, \textit{M. Kim} [Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006); Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)]. proposed a way to extend Chabauty-Coleman approach to curves having \(r\ge g\). A different method has been introduced in [\textit{J. S. Balakrishnan} et al., J. Reine Angew. Math. 720, 51--79 (2016; Zbl 1350.11067)] for hyperelliptic curves with a model \(y^2=f(x)\) where \(f\) is a monic polynomial of odd degree without multiple roots, when \(K={\mathbb{Q}}\) and \(r=g\). This method involves \(p\)-adic Arakelov theory and intersection theory on arithmetic surfaces. In the paper under review, the authors extends this method to the case of an algebraic number field \(K\). They also consider bielliptic curves of genus \(2\), extending to number fields the result proved for \(K={\mathbb{Q}}\) in [\textit{J. S. Balakrishnan} and \textit{N. Dogra}, Duke Math. J. 167, No. 11, 1981--2038 (2018; Zbl 1401.14123)]. They illustrate their method by giving several examples, some with \(r\) up to \(4\), where the approach of [\textit{S. Siksek}, Algebra Number Theory 7, No. 4, 765--793 (2013; Zbl 1330.11043)] fails. They describe the algorithm for elliptic curves and for curves of genus \(2\) over quadratic fields. quadratic Chabauty; bielliptic curves; \(p\)-adic heights Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points, Arithmetic varieties and schemes; Arakelov theory; heights, Elliptic curves over global fields, Heights Explicit quadratic Chabauty over number fields	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A famous theorem of A. Weil gives an upper bound for the number of rational points on curves over finite fields. A curve is called maximal if it attains this upper bound, the cardinality of the finite field being necessarily a square (say \(q^2)\). The genus \(g\) of a maximal curve satisfies \(g\leq q(q-1)/ 2\). The paper deals with the classification problem for maximal curves of a given genus. The first result on this problem is due to \textit{H. G. Rück} and \textit{H. Stichtenoth} [J. Reine Angew. Math. 457, 185-188 (1994; Zbl 0802.11053)] and they showed that the curve \(X^{q+1}+ Y^{q+1}= Z^{q+1}\) is the unique maximal curve with \(g= q(q-1)/ 2\). The main tool here is the action of Frobenius on the Jacobian variety of a maximal curve. The main result of the paper under review is the classification of the curve \(X^m+ Y^m= Z^m\) with \(m= (q+1)/ 2\) as the unique maximal curve with genus \(g= (q-1) (q-3)/8\) having a nonsingular plane model. Besides the action of Frobenius on the Jacobian variety, the other main tool here is the theory of Frobenius orders of morphisms developed by \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc. (3) 52, 1-19 (1986; Zbl 0593.14020)]. finite fields; rational points; maximal curves A. Cossidente, J. W. P. Hirschfeld, G. Korchmáros, F. Torres, On plane maximal curves, Compositio Math. Curves over finite and local fields, Arithmetic ground fields for curves, Finite ground fields in algebraic geometry On plane maximal curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theorem of Gross and Zagier relates the Néron-Tate heights of Heegner points on modular curves to the central derivative of certain \(L\)-functions. The arithmetic Gan-Gross-Prasad conjecture is a generalization of this theorem to higher-dimensional Shimura varieties. The author has proposed an approach to establish the conjecture using an ``arithmetic'' relative trace formula. The paper is a major breakthrough in this approach. The paper proves the fundamental lemma for the relative trace formula, with the assumption that the local field is \({\mathbb Q}_p\) where \(p\) is a large enough prime. The argument should work for all local fields with large residue characteristic, in the general case there are just some claims that are expected to be true but need to be checked. A relative trace formula is a distribution which can be decomposed into a sum of orbital integrals. The fundamental lemma are identities for the orbital integrals over a local field, when the orbital integrals are associated to a base function (identity element of Hecke algebra). Once such identities are established, it is strong evidence that the trace formula approach will work, and some partial results can follow with relatively small amount of extra work. The reviewer does not claim to understand the details of the proof of the fundamental lemma, however the brilliance of the idea behind it is clear. Previously Jacquet introduced an idea to use Fourier transform in the proof of fundamental lemma. The current paper further developed the idea, and uses Weil representation (which encodes Fourier transform as Weyl group action) in the proof. This is combined with a local-global argument where the local orbital integral identity is derived from a global one, and an induction argument. The paper also used the same argument to reprove the fundamental lemma for Jacquet-Rallis relative trace formula that is used for Gan-Gross-Prasad conjecture for unitary groups. That fundamental lemma was originally proved using arithmetic geometric method by \textit{Z. Yun} [Duke Math. J. 156, No. 2, 167--227 (2011; Zbl 1211.14039)] for positive characteristic case and extended to any local field with large residue characteristic using a standard argument in logic. In comparison, the proof presented in current paper can be considered an elementary proof. arithmetic fundamental lemma; arithmetic Gan-Gross-Prasad conjecture; Kudla-Rapoport special divisor; relative trace formula; unitary Shimura variety; Weil representation Theta series; Weil representation; theta correspondences, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Algebraic cycles, Modular and Shimura varieties Weil representation and arithmetic fundamental lemma	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book originates from a series of lectures and seminar talks for graduate students at various universities in Japan. It is a research monograph describing research by the author on finite branched coverings of projective complex manifolds in connection with the theory of algebraic functions of several complex variables. The author presents a theory generalizing earlier work in one complex variable, in particular work of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)]. The book has three chapters: 1. ``Branched coverings of complex manifolds'', 2. ``Fields of algebraic functions'', 3. ``Weil-Tōyama theory''. The main result in chapter 1 is a theorem providing a suitable sufficient condition for the existence of a finite Galois covering of a projective manifold, branched over a given divisor. In the case of compact Riemann surfaces, this originates in a problem posed by Fenchel. In addition, chapter 1 contains many well chosen examples of finite branched coverings. -- Chapter 2 studies finite abelian coverings using the theory of currents developed by \textit{G. de Rham} and \textit{K. Kodaira} in ``Harmonic integrals'', Lecture Notes Inst. Adv. Study (Princeton 1950). There are results providing necessary and sufficient conditions for the existence of a finite abelian covering onto a projective complex manifold branched over a given divisor. Furthermore, the set of all isomorphism classes of finite abelian branched coverings of a projective complex manifold is described using the notion of rational divisor classes. The results can be considered as higher dimensional generalizations of results of Iwasawa from 1952. -- Chapter 3 studies similar questions for finite Galois coverings, but according to the author the results obtained are not completely satisfactory. The book should be a good source for inspiration to further work. finite branched coverings of projective complex manifolds; algebraic functions; finite Galois coverings M. NAMBA, Branched coverings - algebraic functions, Pitman Research Notes in Mathematics Series 161. Zbl0706.14017 MR933557 Coverings of curves, fundamental group, Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Low-dimensional topology of special (e.g., branched) coverings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization Branched coverings and algebraic functions	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an irreducible smooth projective curve of genus \(g\) defined over an algebraically closed field \(k\). Let \(r\geq 2\) be an integer, assume that \(r>3\) if \(g=2\). Let \({\mathcal N}(r)\) (respectively \(N(r)\)) denote the moduli stack (respectively moduli space) of stable PGL\((r,k)\)-bundles on \(X\). The authors prove that the natural map \({\mathcal N}(r) \to N(r)\) is an isomorphism outside a closed subset of codimension three. This identifies the Brauer group of \({\mathcal N}(r)\) with the Brauer group \(\mathrm{Br}(N(r))\) of \(N(r)\). The moduli space \(N(r)\) has \(r\) irreducible components \(N(r)_i\) parametrized by \({\mathbb Z}/r{\mathbb Z}\). Let \(M(r,L)\) be the moduli space of vector bundles of rank \(r\) with determinant isomorphic to \(L\). Then \(N(r)_i\) is the quotient of \(M(r,L), d(L)=i\), by the \(r\)-torsion points of the Jacobian. Using the Leray spectral sequence of the quotient map, the authors show that there is a surjective homomorphism \(\mathrm{Br}(N(r)_i) \to \mathrm{Br}(M(r,L))\) and explicitly compute the kernel of this map. It is known that \(\mathrm{Br}(M(r,L)) \cong {\mathbb Z}/h{\mathbb Z}\), where \(h\) is the greatest common divisor of \(r\) and \(i\). Brauer group; moduli stack; stable projective bundles I. Biswas and A. Hogadi, \textit{Brauer group of moduli spaces of} PGL(\(r\))\textit{-bundles over a curve}, arXiv:0904.4640. Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli Brauer group of moduli spaces of PGL\((r)\)-bundles over a curve	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This reproduction of the 1979 edition of André Weil's (1906-1998) Œuvres scientifiques covers more than 1500 pages. The articles and books are quoted in chronological order along the three volumes. For almost all references André Weil adds explicit and enlightening comments, the corresponding manuscript having been read in advance by Jean-Pierre Serre. The periods concerned by the three volumes are respectively 1926-1951, 1951-1964, 1964-1978. The Collected Papers are not the Opera omnia of the great mathematician. In particular they stop at the year 1978. In volume I a list of eleven conferences by Weil at the Bourbaki Seminar is dressed. Most articles are written in French. We tried to feature flashes on the various contributions ranged by general themes in each of the three volumes. Surprisingly Weil claimed himself to be an algebraist. But he could not have accepted Salomon Lefschetz's statement: If it is just turning a winch, it is algebra; if an idea comes in, it is topology. References to Bourbaki are not frequent in the collection. The interested reader should consult the Bourbaki Archives now available at the Paris Academy of Sciences. They also contain indications on Weil's living conditions during the times of World War II. Volume I Geometry / Number Theory [1926] The very first mentioned article is about surfaces with negative curvature. Weil extends a result due to Carleman: For a surface \(S\) with boundary \(L\) the inequality \(S \leq L^2/4\pi \) holds. Weil pays tribute to Hadamard's seminar at Collège de France. [1948c] Weil considers this text to be a mere exercise. For more information he refers to his \textit{Basic number theory}. [1939a] The purpose of this article is to explain analogies between the field of algebraic numbers and the field of algebraic functions, by elementary methods. Algebra [1927c,1928] In order to verify that reliability on earlier great mathematicians may lead to useful results, Weil associates an algebraic curve \(C\) to a finite algebraic field {k} of numbers determined by an equation with coefficients in \(k\). He is able to obtain a precious tool for the study of diophantian equations. Geometry on algebraic curves is about points or systems of points that remain invariant with respect to birational transformations. Weil relates how he got acquainted with Mordell's achievements. [1929] In 1922 Mordell announced that for a cubic \(C\) of gender 1, given by an equation with rational coefficients, the points with rational coordinates located on \(C\) may be deduced rationally from a finite number of them. Weil succeeds in replacing that algebraic theorem by a statement of essentially arithmetical flavor. [1932c] From Weil: I had not given up trying to prove the Mordell conjecture; I did not even loose hope to come closer to that aim [\dots] by a careful analysis and a thorough use of the tools I had performed in my thesis. [1934a] The note is written like a letter addressed to Hecke by Weil and Chevalley. It follows methods due to Artin and Herbrand. [1942] This long letter to Artin is introduced by the following lines: I have now reached a stage in my work on correspondences where I feel it will be helpful if I make a general survey of the theory, for you and a few such people. This seems all the more desirable, as I now find that the final writing up is going to involve a recasting of the intersection theory in algebraic geometry (since I am not altogether pleased with v.d.Waerden's treatment of this subject); this means that things may not get ready for quite a while. [1934b,c] For both these articles the framework concerns the homomorphic association of a group of matrices to a given group \(G\). One obtains a linear representation of \(G\). É. Cartan observes that by the way a remarkable fact is established: A unitary group for which the set of all traces is finite, is by itself finite. [1936i] The article contains remarks made by Weil on technical results due to Chevalley; the considered groups may not satisfy Hausdorff's axiom. [1951b] This is a very technical paper; Weil acknowledges conversations with Chevalley and Shapiro. He regrets the inappropriate terminolgy Weil group, so much more because there could be a confusion with the Weyl group in the theory of semisimple groups. [1935b] As soon as 1926-27 during his stay in Göttingen Weil got acquainted with the theory of ideals in the ring of polynomials and foundations of algebraic geometry. He could improve on his thesis. The Bourbaki group obtained the publication of this article, in memory of Jacques Herbrand. [1946a] From the introduction to the present publication: Algebraic geometry, in spite of its beauty and importance, has long been held in disrepute by many mathmaticians as lacking proper foundations. Our method of exposition will be dogmatic and unhistorical throughout, formal proofs, without references, being given at every step. [1938b] The expository article on different ways to think about the Riemann-Roch property was followed by comments made by Weil. He acknowledged that the subject had been one of his favorite topics at least in its classical form as far as algebraic curves are considered. He states that he really did not take part in later investigations. [1939b] Weil estimates that among finite groups the groups with \(p^n\) elements, \(p\) being a prime number, range among the most important; they show up in the proof of Sylow's theorem. Analogies with nilpotent Lie groups are not completely understood. Weil announces some new results. In his comments he also mentions that in the text a mistake was detected by Michel Lazare: a famous `One concludes easily'! [1943b] In his comments Weil provides some indications on misleading terminologies. [1940b] Due to circumstances the publication of this article was premature. In the recension it said that except an unproven `important lemma' the article could not provide any information to German algebrists! [1947b] In his comments Weil speaks about the importance for him of results due to de Rham and Hodge. He also mentions the discussions he had with H. Cartan while both of them hold a position at the Strasbourg university. He reports on a story also related by H. Cartan. Unsatisfied by the usual presentation of the general Stokes formula, they agreed on a conclusion: We should write once and for all a correct proof of that formula and then publish it in a book. Was it the birth of Bourbaki? [1949e] The theory of fibre spaces, and in particular the theory of homogenuous functions, was uprising; its cohomological aspect had become of great importance. The Bourbaki committee demanded collaborators for reports on that subject. It was suggested to call Weil algebra a certain graded differential algebra. Weil said he would prefer the teminology universal algebra attached to the Lie algebra \(A(G)\) of a group \(G\). [1941] Weil describes the state of the arts among mathemticians around Prrinceton. Should one build mathematics on set theory, as done by Bourbaki? Could that be realized for probability theory, differential geometry, algebraic geometry? Weil claimed that any remedy for that dangerous situation could come only from a strict adherence to the principles of algebra. [1948a] After Weil the develoment of geometry on algebraic varieties, the varieties deduced from algebraic curves, did always occupy an outstanding position due to the importance of the particular properties and to the simplicity of the generation method; so studying these objects is a natural introduction to any general theory. [1948b] This paper is the continuation of the preceding one. [1949d] Riemann had informed Hermite that the periods of \(2n\) periodical meromorphic functions of \(n\) variables satisfying certain relations and inequalities enable the possibility of formulating thêta series. Related work performed by Weierstraß, Picard, Poincaré, Appell contributes to the generality of Abelian varieties. Well announces simpler versions. [1949b] The title may sound elementary. Weil reports that his study starts indeed from a reading of wholly elementary procedures due to Gauß. But he relies on many other mathematicians' achievements and directly on Dolbeault for specific questions. The outcome is very impressive and conjectures remain. [1949c] From the author's introduction: The notion of abstract variety makes it possible for the algebraic geometers to imitate closesly the definitions and procedures which have been so frutfully applied by modern topologists to the theory of fibre bundles. [1950b] This long article contains Weil's address to the 1950 ICM Congress in Cambridge/USA. It may be enlightening to quote from his comments and his talk in a nonchronological order. Weil's wish was to develop algebraic geometry over rings, the ring of integers being the most elementary case. He does not claim to having been the pioneer, but is satisfied to see the outcome in Shimura's work on local rings and the theory of schemes by Grothendieck, his students and successoras. And now the quotation from the beginnning of Weil's speach, expressing a naive observation. The preceding speaker concluded his address with a reference to Dedekind and Weber. It is therefore fitting that I should begin with a homage to Kronecker. There appears to have been a certain feeling of rivalry, both scientific and personal, between Dedekind and Kronecker during their life-time; this developed into a feud between their followers, which was carried on until the partisans of Dedekind, fighting under the banner of the purity of algebra, seemed to have won the field, and to have exterminated or converted their foes. [1951a] Weil's motivation is providing an acount of results due to Northcott, Siegel and himself, and further settle algebraic foundations for the theory of arithmetics on algebraic varieties. [1950a] Weil intends to write a survey combining known facts about algebraic varieties and others likely to be. An Abelian variety is an abstract `complete' variety in which is defined a law of composition making it a group. Analysis [1932a] Weil presents a new proof of Poincaré's theorem concerning a system of curves defined on a ring-shaped surface by a differential equation of first order. In Hadamard's seminar importance of ergodicity is stressed. It recommends the study along the achievemenets due to Poincaré, the physicists Maxwell and Boltzman, as well as von Neumann and G.D. Birkhoff later on. [1932b] Weil became fond of functions on several complex variables, motivated by his famous theorem (going back actually to Cauchy) on the convergence radius of a Taylor series. [1935d] The article is an elaboration of [1932b]. [1935c] Making use of von Neumann's definition of almost periodic functions on a group, Weil was able to extend a homomorphic image of the group to its closure which is a compact group. The relation with Haar's measure will anounce the beginning of abstract Harmonic Analysis. [1935e] The article is insignificant in Weil's eyes. [1936a] The article is a minor contribution by Weil at the first (and last) International Topology meeting held in Moscow in 1935, with Alexandroff as chief organizer. [1936b] The text about arithmetics on algebraic varieties is the written version of talks delivered by Weil after the just mentioned Moscow meeting. He acknowledges being unable to understand one specific page and hence concludes that it must be a mistake! [1935a] This draft appears in its final form [1938a] three years later. [1938a] Weil states that some of the most brilliant advances in modern mathematics have been accomplished in the domains of arithmetics, algebraic varieties and topology. These topics show analogies and very often rely on groups that happen to be Abelian; the latter restriction is frequently the origin of success. In his comments Weil says that nowadays one may claim that this very long article inaugurated the study of fibre vector spaces on an algebraic curve of dimension \(\geq\) 1. [1936d] The article is the summary of a talk delivered at the Geneva university. The existence and uniqueness of an invariant measure on a locally compact group is explained. Weil extends the statement to homogeneous spaces. [1946b] Weil had thought being able to detect in Siegel's computing of volumes some opportunities to strengthen his theory of an invariant measure on homogeneous spaces and to find nontrivial examples. [1943a] In this long carefully written technical article, the coauthors rely to a certain extent on ideas expressed by Ahlfors. In the later comments Weil acknowledges that their results did not go so far as Chern's pioneering work in a new area of differential geometry. Their article makes use of embeddings, whereas Chern's formulation is intrinsic. General Topology [1936fg] During his Moscow stay Weil had opportunities to discuss with Pontrjagin and he became aware of the necessity to have a book on fundamentals of a theory on topological groups. This had to be done before the publication of Weil's book on Integration Theory. He introduced for instance the notions of complete spaces, bicompact spaces (later called compact spaces). Yet the most important event was the rise of the Bourbaki group of mathematicians with Weil's influential membership. One of the earliest books by Bourbaki was indeed the issue on Topology. [1937] The section reproduces all the 39 pages of Weil's book with explicit topic stated in the title: \textit{Sur les espaces à structure uniforme et sur la topologie générale.} The comments made by the author are reduced to the following statement: Forty years later one may smile on my eagerness to get rid of the countable situation. Chase it through the door, it will reenter through the window! [1945] Corrections have first been pointed out by H. Samelson. [1936h] Weil got interested in elliptic functions on the in that time popular \(p\)-adic groups. His results come close to the classical case. Functional Analysis [1927a,b] Weil examines essential differences between the sets of continuous functions and the set of square summable functions (Hilbert spaces). He mentions Volterra \textit{la cara e buona imagine paterno} who launched viewpoints exceeding his own results. It is also reported Courant having attributed to Weil the statement \textit{unproduktiv}. [1940c] The reading of this instructive text should be recommended to historians of mathematics. It provides an extensive survey of the interplay between probability theory and measure theory. More generally it furnishes a clew for the comprehension of modern axiomatized mathematics methods. The pioneering works of Lebesgue and Kolmogoroff are stressed. Yet Weil recognized later that, possibly boosted by his study on the Haar measure for groups, his reduction to locally compact spaces was too restrictive. But it had been more or less adopted by Bourbaki in the first editon of his Theory on Integration. [1940d] This book may be considered to be one of the first classics in Abstract Harmonic Analyis, along with the treatises by Pontrjagin, Loomis, and a few other authors. In his comments Weil reveals related historical facts. He also makes some surprising statements on himself: By temperament I was born arithmetician and algebrist, somehow geometer, not at all analyst. The real analyst, to feel himself comfortable, must possess instruments guaranteeing precision [dots] For me, as soon as one has to do with analysis, my instinct leads me to reduce this kit to an as small as possible number of strong useful tools; after that the algebrist may emerge again. General [1936c] Weil describes personal views on the organisation of studies in India. [1936e] Weil comments on examination procedures of Mathematiques spéciales students. [1938c] Weil explains his personal opinion on \textit{French Science} around 1938. [1940a] On March 26, 1940, André Weil wrote a letter to his sister Simome at the time he wanted to prepare a draft for a historical study of arithmetics and algebra theories starting with Gauß's Disquisitiones. He says to Simone: Maybe you get the feeling of the beginning; you wll surely not understand what comes after. [1947a] The article is an important contribution to the collective volume \textit{Les grands courants de la pensée mathématique} organized and edited by François Le Lionnais. Weil had often tried to avoid delays of that publication. His article examines essentially the possible evolutions of many mathematical topics; towards the end philosophical and epistemological considerations are made. [1949a] This article on applied mathematics is Weil's answer to a problem suggested to him by Claude Lévi-Strauss. Weil; Bourbaki; Artin; Chevalley; Shimura; Taniyama; Hodge; Kähler; Fermat; Ehresmann Collected or selected works; reprintings or translations of classics, History of mathematics in the 20th century, History of number theory, History of algebraic geometry, Proceedings, conferences, collections, etc. pertaining to number theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry Œuvres scientifiques. Collected papers. Vol. I (1926-1951)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Deformation theory in algebraic geometry is a basic tool that is just as old as the idea of classifying algebraic varieties. In his famous memoir on abelian functions from 1857, Riemann initiated the study of deformations of the complex structure of compact one-dimensional complex manifolds of given topological genus \(g\), that is of compact Riemann surfaces (or non-singular complex projective curves) of genus \(g\). The deformation of complex algebraic surfaces seems to have been considered first by Max Noether in 1888. However, a systematic deformation theory for higher-dimensional complex manifolds could be developed only as late as in 1958 and in the years thereafter, thus even more than 100 years after Riemann's pioneering memoir, and with the modern conceptual framework of sheaf theory, sheaf cohomology, and Hodge theory as a crucial ingredient [cf.: \textit{K. Kodaira}, ``Complex manifolds and deformation of complex structures.'' Grundlehren der Mathematischen Wissenschaften, 283. New York etc.: Springer-Verlag. X, 465 (1986; Zbl 0581.32012)]. This modern approach to complex-analytic deformation theory is well-known as Kodaira-Nirenberg-Spencer-Kuranishi theory, and its immediate effect on the development of classification theory in algebraic geometry has been equally revolutionary. In fact, A. Grothendieck immediately realized the functorial character of the Kodaira-Spencer theory of infinitesimal deformations of complex-analytic manifolds, and he outlined an analogous approach within his newly created foundational framework of algebraic schemes and formal algebraic geometry right away. These fundamental ideas, which eventually led to what is now called algebraic deformation theory, were published in a series of Bourbaki seminar expositions collected in his celebrated ``Fondements de la Géométric Algébrique'' [\textit{A. Grothendieck} ``FGA'', Secretariat Math. Paris (1962; Zbl 0239.14002)]. In the sequel, algebraic deformation theory has rapidly grown into a vast and central topic in modern abstract algebraic geometry, due to its crucial importance in regard to variational problems, including local properties of moduli spaces of varieties, vector bundles, and singularities. In its present state of art, algebraic deformation theory is highly formalized, both conceptually and methodically rather involved, widely ramified within algebraic geometry, and therefore not easily accessible to non-experts in the field. Although being of such importance and ubiquity in variational algebraic geometry, and while still developing rapidly, algebraic deformation theory had so far not yet found an adequate reflection in the relevant textbook literature. This fact made it rather difficult for non-specialists to find a solid orientation in this vast field of contemporary mathematical research, all the more so as numerous subtle technicalities and allegedly well-known results are scattered in the huge literature as ``folklore'', without rigorous and detailed proofs. The book under review is an attempt to partially fill this bothersome gap in the literature, and to provide a largely self-contained and comprehensive account of deformation theory in classical algebraic geometry, with complete proofs of those results and techniques that are needed to fully understand the local deformation theory of algebraic schemes over an algebraically closed field. This includes the careful explanation of those basic tools that are indispensable, for example, in the local study Hilbert schemes, Quot schemes, Picard schemes, and other classifying objects in this context. In this vein, and for the first time in the literature, the author compiles some of the many folklore results scattered in the literature, with detailed and systematic proofs, which must be seen as a just as valuable as rewarding contribution towards a solid foundation of algebraic deformation theory, and as an utmost useful service to the mathematical community likewise. As for the contents, the present book consists of four chapters, each of which is divided into several sections and subsections. In addition, and for the convenience of the non-expert reader, there are five appendices devoted to some basic facts from commutative algebra and algebraic geometry which are used throughout the text. After a brief introduction, in which an outline of the complex-analytic infinitesimal deformation theory à la Kodaira-Nirenberg-Spencer-Kuranishi serves as an explanation of the logical structure of algebraic (and functorial) deformation theory, Chapter 1 introduces the reader to infinitesimal deformations in an elementary fashion. The first section discusses extensions of rings and algebras, together with their generalizations to schemes over a base scheme, whereas the second section explains locally trivial deformations of schemes. This includes infinitesimal deformations of non-singular affine schemes, automorphisms of deformations and their extendability properties, first-oder locally trivial deformations, higher-order deformations, and the related (cohomological) obstruction theory for deformations. Chapter 2 provides the foundations of formal deformation theory. Starting with the concepts of formal smoothness and relative obstruction spaces for ring extensions, the author reconsiders infinitesimal deformations of algebraic schemes via M. Schlessinger's theory of functors of Artin rings, culminating in Schlessinger's famous theorem on the existence of (semi-) universal formal deformations [cf.: \textit{M. Schlessinger}, Trans. Am. Math. Soc. 130, 208--222 (1968; Zbl 0167.49503)]. This is followed by a discussion of deformation functors and local moduli functors in general, including their obstruction spaces as well as concrete applications to the deformation theory of algebraic surfaces in characteristic zero. In this regard, the author provides a largely self-contained treatment of formal deformation theory along the classical approach, that is without introducing cotangent complexes (à la L. Illusie) or methods of differential graded Lie algebras as more recent tools in deformation theory. Chapter 3 illustrates the general theory of deformation functors by several concrete examples. With the single exception of deformations of algebraic vector bundles, which have already been exhaustively treated in several recent research monographs like the book of \textit{D. Huybrechts} and \textit{M. Lehn} [``The geometry of moduli spaces of sheaves''. Aspects Math. E 31 (1997; Zbl 0872.14002)], the author describes in great detail the most important deformation functors in current algebraic geometry, mainly by carefully verifying Schlessinger's conditions for the existence of (semi-)universal deformation spaces in the respective cases. Moreover, the first-order deformations, i.e., the tangent spaces of these functors, as well as the corresponding obstruction spaces are thoroughly analyzed, thereby revealing a similar pattern in almost all of these exemplary cases. As for the concrete examples treated in this chapter, the author discusses the deformation functor of an affine scheme with at most quotient singularities, the local (relative) Hilbert functor of closed subschemes of a scheme, the local Picard functor of a scheme, deformations of sections of an invertible sheaf on a scheme, deformations of various types of morphisms of schemes, deformations of a closed embedding, and (co-)stable subschemes. Chapter 4 gives a thorough introduction to Hilbert schemes, Quot schemes, and flag Hilbert schemes. As the author points out, these objects are needed to construct important examples of global deformations, and to study their local behaviour in the framework of the main theme of the present text. Besides, until now it was rather difficult to give precise references for many results on the geometry of these classifying objects, just like in local deformation theory, and this very fact has been another reason for the author to include that chapter in his book. Together with the very recent monograph ``Fundamental Algebraic Geometry: Grothendieck's FGA Explained'' by \textit{B. Fantechi} et al. [Math. Surv. Monogr. 123 (2005; Zbl 1085.14001)], this chapter in the book under review provides the only (and overdue) reasonably comprehensive exposition on Grothendieck's Hilbert and Quot schemes. Apart from a general introduction to Castelnuovo-Mumford regularity, flattening stratifications, Hilbert schemes, Quot schemes, flag Hilbert schemes, and Grassmannians, together with applications to families of projective schemes, there is a concluding section on plane curves, their equisingular infinitesimal deformations, and their so-called Severi varieties. The author's approach to the proof of existence of nodal curves with an arbitrary number of singularities is very original and apparently new. Based on the use of multiple point schemes, this example strikingly illustrates the power of algebraic deformation theory even in classical curve theory. The five appendices at the end of the book collect some basic standard topics and are titled as follows: A. Flatness; B. Differentials; C. Smoothness; D. Complete intersections; E. Functorial language. Most of the results presented here come with full proofs, which further strengthens the already high degree of self-containedness of the book. There are no explicit exercises or working problems accompanying the text, but there is a wealth of illustrating concrete examples, additional remarks, historical annotations, and hints to further reading. The bibliography includes 190 references, ranging from the very classical up to the most recent articles and books, and both a very carefully compiled list of used symbols and a just as thorough alphabetical index considerably enhance the value of the entire treatise. Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. The author's style of writing captivates by its high degree of comprehensiveness, completeness, rigour, sytematical exposition, creative originality, lucidity, and user-friendliness in a like manner. The entire text is kept at a level that makes it suitable for graduate students with a solid background in commutative algebra, homological algebra, and basic algebraic geometry. But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on. textbook; formal methods; local deformation theory; local moduli; Hilbert schemes; Picard functors E. Sernesi, \textit{Deformations of algebraic schemes}. Springer 2006. MR2247603 Zbl 1102.14001 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Formal methods and deformations in algebraic geometry, Local deformation theory, Artin approximation, etc., Infinitesimal methods in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Deformations of algebraic schemes	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a non-singular complex curve of genus \(g>1\). For a fixed line bundle \(\xi\) over \(C\) of given degree \(d\), denote by \({\mathcal M}(2,\xi)\) the (coarse) moduli space of isomorphism classes of rank-2 stable vector bundles over \(C\) with determinant line bundle isomorphic to \(\xi\). By a result by \textit{J.-M. Drezet} and \textit{M. S. Narasimhan} [Invent. Math. 97, No 1, 53-94 (1989; Zbl 0689.14012)], the Picard groups \(\text{Pic}{\mathcal M}(2,\xi))\) are freely generated by the line bundle \({\mathcal O_ M}(\Theta)\) associated with an ample divisor \(\Theta\) in \({\mathcal M}(2,\xi)\). This divisor \(\Theta\) is called the generalized theta divisor of \({\mathcal M}(2,\xi)\), and its global sections \(f\in H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\) are called the generalized theta functions on \({\mathcal M}(2,\xi)\). Recent developments in conformal quantum field theory gave rise to conjectures about the dimension of the spaces \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) of generalized theta functions of order \(k\) [cf. \textit{E. Verlinde} and \textit{H. Verlinde}, ``Conformal field theory and geometric quantization'', Prepr. PUPT-89/1149 (1989)]. In the present paper, the author provides a partial verification of these conjectures, i.e., of the so-called Verlinde formulae. More precisely, he proves that in the special case of \(k=1\) and \(d=\deg\xi\) an odd integer, the conjectured formula \[ \dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))=2^{g-1}\centerdot (2^ g-1) \] indeed holds true. Moreover, using previous results of \textit{A. Beauville} [Bull. Soc. Math. Fr. 119, No. 3, 259-291 (1991)], the author succeeds in exhibiting an explicit base for the vector space \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\). The method of proof, in the particular case under investigation, is based upon the handy description of the moduli space \(M(2,\xi)\) for hyperelliptic ground curves, which is due to \textit{U. V. Desale} and \textit{S. Ramanan} [Invent. Math. 38, 161-185 (1976; Zbl 0323.14012)], and on a comparison argument with respect to the number of odd theta characteristics on the base curve \(C\). A general verification of the conjectured Verlinde formulae, i.e., for \(\dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) with \(k\) and \(d=\deg \xi\) arbitrarily chosen, has recently been obtained by \textit{A. Szenes} and \textit{A. Bertram} [cf. ``Hilbert polynomials of moduli spaces of rank-2 vector bundles'', I, II (Harvard-University, September 1991 and November 1991)]. An announcement and sketch of their general results was published by A. Szenes, after the appearance of the present article [cf. \textit{A. Szenes}, Int. Math. Res. Not. 1991, No. 7, 93-98 (1991)]. coarse moduli space of rank-2 stable vector bundles; generalized theta divisor; generalized theta functions; odd theta characteristics Y. Laszlo, La dimension de l'espace des sections du diviseur thêta généralisé , Bull. Soc. Math. France 119 (1991), no. 3, 293-306. Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Theta functions and abelian varieties, Families, moduli of curves (algebraic) Dimension of the space of sections of the generalized theta divisor.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity \(C\) and for the so called divisorial filtration for a modification of \(({\mathbb C}^2,0)\) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ \(C\) or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious ``motivic version'' of an A'Campo type formula. Campillo, A., Delgado, F., Gusein-Zade, S.: Multiindex filtrations and motivic Poincaré series. Monatshefte. Math. 150, 193-209 (2007) Singularities of curves, local rings, Singularities in algebraic geometry, Complex singularities Multi-index filtrations and generalized Poincaré series	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper consists of an introduction to set the stage and to announce the main result (and some of its shortcomings), six sections and references. The first section gives a brief overview of the construction of abelian schemes over Hilbert modular surfaces. In the second section the main object of study, the fundamental relative motive \(\mathcal V^{p,q}\), is introduced. Its various realizations are discussed. In particular, there is a comparison result between the Betti realization and intersection cohomology. The third section deals with the Betti and de Rham realizations. The fourth section gives an overview of \(L\)-functions. The results of Brylinski-Labesse and Chai are discussed. The exposition is quite detailed and technical. In the fifth section the main result (cf. \textit{infra}) is presented. The proof of this result takes quite some place. As usual an important role is played by Beilinson's Eisenstein map. This is explained in the sixth section. Let \(F/{\mathbb Q}\) be a real quadratic field with ring of integers \(\mathcal O_F\). Let \(G/{\mathbb Q} \hookrightarrow\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\) such that the maps \(G\rightarrow{\mathbb G}_m\), \({\mathbb G}_m\hookrightarrow\text{Res}_ {F/{\mathbb Q}}{\mathbb G}_{m,F}\) and \(\text{det}:\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\rightarrow \text{Res}_{F/{\mathbb Q}}{\mathbb G}_{m,F}\) lead to a Cartesian square. The group \(G\) gives rise to a Shimura variety \(S({\mathbb C})\) with a \(G({\mathbb A}_f)\) action. Let \(V^{p,q}=\text{Sym}^pV_2^{\vee}\otimes\text{Sym}^qV _2^{\vee}\), where \(V_2^{\vee}\) is the dual of the standard representation of \(\text{GL}_2\). It is known that intersection cohomology with respect to the Baily-Borel compactification decomposes into \(G({\mathbb A}_f)\)-isotypic components \(H^2_{\text{ét}}(\pi_f)\) which are \(\text{Gal} (\overline{\mathbb Q}/{\mathbb Q})\)-modules. An important result due to J. L.~Brylinski and J.-P.~Labesse says that, for almost all \(p\), the local \(L\)-functions \(L_p(s,H^2_{\text{ét}}(\pi_f))\) can be computed as special values of some automorphic \(L\)-function. This \(L\)-function has simple zeros at \(s=1-n\), \(p+2\leq n\leq p+q+2\) for certain \(\pi_f\), assuming \(p\geq q>0\). It turns out to be appropriate to work with relative motives. Let \(\mathcal A/S\) be the universal abelian scheme over \(S\). \(\mathcal A\) comes equipped with an \(\mathcal O_F\)-action. As a matter of fact, one is interested in constructing elements in \(H^{p+q+3}_{\mathcal M}(\mathcal A^{p+q},\overline{\mathbb Q}(n))\). A basic role is played by the direct summand \(\mathcal V^{p,q}\) of \(R(\mathcal A^{p+q}/S)\), built as suitable \(\text{Sym}^p\otimes\text{Sym}^q\). In Beilinson's philosophy, for a pure (Chow) motive \(M\) with coefficients in \(\overline{\mathbb Q}\) and \(L\)-function \(L(s,H^i_{ \text{ét}}(M))\), one has, for \(n>i/2+1\), regulator maps \[ r_{\mathcal H}:H^{i+1} _{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\rightarrow H^{i+1}_{\mathcal H}(M\otimes {\mathbb R},{\mathbb R}(n)) \] such that there exist \(\overline{\mathbb Q}\)-subspaces \(\mathcal K(i,n) \subset H^{i+1}_{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\) such that \[ r_{\mathcal H} (\mathcal K(i,n))=L(n,H^i_{\text{ét}}(M))\det\mathcal{DR}(i,n), \] where \(\mathcal{DR} (i,n)\) is Deligne's \(\overline{\mathbb Q}\)-structure. In the underlying situation, write \(r_{\mathcal H}:H^{\bullet}_{!\mathcal M}(\mathcal V^{p,q},{\mathbb Q}())\rightarrow H^{\bullet}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R},{\mathbb R}())\) for Beilinson's regulator map. For \(S=S(\pi)\cup\{\ell\}\), let \({L_S(s,H^2_{\text{ét}} (\pi_f)^{\vee}))=\prod_{p\not\in S}L_p(s,H^2_{\text{ét}}(\pi_f)^{\vee})}\). The main result may be stated as follows: Let \(\pi\in\text{coh} (V^{p,q})\) be stable or \(\varepsilon(\pi_f)=-1\); then there is a \(G({\mathbb A}_f)\) -submodule \(\mathcal K(p,q,n)\subset H^{p+q+3}_{\mathcal M}(\mathcal V^{p,q}, {\mathbb Q}(n))\) for all \(p+2\leq n\leq p+q+2\) and \(p\geq q>0\), such that: (i) \(r_{\mathcal H}(\mathcal K(p,q,n))\subset H^{p+q+3}_{\mathcal H}(\mathcal V^{p,q}\otimes {\mathbb R},{\mathbb R}(n))\); (ii) \(r_{\mathcal H}(\mathcal K(p,q,n))(\pi_f)=L_S(n,H^2_ {\text{ét}}(\pi_f))\cdot\mathcal{DR}(p,q,n)(\pi_f)\) (equality of \(\overline{\mathbb Q}\)-subvector spaces in \({\mathbb R}\otimes_{\mathbb Q}\overline{\mathbb Q}\)). \(K(p,q,n)\) can be explicitly constructed. In (ii) \(\mathcal{DR}(p,q,n)(\pi_f)\) is Deligne's \(\overline{\mathbb Q}\)-structure on absolute Hodge cohomology \(H^{p+q+3}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R}, {\mathbb R}(n))(\pi_f)\). higher regulators; Hilbert modular surface; motivic cohomology; absolute Hodge cohomology Kings, G., Higher regulators, Hilbert modular surfaces, and special values of \textit{L}-functions, Duke Math. J., 92, 1, 61-127, (1998) Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This note extends the proper base change theorem for torsion sheaves to the following. Given noetherian schemes with Y excellent, a proper morphism \(\pi:\quad Y\to X,\) a morphism \(f:\quad X'\to X\) whose pullback \(f':\quad Y'\to Y\) is normal, and an abelian sheaf F on Y, the base change morphisms \(f^(R^{\bullet}\pi_F)\to R^{\bullet}(\pi_'f'{}^F)\) are isomorphisms. The proof uses the original theorem, standard results on étale cohomology, and results from SGA 4 (Sém. Géométrie algébrique 4). proper base change theorem; étale cohomology Deninger, C., A proper base change theorem for non-torsion sheaves in ètale cohomology, J. Pure Appl. Algebra, 50, 231-235, (1988) Étale and other Grothendieck topologies and (co)homologies, Families, fibrations in algebraic geometry A proper base change theorem for non-torsion sheaves in étale cohomology	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Shimura-Taniyama-Weil conjecture states that every elliptic curve over rationals appears as a factor of the Jacobian of a modular curve. The Taylor-Wiles (also Diamond, Breuil, Conrad) proof of this conjecture uses the marvelous interaction of local Frobenius elements inside the absolute Galois group to show that the deformation rings are small enough to be isomorphic to Hecke algebras. Most ingredients of Wiles' proof (except for the construction of modular Galois representations and a detailed description of arithmetic moduli of elliptic curves) are covered (to some extent) by the author's book [\textit{H. Hida}, Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, No. 69, Cambridge University Press (2000; Zbl 0952.11014)]. The present book provides the missing ingredients, as well as an outline of some new modularity results concerning two-dimensional Galois representations. Prerequisites for the reader are rather high (for example, some experience with arithmetic algebraic geometry). Complete proofs (or detailed references) of all statements are given, hence the book may be adressed to graduate students working in number theory and arithmetic algebraic geometry. The book is divided into five chapters. Algebro-geometric methods (Cartier duality, cohomology of coherent sheaves, descent etc.) are recalled in the preliminary Chapter 1, which are then used in studying classification problems of elliptic curves. In Chapter 2 the author gives a comprehensive account of the theory of moduli spaces of elliptic curves. The first five sections contain an explanation of the theory of divisors on curves, geometric theory of elliptic curves, elliptic curves over \(p\)-adic and complex numbers. Elliptic curves with level \(N\)-structures are studied in section 2.6. Regularity of the moduli schemes (Katz-Mazur theorem) is established in section 2.8 (Theorem 2.8.2). Let \(p> 3\) be a prime, \((p,N)=1\). The main result of section 2.9 (Proposition 2.9.6) says that the \(p\)-ordinary moduli scheme is irreducible. Chapter 3 concerns the so-called ``control theorems''. In section 3.1 the horizontal control theorem (Theorem 3.15) has been proved. This result was used in [\textit{H. Hida}, Modular forms and Galois cohomology (loc. cit.)] for the identification of a (minimal) universal Galois deformation ring with an appropriate Hecke algebra. Vertical control theorems are proved in section 3.2, which describe specialisations of \(p\)-adic deformation of classical modular forms (Theorem 3.2.15, Lemma 3.2.18). They allow to identify the \(p\)-ordinary Galois deformation ring with a big \(p\)-adic Hecke algebra, and are useful in many other arithmetic applications. In Chapter 4 the author constructs modular Galois representations (and studies their ramifications) using Jacobians of modular curves (Theorem 4.2.7). Such representations are first constructed for modular forms of weight 2 by the method employed by \textit{G. Shimura} in his book [Introduction to the arithmetic theory of automorphic functions, Iwanami-Shoten and Princeton University Press (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)], and a general situation is reduced to this case. Let us recall that there are (another) two ways to construct modular Galois representations directly: geometric (Deligne), and analytic (Langlands). In the last Chapter 5 the author discusses modularity problems: the description of the original Wiles' treatment (Theorems 5.2.1 and 5.2.6), and the author's new results concerning elliptic \(\mathbb{Q}\)-curves (section 5.2.3). control theorems; Shimura-Taniyama-Weil conjecture; elliptic curve; modular curve; deformation rings; Hecke algebras; modular Galois representations; moduli spaces of elliptic curves; modular forms Hida, H.: Geometric modular forms and elliptic curves. World Scientific, Singapore (2000) Galois representations, Elliptic curves over global fields, Research exposition (monographs, survey articles) pertaining to number theory, \(p\)-adic theory, local fields, Holomorphic modular forms of integral weight, Local ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Geometric modular forms and elliptic curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(N\) be a natural number and let \(X_0(N)\) be the modular curve with respect to \(\Gamma_0(N)\). Let \(J_0(N)\) be the Jacobian variety of \(X_0(N)\) and let \(\mathcal{C}(N)\) be the \(\mathbb{Q}\)-rational cuspidal divisor class group. When \(N\) is a prime, \textit{B. Mazur} [Publ. Math., Inst. Hautes Étud. Sci. 47, 33--186 (1977; Zbl 0394.14008)] proved that the two groups \(\mathcal{C}(N)\) and \(J_0(N)(\mathbb{Q})_{\mathrm{tor}} \) coincide. For general cases, however, it is still an open problem whether the two groups are equal. Let \(C_0(N)\) be the closed subset of \(X_0(N)\) consisting of all cusps and let \(\bar{J}_0(N)\) denote the generalized Jacobian of \(X_0(N)\) with modulus \(C_0(N)\) in the sense of \textit{J.-P. Serre} [Algebraic groups and class fields. Transl. of the French edition. New York etc.: Springer-Verlag (1988; Zbl 0703.14001)]. In this paper the authors investigate the structure of its torsion subgroup \(\bar{J}_0(N)(\mathbb{Q})_{\mathrm{tor}}\), and observe that \(\bar{J}_0(N)(\mathbb{Q})_{\mathrm{tor}}\) is unexpectedly smaller than \(J_0(N)(\mathbb{Q})_{\mathrm{tor}} \) by proving the following result. Let \(p\) be a prime number and \(n\) be a positive integer. Then (1) \(\bar{J}_0(p)(\mathbb{Q})_{\mathrm{tor}}\) is a cyclic group of order 2. (2) If \(p\not\equiv 11\pmod {12}\) and \(p>5\), then \(\bar{J}_0(p^n)(\mathbb{Q})_{\mathrm{tor}}\) is isomorphic to the trivial group up to \(2p\)-torsion. (3) The statement (2) holds without the assumption \(p\not\equiv 11\pmod {12}\) but up to \(6p\)-torsion. Furthermore they determine the structure of \(\bar{J}_0(p^n)(\mathbb{Q})_{\mathrm{tor}}\) more precisely under the hypothesis that \(J_0(p^n)(\mathbb{Q})_{\mathrm{tor}}=\mathcal{C}(p^n)\). modular curves; cuspidal divisor classes; generalized Jacobian varieties; torsion points Jacobians, Prym varieties, Elliptic and modular units Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the trace formula approach to the lifting problems of automorphic representations of reductive groups, the matching of local orbital integrals has been reduced to that for the unit element of the Hecke algebra related to relevant groups, which is usually called the fundamental lemma. In general, the fundamental lemma is one of the most difficult open problems in the trace formula approach to the theory of automorphic representations. The fundamental lemma of Jacquet and Ye is the one in the relative trace formula approach to the quadratic base change problem for \(\text{GL}(n)\) [\textit{H. M. Jacquet} and \textit{Y. Ye}, Bull. Soc. Math. Fr. 120, 263--295 (1992; Zbl 0785.11032)]. An equivalent formulation of the fundamental lemma of Jacquet and Ye is an identity of two distributions, called Jacquet and Ye's conjecture. One of the distributions is of generalized Kloosterman integral type and is defined over \(\text{GL}(n,E)\), and the other distribution is of relative Kloosterman integral type and is defined over \(\text{GL}(n,F)\), where \(E/F\) is a quadratic extension. Jacquet and Ye verified their conjecture for \(\text{GL}(2)\) and \(\text{GL}(3)\) over a local non-Archimedean field of characteristic zero. The aim of the paper under review is to prove the above conjecture of Jacquet and Ye for \(\text{GL}(n)\) over local fields of positive characteristic. The main idea is to interpret the function-field analogies of the distributions of Kloosterman type defined by Jacquet and Ye as the trace of the Frobenius endomorphism over the \(l\)-adic cohomology of a certain algebraic variety, in the sense of Grothendieck. The conjectural identity of Jacquet and Ye in the function-field case becomes a certain quasi-invariant property of the trace with respect to a simply defined involution. It is a beautiful geometric argument. The original conjecture of Jacquet and Ye over local fields of characteristic zero is still open for \(\text{GL}(n)\), \(n\geq 4\). Ngô báo châu, \textit{Le lemme fondamental de Jacquet et Ye en caractéristique positive}, \textit{Duke Mathematical Journal}\textbf{96} (1999), 473-520. Representation-theoretic methods; automorphic representations over local and global fields, Étale and other Grothendieck topologies and (co)homologies The fundamental lemma of Jacquet and Ye in positive characteristic	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives with rational coefficients over a finite field when one assumes the Tate conjecture. The main result of the the paper under review is a similar result for motivic complexes. In a first section the statements of the Tate conjecture and of the generalised Tate conjecture are recalled and proven to be equivalent. The construction of ``the'' category of motivic complexes over a finite field \(k\) proceeds then in essentially two steps. In a first step a semisimple Tannakian category of motives \(\mathcal{M}_{\mathrm{num}}(k)\) over \(k\) is constructed. This is essentially done by modifying the commutativity constraint in Grothendieck's category of pure motives (for numerical equivalence), using the Weil conjectures proven by Deligne. The group of Weil \(q\)-numbers can be identified with a subgroup of the fundamental group \(\pi(\mathcal{M}_{\mathrm{num}}(k))\), and it is shown that equality holds if and only if the Tate conjecture holds over \(k\). The second step of the construction consists of showing that the bounded derived category \(\mathcal{D}^{\mathrm{b}}(\mathcal{M}_{\mathrm{num}}(k))\) is essentially the only candidate for a good triangulated category of motivic complexes. This category indeed has the hoped for properties if and only if the Tate conjecture holds over \(k\) and numerical equivalence coincides with rational equivalence (always for rational coefficients). In a last section, the (conditional) results are used to produce an explicit description of Beilinson's algebra of correspondences at the generic point \(A(X)\), for a smooth projective variety \(X\) over \(k\). Reviewer's remark: The reader might wish to consult the appendix (12p.) in the version of the paper provided on the first author's webpage, which fills in some details and proofs. Motives; Motivic complexes; Tate conjecture (Equivariant) Chow groups and rings; motives, Heights, Results involving abelian varieties, Complex multiplication and abelian varieties, Other abelian and metabelian extensions Motivic complexes over finite fields and the ring of correspondences at the generic point	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If \(Y\) is a geometrically connected scheme of finite type over a field \(k\), we have a surjective morphism (as profinite groups): \(\text{ pr}_Y \pi _1(Y,) \rightarrow G_k \), where \(\) is a suitable geometric point, \(\pi _1\) is the étale fundamental group and \(G_k\) is the absolute Galois group \(\text{Gal} (k^{\text{sep}}/k)\). Grothendieck conjectured that when \(k\) is finitely generated over \({\mathbb Q}\) and under suitable assumptions on \(Y\) (\(Y\) anabelian), the data \((\pi _1 (Y,),\text{ pr}_Y)\) determine (functorially) the isomorphism class of \(Y\). One case where this conjecture is supposed to be true is for hyperbolic curves, i.e. when \(Y\) is a smooth curve with \(2-2g-n<0\), where \(g\) is the genus of the smooth compactification \(X\) of \(Y\) and \(n\) is the cardinality of \(S(\overline k)\), with \(S = X-Y\). In this paper the conjecture is shown to be true for affine (i.e. such that \(n>0\)) hyperbolic curves. More precisely what is shown is : Let \(U_1\) and \(U_2\) be two affine hyperbolic curves over a field \(k\) finitely generated over \({\mathbb Q}\), then if there exists an isomorphism \({\mathcal F}: \pi _1 (U_1,) \rightarrow \pi_1(U_2,*)\) with \(\text{ pr}_{U_1}=\text{ pr}_{U_2}\circ {\mathcal F}\), \(U_1\) is isomorphic to \(U_2\) (as schemes over \(k\)). The proof of the main theorem requires a lot of work and different techniques (study of the function fields and the Galois groups and their connections, study how to recover properties of the curve from data on the fundamental groups); this leads to results interesting per se, like the ones about affine (not necessarily hyperbolic) curves or about curves over discrete valuation rings. hyperbolic curves; fundamental group; Galois group; Grothendiek conjecture Tamagawa A., The Grothendieck conjecture for affine curves, Compos. Math. 109 (1997), no. 2, 135-194. Coverings of curves, fundamental group, Global ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry The Grothendieck conjecture for affine curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first result provides a generalization of Deligne formulae to the case where \(n \geq 0:\) Theorem 1. Let \(\pi : X \rightarrow B\) be a proper flat morphism of integral schemes of relative dimension \(n \geq 0\) and let \(L \rightarrow X\) be a line bundle which is very ample on the fibers. \noindent 1) There is a canonical functorial isomorphism \[ \lambda_{n+1}(L, X, B) = \langle L,..., L \rangle _{X/B}. \] 2) If \(X\) and \(B\) are smooth, and \(K\) is the relative canonical line bundle of \(X \rightarrow B,\) then there is a canonical functorial isomorphism \[ \lambda^{2}_{n}(L, X, B) = \langle L^{n}K^{-1}, L,..., L \rangle _{X/B}, \] where the right sides of 1) and 2) are Deligne pairings of \(n + 1\) line bundles. The authors establish formulas for the first two terms in the Knudsen-Mumford expansion for \(\det(\pi_{*}L^{k})\) in terms of Deligne pairings of \(L\) and the relative canonical bundle \(K.\) As a corollary, they showed that when \(X\) is smooth, the line bundle \(\eta\) associated to \(X \rightarrow B,\) which was introduced by \textit{D.H. Phong} and \textit{J. Sturm} [Commun. Anal. Geom. 11, No. 3, 565--597 (2003; Zbl 1098.32012)], coincides with the CM bundle defined by \textit{S. Paul} and \textit{G. Tian} [CM stability and the generalized Futaki invariant, \url{arXiv: math.AG/0605278}]. line bundle; Deligne pairings; Knudsen-Mumford expansion D.\ H. Phong, J. Ross and J. Sturm, Deligne pairing and the Knudsen-Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475-496. Schemes and morphisms, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Analytic sheaves and cohomology groups, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Global differential geometry of Hermitian and Kählerian manifolds Deligne pairings and the Knudsen-Mumford expansion	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author introduces some complex of étale sheaves on a two-dimensional proper (not necessarily regular) \(\mathbb Z\)-scheme. This complex is a substitute of the complex \(\mathbb Z(2, X)\) introduced by \textit{S. Lichtenbaum} [in: Number theory, Proc. Journ. arithm., Noodwijkerhut 1983, Lect. Notes Math. 1068, 127-138 (1984; Zbl 0591.14014)]. Modulo Gersten's conjecture (which is known for regular surfaces over a finite field) his complex is quasi-isomorphic to Lichtenbaum's one (up to 2-torsion). The main result is a duality theorem for constructible sheaves using the complex as a dualizing sheaf. The theorem extends the Artin-Verdier duality for regular one-dimensional schemes and Deninger's duality theorem for singular schemes to the case of two-dimensional schemes. arithmetic surfaces; Artin-Verdier duality; Deninger duality; two dimensional scheme M. Spieß, ''Artin-Verdier duality for arithmetic surfaces,'' Math. Ann., vol. 305, iss. 4, pp. 705-792, 1996. Étale and other Grothendieck topologies and (co)homologies, Arithmetic varieties and schemes; Arakelov theory; heights Artin-Verdier duality for arithmetic surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of the Jacobian on smooth plane curves are of key interest, mainly from Fermat and Klein curves. For degree 4, which are smooth plane quartics, the stratification by automorphism appears natural different families to study over the algebraic closure. This study is over an algebraic closed field of zero characteristic \(K\). One family is the named Kuribayashi curves, with \(a\in K\), defined by \[C_a: X^4+Y^4+Z^4+a(X^2Y^2+Y^2Z^2+Z^2X^2)=0\] which in order to be non-singular \(a\notin\{-1,\pm 2\}\) and to have exact automorphism group equal to \(S_4\) we impose \(a\neq 0\) and \(a\neq \frac{-1+\pm\sqrt{-7}}{6}\). We remind to the reader that Henn in 1976 [Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3, Inagural-Dissertation, Heidelberg] discovered such families by automorphism on quartics (and Kuribayashi with his coauthors are publishing results on this direction from 1978). Thus is interesting to study the Jacobian of such family as it is for Fermat or Klein quartic curves. \textit{L. Kulesz} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 503--506 (1999; Zbl 0954.11021)], proved that the Jacobian of \(C_a\) is isogenous (isogeny defined in a maximal \(\mathbb{Z}/2\times\mathbb{Z}/2\)-Galois extension where the element \(a\) is defined) to the product of \(E_a^3\) with \(E_a\) the elliptic curve defined by \[y^2=x(x-1)(x+a+1).\] The authors assume that a sextactic point \(P\) appears in a fix smooth plane curve \(C\) of degree \(d\geq 4\) (we refer to the paper under review for the definition of sextactic point). Consider the natural map from \(C\) to its Jacobian given by \(\varphi:[Q]\mapsto [Q-P]\). We recall that there is a finite set of sextactic point on \(C\). And there are sextactic points in \(C_a\) if and only if \((a-14)(a^3+68 a^2-91 a+98)=0\) from [\textit{K. Alwaleed} and \textit{F. Sakai}, Saitama Math. J. 26, 67--82 (2009; Zbl 1202.14032)], and the case of the image of the group generated by the sextactic points for \(C_{14}\) under \(\varphi\) is obtained explicit in the preprint by \textit{K. Alwaleed} [``Group generated by total sextactic points of the Kuribayashi quartic curve'', Preprint]. Thus, in the paper under review, the authors deal the \(C_a\)'s such that \((a^3+68 a^2-91 a+98)=0\) obtaining first that any sextactic point on the Jacobian via \(\varphi\) has order 8 or a divisor of 8, and more precisely such finite torsion group of the Jacobian of \(C_a\) generated by the sextactic points via \(\varphi\) (where we recall that we have fixed one of such three values of \(a\) satisfying \(a^3+68 a^2-91 a+98=0\)) is isomorphic to \((\mathbb{Z}/8)^3\). Jacobian Kuribayashi curves; sextactic points Curves of arbitrary genus or genus \(\ne 1\) over global fields, Special algebraic curves and curves of low genus On the Jacobian of Kuribayashi curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a number field and \(X\) a smooth complete \(k\)-variety. Based on the reciprocity law for the Brauer group of \(k\), Manin isolated a condition which is necessary for the Hasse principle for \(X\) to hold. A similar condition exists for weak approximation. Both have been widely studied under the name (Brauer-) Manin obstruction. These obstructions account for most of the known counterexamples to the Hasse principle [resp. to weak approximation]. Therefore people have been trying to find classes of varieties for which it could be proved that the Manin obstruction is the only one. The paper under review adds new examples of this type. To rephrase its main technical result, consider a dominant morphism \(V\to \mathbb{A}^n_k\) of \(k\)-varieties whose generic fibre \(V_\eta\) is geometrically integral, and such that a smooth projective model of \(V_\eta\) has geometrically finite Brauer group and torsion-free Picard group. Assume that, for all \(k\)-points \(x\) in a non-empty open subset of \(\mathbb{A}^n_k\), the Manin obstruction to the Hasse principle [resp. to weak approximation] it the only one for the smooth projective models of the fibre \(V_x\). Then, under additional technical conditions, which in particular guarantee the existence of a smooth \(\overline k(D)\)-point of \(V \times_k\overline k(D)\) for almost all lines \(D\subset\mathbb{A}^n_k\), it is proved that the Manin obstruction is the only one for the smooth projective models of \(V\). Examples are given which are covered by this theorem, e.g., certain fibrations in conics or in Châtelet surfaces. This result generalizes a similar theorem about fibrations over the line, due to the same author [\textit{D. Harari}, Duke Math. J. 75, No. 1, 221-260 (1994; Zbl 0847.14001)]. A central technical tool is the study of the Brauer groups of the fibres of a smooth projective morphism \(V\to U\), where \(U \subset \mathbb{P}^1_k\) is open. If the generic fibre \(V_\eta\) is geometrically integral and has geometrically finite Brauer group and torsion-free Picard group, it is shown that \(\text{Br} (V_\eta)/ \text{Br} k(t) \cong\text{Br}(V_x)/ \text{Br} k\) for all \(x\) in a Hilbertian subset of \(\mathbb{P}^1(k)\). Again, this sharpens a result from the above-mentioned paper. Brauer-Manin obstruction; Hasse principle; weak approximation; Brauer group; Picard group Harari, D., \textit{flèches de spécialisation en cohomologie étale et applications arithmétiques}, Bull. Soc. Math. France, 125, 143-166, (1997) Global ground fields in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Brauer groups of schemes, Varieties over global fields Arrows of specializations in étale cohomology and arithmetical applications	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is an application of a new algebraic reformulation of the Picard group \(\text{pic} (G)\) of a quasi-compact subset \(G \subseteq X = \text{Spec} A\) for a commutative ring \(A\). A torsion theoretic (algebraic) proof is given of \(A\). Grothendieck's theorem that a complete intersection that is factorial in co-dimension 3 is factorial. Our proof is along the lines of Grothendieck's SGA2 proof, but eliminates the need for spectral sequences and (formal) sheaf theory. All torsion theoretic details are given in a lengthy appendix, including the proof of the long standing conjecture that \(Q_G = \varinjlim Q_U\) for \(G\) quasicompact and \(U\) open, where \(Q_G\) is defined in the following way: Denote by \(F_G = \{I \subseteq A \|I \nsubseteq\) any \(p \in G\}\) the torsion filter corresponding to \(G\). \(F_G\) is a directed set under reverse inclusion so we can form, for any \(A\)-module \(M\), the \(A\)-module \(P_G (M) : = \varinjlim_{I \in F_G} \Hom (I,M)\) and \(Q_G (M) : = P_G (P_G(M))\). This yields the interesting result that restriction of an \({\mathcal O}_X\)-Module to a quasi-compact, generically closed subset does not require the sheafification process. Picard group; complete intersection; factorial; torsion Call, F.: A theorem of Grothendieck using Picard groups for the algebraist. Math. scand. 74, 161-183 (1994) Torsion modules and ideals in commutative rings, Picard groups, Linkage, complete intersections and determinantal ideals A theorem of Grothendieck using Picard groups for the algebraist. -- Appendix	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a split reductive group scheme over \(\mathbb{Z}\) (recall that for any algebraically closed field \(k\) there is a bijection \(G\mapsto G\otimes k\) between isomorphism classes of such group schemes and isomorphism classes of connected reductive algebraic groups over \(k\)). Let \(B\) be a Borel subgroup of \(G\). Let \(S\) be a scheme and \(X\) a smooth proper scheme over \(S\) with connected geometric fibers of pure dimension 1. Our goal is to prove the following theorems. Theorem 1. Any \(G\)-bundle on \(X\) admits a \(B\)-structure after a suitable surjective étale base change \(S'\to S\). Theorem 2. Any \(G\)-bundle on \(X\) becomes Zariski-locally trivial after a suitable étale base change \(S'\to S\). Theorem 3. Suppose that \(G\) is semisimple. Let \(D\) be a subscheme of \(X\) such that the projection \(D\to S\) is an isomorphism. Set \(U:=X\setminus D\). Then for any \(G\)-bundle \(F\) on \(X\) its restriction to \(U\) becomes trivial after a suitable faithfully flat base change \(S'\to S\) with \(S'\) being locally of finite presentation over \(S\). If \(S\) is a scheme over \(\mathbb{Z}[n^{-1}]\) where \(n\) is the order of \(\pi_1(G(\mathbb{C}))\) then \(S'\) can be chosen to be étale over \(S\). Zariski-locally trivial bundle; reductive group scheme; \(G\)-bundle V. Drinfeld and C. Simpson, \(B\)-structures on \(G\)-bundles and local triviality, Math. Res. Lett. 2 (1995), 823--829. Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] \(B\)-structures on \(G\)-bundles and local triviality	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author is generalizing to the relative setting classical results as the theorem of Rojtman (the smooth case) and its generalization to singular varieties of Levine. His main theorem is: Theorem. Let \(X\) be a smooth affine \(k\)-variety of dimension at least 2, \(D\) an effective divisor on it. Then the Chow group of zero cycles on \(X\) with modulus \(D\), \(\mathrm{CH}_0(X\|D)\) is torsion free except possibly for \(p\)-torsion if the characteristic of \(k\) is \(p>0\) The main idea of the proof is based on generalizing to the relative setting of the arguments of the paper [\textit{M. Levine}, Proc. Symp. Pure Math. 46, 451--462 (1987; Zbl 0635.14007)]. This paper is a continuation of the author's previous paper on torsion 0-cycles [\textit{F. Binda} et al., J. Algebra 469, 437--463 (2017; Zbl 1374.14007)]. Algebraic cycles, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Motivic cohomology; motivic homotopy theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Torsion 0-cycles with modulus on affine varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [J. \(K\)-Theory 12, No. 1, 183--201 (2013; Zbl 1291.19005)], \textit{G. Banaszak} and the author obtained the sufficient condition for the validity of the local to global principle for étale \(K\)-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale \(K\)-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by \textit{S. Barańczuk} [Expo. Math. 35, No. 2, 206--211 (2017; Zbl 1391.37089)]. We show that all our results remain valid for Quillen \(K\)-theory of \(\mathcal{X}\) if the Bass and Quillen-Lichtenbaum conjectures hold true for \(\mathcal{X}\). algebraic curve; étale \(K\)-theory; curve; Hasse principle \(K\)-theory of schemes, Galois cohomology, Relations of \(K\)-theory with cohomology theories, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Hasse principle, weak and strong approximation, Brauer-Manin obstruction Note on linear relations in Galois cohomology and étale \(K\)-theory of curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be an algebraically closed field of characteristic zero and \(C/K\) be a smooth projective curve over \(K\) of genus \(g\geq 2\). A point \(P\) on \(C\) is a Weierstrass point if there exists a holomorphic differential form on \(C\) whose order at \(P\) is \(\geq g\). Let \(J\) be the Jacobian of \(C\). The Weierstrass subgroup of \(C\) is defined as being the subgroup of \(J(K)\) generated by the Weierstrass points. It is a geometric invariant of \(C\). The authors are interested in the determination of the structure of this group. In some particular cases, for instance Fermat curves, these groups have been found to be torsion. M. Girard has previously obtained the first examples where this group has positive rank. Concerning the generic genus-\(3\) curve, she has proved that this rank is at least \(11\). In this paper, the authors complete this result by showing that there are no relations between the Weierstrass points on the generic genus-\(3\) curve. They also obtain this fact for generic curves of any genus \(g\geq 3\). They show that the Weierstrass subgroup of the generic curve of genus \(g\geq 3\) has maximal rank, and is isomorphic to \(\mathbb Z^{g(g^2-1)-1}\). Projective curves; Weierstrass points Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Computational aspects of algebraic curves The Weierstrass subgroup of a curve has maximal rank	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an algebraic curve defined over \(\overline{\mathbb Q}\) and contained in the algebraic torus \(G_m^n\). We have the following result of Bombieri, Masser and Zanier: If \(X\) is not contained in the translate of a proper algebraic subgroup, then only finitely many points of \(X\) are contained in an algebraic subgroup of \(G_m^n\) of dimension \(n-2\). In this paper the previous result is generalized to show that there are at most finitely many points of \(X\) that are ``close'' to an algebraic subgroup of dimension \(n-2\), where the notion of close is given in terms of the Weil height. For any subset \(H \subset G_m^n\) and any \(\varepsilon \in \mathbb R\) the author defines the truncated cone: \[ C(H,\varepsilon)=\{ab; a \in H, b \in G_m^n, h(b) \leq \varepsilon(1+h(a)) \}, \] where \(h(.)\) is denoting the Weil height on \(G_m^n\). Also for \(0 \leq m \leq n\), the special subset \(H_m\) is defined as the points of \(G_m^n\) that are contained in an algebraic subgroup of dimension at most \(m\). With notation as above the result of Bombieri, Masser and Zanier states that \(X \cap H_{n-2}\) is finite. Also Kronecker's theorem characterizing the points with Weil height zero as the torsion points, implies that \(C(H_m,0)=H_m\). The main result of this paper is: Theorem: Let \(X \subset G_m^n\) be an irreducible closed algebraic curve defined over \(\bar{\mathbb Q}\), if \(X\) is not contained in a proper coset there exist an \(\varepsilon > 0\) effective and depending only on \(h(X), \deg(X),\) and \(n\) such that \(X \cap C(H_{n-2},\varepsilon)\) is finite with cardinality bounded effectively in terms of \(h(X)\), \(\deg(X)\) and \(n\). The theorem is proven in two steps: first showing that the height of points in \(X \cap C(H_{n-2},\varepsilon)\) is bounded for \(\varepsilon\) small and then showing that a subset of bounded height is finite for small \(\varepsilon\). The second part proving the finiteness statement can be generalized to higher dimensional varieties in \(G^n_m\). multiplicative group; algebraic curves; height functions; finiteness result; rational points P. Habegger, A Bogomolov property for curves modulo algebraic subgroups , Bull. Soc. Math. France 137 (2009), 93--125. Heights, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Arithmetic ground fields for curves, Counting solutions of Diophantine equations, Rational points, Global ground fields in algebraic geometry A Bogomolov property for curves modulo algebraic subgroups	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a nonsingular projective curve of genus \(g\) and let \(J(C)\) denote its Jacobian variety. Let \(A_\bullet(J(C))_\mathbb{Q}\) denote the group of rational algebraic cycles modulo algebraic equivalence, graded by dimension. The so-called tautological ring \({\mathcal R}(C)\) is defined to be the subgroup of \(A_\bullet(J(C))_\mathbb{Q}\) containing the class of \(C\) and stable under the Fourier transform, the intersection product, the Pontryagin product, pullbacks and pushforwards through multiplication maps by integers. In this paper the author constructs a set of generators of \({\mathcal R}(C)\), which enables us to control all the possible structures of the tautological ring. Let \([A_d(J(C))_{\mathbb{Q}}]_x= \{W\in A_d(J(C))_{\mathbb{Q}}; m^* W= m^{2g-2d-s} W\) for any \(m\in\mathbb{Z}\}\). He takes the set of \(s\)-components \(C_{(s)}\in[A_1(J(C))_{\mathbb{Q}}]_s\) of the curve \(C\) as a starting point, and constructs inductively a certain cycle \(\lambda_{\{i_1,\dots, i_d\}}\in [A_d(J(C))_{\mathbb{Q}}]_{i_1+\cdots+ i_d}\) which is annihilated by the principal polarization divisor and is congruent to \(C_{(i_1)}\cdots C_{(i_d)}\) modulo the subspace spanned by elements of lower degree. As an application he gives an exhaustive description of \({\mathcal R}(C)\) for all the possibilities that may occur in the cases when \(g\leq 9\). algebraic cycles; Chow group; Jacobian doi:10.1007/BF03191366 Algebraic cycles, (Equivariant) Chow groups and rings; motives, Jacobians, Prym varieties, Subvarieties of abelian varieties Tautological cycles on Jacobian varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y\) be a curve over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(W\) be a complete local ring with residue field \(k\) and \(G\) a finite flat group scheme over \(W\) which acts faithfully on \(Y\). Let \(\text{Def}(Y,G)\) be the functor which associates to a local Artinian \(W\)-algebra \(R\) with residue field \(k\) the set of isomorphism classes of \(G\)--equivariant deformations of \(Y\) to \(R\). The functor \(\text{Def}(Y,G)\) is studied using cohomological methods. The first 3 chapters of the paper give an exposition of certain cohomological methods for studying equivariant deformations of curves with group scheme actions. This theory is applied to the case of three point covers with bad reduction. There are three appendices including Picard stacks, the cohomology of affine group schemes and some spectral sequences. equivariant deformation; group schemes; cotangent complex [11] S. Wewers, `` Formal deformation of curves with group scheme action {'', \(Ann. Inst. Fourier (Grenoble)\)55 (2005), no. 4, p. 1105-1165. Cedram \| &MR 21 \| &Zbl 1079.} Local deformation theory, Artin approximation, etc., Coverings of curves, fundamental group, Deformations and infinitesimal methods in commutative ring theory Formal deformation of curves with group scheme action	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author establishes a Grothendieck-Riemann-Roch theorem for Deligne-Mumford stacks. More precisely: He defines ``cohomology with coefficients in representations'' of a Deligne-Mumford stack, constructs a Riemann-Roch transformation from \(K\)-theory to this new cohomology theory and proves a GRR theorem for this Chern character. The cohomology \(H^\bullet_{\text{rep}}(F, \ast)\) with coefficients in representations of a stack \(F\) is defined as the étale cohomology \(H^\bullet((I_F)_{\text{et}}, \Gamma(\ast))\) of the ramification stack \(I_F = F \times_{F\times F} F\) where \(\Gamma\) is a cohomology theory in the sense of \textit{H. Gillet} [Adv. Math. 40, 203-289 (1981; Zbl 0478.14010)]. The main tool in the construction of the Riemann-Roch transformation is the so-called dévissage theorem for \(G\)-theory which may be viewed as a generalization of \textit{A. Vistoli}'s decomposition theorem for equivariant \(K\)-theory [Duke Math. J. 63, No. 2, 399-419 (1991; Zbl 0738.55002)] and which is proved by generalizing the étale descent theorem [see \textit{R.~W. Thomason} and \textit{Th. Trobaugh}, in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)] from schemes to stacks. The proof of the GRR theorem runs through the usual patterns (deformation to the normal cone, Chow envelopes, \(\ldots\)). Grothendieck-Riemann-Roch theorem; Deligne-Mumford stack; cohomology with coefficients in representations; equivariant \(K\)-theory; Vistoli's decomposition theorem; étale descent; homological descent; Chow envelope Bertrand Toën, ``Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford'', K-Theory18 (1999) no. 1, p. 33-76 Riemann-Roch theorems Riemann-Roch theorems for Deligne-Mumford stacks	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author states: ``The goal of this note is to describe the Grothendieck-Serre duality on an arithmetic surface after the fixing of a horizontal divisor on this surface. We apply this description to the generalization of $\theta$-invariants from the case of arithmetic curves to the case of arithmetic surfaces.'' Pontryagin dual group; ind-Euclidean lattice; pro-Euclidean lattice; Arakelov degree Arithmetic varieties and schemes; Arakelov theory; heights, Adèle rings and groups Grothendieck-Serre duality and theta-invariants on arithmetic surfaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a survey of the author's work with [\textit{A. J. De Jong, X. He} and \textit{J. M. Starr}, Publ. Math., Inst. Hautes Étud. Sci. 114, 1--85 (2011; Zbl 1285.14053)] on Serre's conjecture II in the case of the function field \(K(S)\) of a surface \(S\) over an algebraically closed field \(k\): every \(G\) torsor over \(K(S)\) of a connected, simply connected, and semisimple algebraic group \(G\) over \(k\) is trivial, and more generally, a theorem about the existence of rational points for varieties defined over \(K(S)\). The proof uses a geometric argument involving rational simple connectedness. The general principle is that a variety over the field \(K(S)\) should have a rational point if it is ``rationally simply connected'' and if the elementary obstruction vanishes. However note that the definition of ``rational simple connectedness'' has been worked out only in the case of homogenous spaces ([Zbl 1285.14053] and [\textit{Y. Zhu}, ``Homogeneous fibrations over curves'', \url{arXiv:1111.2963}], and complete intersections [Zbl 1285.14053]. Roughly speaking, rational simple connectedness is the analogue of simple connectedness in topology, just as rational connectedness can be thought of as the analogue of connectedness. So part of the definition involves defining what it means for the moduli space of rational curves to be rationally connected. The actual definition is too technical to state in a review. The idea of producing the rational point is the following (again this is an over-simplified account of the proof). One first finds a model of the variety over \(K(S)\) as an algebraic fibration over the surface \(S\). Then using the Lefschetz fibration, one can write this as a family of varieties over a curve defined over \(\kappa\), the function field of a curve (in this case, \(\mathbb{P}^1\)). It suffices to show that there is a section (over \(\kappa\)) of the family. To do this, one studies the space of sections and the Abel-Jacobi map to the Picard variety of the curve. If one can prove that this is surjective with rationally connected fibers, then one can conclude the existence of a section, which is the same as a rational point over \(\kappa\) of the space of sections, by a theorem of \textit{T. Graber, J. Harris} and \textit{J. M. Starr} [J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)]. rationally connected varieties; semisimple algebraic groups; torsors Starr, J. M., Rational points of rationally simply connected varieties, Variétés rationnellement connexes: aspects géométriques et arithmétiques, 31, 155-221, (2010), Soc. Math. France: Soc. Math. France, Paris Rational and unirational varieties, Compactifications; symmetric and spherical varieties, Structure of families (Picard-Lefschetz, monodromy, etc.), Galois cohomology Rational points of rationally simply connected varieties	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review deals with the problem of computing all integer solutions \((x, y)\) to the hyperelliptic equation \(y^2 = f(x)\), where \(f(x)\) is a separable polynomial of degree at least 3 with integer coefficients. \textit{R. F. Coleman}'s interpretation [Duke Math. J. 52, 765--770 (1985; Zbl 0588.14015)] of the method of Chabauty, allowing one to determine the rational points on a curve whose Jacobian has Mordell-Weil rank less than its genus. Over the last decade, \textit{M. Kim} [Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006)] has initiated a program aimed at removing this restricting on rank, allowing the study of rational points on hyperbolic curves through the use of nonabelian geometric objects generalizing the role of the Jacobian in the Chabauty-Coleman method. In this frame, the authors gave a method which they call \textit{quadratic Chabauty} [J. Reine Angew. Math. 720, 51--79 (2016; Zbl 1350.11067)] based on \(p\)-adic height pairings to \(p\)-adically approximate the set of integer solutions of equation \(y^2 = f(x)\) in the case when the Jacobian of the corresponding algebraic curve has Mordell-Weil rank equal to its genus. In this paper, a full algorithm is presented for the computation in practice of all integer solutions of hyperelliptic equation \(y^2 = f(x)\). It combines the quadratic Chabauty method with the Mordell-Weil sieve. All the necessary computations are described and an analysis of the \(p\)-adic precision which must be maintained throughout the computation is provided. A few examples are given. hyperelliptic equations; integer solution; Chabauty-Coleman method; quadratic Chabauty method; Mordell-Weil sieve J. S. Balakrishnan, A. Besser, and J. S. Müller, Computing integral points on hyperelliptic curves using quadratic Chabauty, Math. Comp. 86 (2017), 1403--1434. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Computer solution of Diophantine equations, Arithmetic varieties and schemes; Arakelov theory; heights Computing integral points on hyperelliptic curves using quadratic Chabauty	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study geometric properties of maximal curves over \({\mathbb F}_{q^2}\) having classical Weierstrass gaps. A maximal curve \(X\) over \({\mathbb F}_{q^2}\) is a projective, geometrically irreducible, nonsingular algebraic curve, whose set of \({\mathbb F}_{q^2}\)-rational points attains the Hasse-Weil bound: \[ \# X({\mathbb F}_{q^2})=q^2+1+2gq. \] For a maximal curve \(X,\) let \(P\in X({\mathbb F}_{q^2})\) and \({\mathcal D}=\| (q+1)P\| .\) The authors investigate the interplay between the canonical divisor and \(\mathcal D\) via Stöhr-Voloch's approach to the Hasse-Weil bound. They prove that the support of the \({\mathbb F}_{q^2}\)-Frobenius divisor associated with \(\mathcal D\) is contained in the union of the Weierstrass points (for the canonical divisor) and the set \(X({\mathbb F}_{q^2}).\) When the curves have classical Weierstrass gaps they show that this is an equality. The paper contains a number of examples of maximal curves with classical Weierstrass gaps. They conclude with an interesting remark and condition concerning maximal curves over \({\mathbb F}_{q^2}\) for which each rational point is not a Weierstrass point. This suggests closer investigation in the case when \(g>1.\) maximal curves; Weierstrass points; Frobenius divisor A. Garcia, F. Torres, On maximal curves having classical Weierstrass gaps Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry, Finite ground fields in algebraic geometry On maximal curves having classical Weierstrass gaps.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main theorem proved in this article is the following: Theorem 1.1. Let \(Y\) be the underlying variety of a connected affine algebraic group. Then two embeddings of the affine line \(\mathbb{C}\) in \(Y\) are equivalent (i.e. the same up to an automorphism of \(Y\)) provided that \(Y\) is not isomorphic to a product of a torus \((\mathbb{C}^)^k\) and one of the three varieties \(\mathbb{C}^3\), \(\mathrm{SL}_2\), and \(\mathrm{PSL}_2\). The case where \(Y=\mathbb{C}^n\) with \(n=2\) or \(n \geq 4\) is classical; see the introduction of the article for an overview. Furthermore, the second author established in a former article a particular case of Theorem 1.1 (see the Main Theorem of [\textit{I. Stampfli}, Transform. Groups 22, No. 2, 525--535 (2017; Zbl 1454.14152)]): two embeddings of the affine line \(\mathbb{C}\) into the underlying variety of \(\mathrm{SL}_n\) are equivalent for all integers \(n \geq 3\). Let us note that, as mentioned in the introduction of the article, the existence of non-equivalent embeddings of \(\mathbb{C}\) into a product of a torus \((\mathbb{C}^)^k\) and one of the three varieties \(\mathbb{C}^3\), \(\mathrm{SL}_2\), and \(\mathrm{PSL}_2\), is a well-known open problem. To prove Theorem 1.1, the authors use a lot of different results related to the theory of affine algebraic groups, their principal bundles, and their actions on algebraic varieties. For instance, one of the main tools they use is the shearing-tool: Let \(X\) and \(X'\) be affine lines embedded in an affine algebraic group G, and let \(H \subseteq G\) be a closed subgroup such that \(G/H\) is quasi-affine. If \(\pi\colon G \to G/H\) restricts to an embedding on \(X\) and \(X'\) and if \(\pi(X)=\pi(X')\), then there exist a \(\pi\)-fiber preserving automorphism of \(G\) that maps \(X\) to \(X'\). automorphisms of algebraic groups; embedding of the affine line; principal bundles Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Automorphisms of surfaces and higher-dimensional varieties, Affine fibrations, Grassmannians, Schubert varieties, flag manifolds Uniqueness of embeddings of the affine line into algebraic groups	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be an algebraically closed field of characteristic zero and \(C\) an integal, nodal curve of arithmetic genus \(g\) over \(K\). Assume \(d\geq 2g-1\), \(d \geq 0\) and \(n \geq 2\) for certain integers \(d\) and \(n\). In this article, the author constructs a smooth compactification \(\bar{M}\) of the moduli space \(M\) of morphisms to projective space \(\mathbb{P}^n\) of a fixed degree \(d\) and computes the top intersection number \(c_1(\mathcal{O}_{\bar{M}}(1))^{(n+1)(d+1-g)+g-1}[\bar{M}]\) of a certain Cartier divisor \(X\) in \(M\) to be \((n+1)^g\). where \(g\) is the arithmetic genus of the curve \(C\). The computation of the above result -- which was known in the case of a smooth curve (see \textit{A. Betram, G. Dasklopoulos} and \textit{R. Wentworth} [J. Am. Math. Soc. 9, No. 2, 529--571 (1996; Zbl 0865.14017)]) -- is made possible by the computation of the Chern character of the degree \(d\) Picard bundle on the natural desingularization \(\tilde{J}^d(C)\) of the compactified Jacobian \(\bar{J}^d(C)\) (i.e. the moduli space of degree \(d\) torsion-free sheaves of rank one on \(C\)). Further applications of this result to Brill-Noether loci are also given. nodal curves; torsion-free sheaves; generalized Jacobian; Picard bundle Bhosle Usha, N, Maps into projective spaces, Proc. Indian Acad. Sci. (Math. Sci.), 123, 331-344, (2013) Vector bundles on curves and their moduli Maps into projective spaces	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors investigate which equivalent definitions for Weierstrass points on smooth curves remain equivalent on Gorenstein curves. It is shown that a Wronskian definition using dualizing differentials is equivalent to a definition involving 1-special subschemes with support at a point. The appearance of 1-special subschemes, instead of just special divisors, is to be expected since the limit of a divisor may be a subscheme, but not a divisor, as smooth curves degenerate to a singular curve. Let X denote an integral, projective Gorenstein curve of arithmetic genus \(g>1.\) An example is given to show that there may not be a locally principal 1-special subscheme (i.e. special Cartier divisor) of degree at most g at a (singular) Weierstrass point P on X. Another example is given to show that there may not exist a morphism \(\phi: X\to {\mathbb{P}}^ 1\) of degree at most g such that \(\phi^{-1}(\phi (P))=\{P\}\). Weierstrass points on smooth curves; Gorenstein curves Widland, C. and Lax, R. F.: Weierstraß points on Gorenstein curves,Pacific J. Math. 142 (1) (1990), 197-208. Riemann surfaces; Weierstrass points; gap sequences, Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Weierstrass points on Gorenstein curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This work comprises a summary of various approaches to the notion of infinitesimal, and a new ''intrinsic'' approach of the author's. The very readable introduction traces the antecedent ideas: a quick nod in the direction of Abraham Robinson, Weil's ''Théorie des points proches sur les variétés différentiables'', the use of nilpotents in algebraic geometry, the étale topology, the rise of topos theory, and synthetic differential geometry. Chapter 0 contains a quick survey of the interpretation of intuitionistic logic in a topos. Chapter 1 is entitled ''Use of Nilpotents''. It covers some of Kock's work on linear algebra in an intuitionistic setting, the Kock-Lawvere axiom for a ring of line type, models of the axiom in algebraic and differential geometry, infinitesimal linearity and the tangent bundle, 1- étale maps, and Reyes' notion of a Fermat ring. Chapter 2 is entitled ''Use of Infinitesimals''. The basic idea of this work is to exploit the observation that in the particular toposes discussed above as models for synthetic differential geometry it so happens that nilpotent elements in the ring of line type are also the elements x satisfying \(NOT(NOT(x=0))\); recall that we are working in an intuitionistic setting. The author defines in general the relation of being infinitesimally close to be the double negation of equality. Of course, it is the fact that this notion is not geometric, i.e. not preserved by inverse image functors, which makes it difficult to work with, and also rather deeper. Infinitesimal invertibility and formal manifolds are examined from this point of view, and comparison is made with the nilpotent approach. Section 2 is particularly interesting; in it a class of toposes is introduced for which a ''Nullstellensatz'' holds, making the interpretation of infinitesimals work more smoothly. Chapter 3 is entitled ''Use of Intrinsic Neighbourhoods''. The idea here is that open neighbourhoods in a topology are interpreted as containing all points ''sufficiently close''. It follows that in an intuitionistic setting we have an intrinsic notion of ''open set'', namely a set closed under the relation of being infinitesimally close. This intrinsic topology is explored in this chapter. Chapter 4 investigates the relationship between local invertibility in the different senses now available, and what they mean in the different models under consideration. There is an appendix on the Dubuc topos. infinitesimal; nilpotents; synthetic differential geometry; intuitionistic logic; tangent bundle; ring of line type; formal manifolds; Nullstellensatz; intrinsic topology; local invertibility; Dubuc topos Penon, Jacques, De l'infinitésimal au local, Diagrammes, 13, suppl., iv+191~pages, (1985) Topoi, Synthetic differential geometry, Abstract manifolds and fiber bundles (category-theoretic aspects), Intuitionistic mathematics, Foundations of algebraic geometry De l'infinitésimal au local. (Thèse de Doctorat d'Etat, Université Paris VII)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article contains a theory of Weierstrass points for higher dimensional schemes. The theory is naturally flexible and well adopted to the many applications of Weierstrass points to arithmetical and geometrical questions. It gives the Weierstrass points as a closed subscheme of the original scheme, defined by the vanishing of sections of locally free sheaves. The scheme of Weierstrass points is naturally stratified by closed subschemes of the same type and they enjoy natural functorial properties. In order to illustrate the uses of our theory and to show how it is different from the families of sets suggested as Weierstrass sets of various types in the literature (Catanese-Schneider, Corti, Iitaka, Ogawa) we give some applications to rational normal scrolls and to hypersurfaces in projective space. Our approach is global and centered around the notion of a wronskian determinant. The construction of the wronskian determinants is the main innovation of the work. It is a section of a locally free sheaf and represents a far reaching generalization of the classical wronskian determinant. A novelty of our approach is that the point of departure is a system of maps of sheaves rather than the underlying scheme. This is important for applications. The main example of such a system is the sheaves of principal parts. The resulting theory is intimately connected with the properties of higher dimensional osculating spaces as defined by Pohl. An important part of the article consists in verifying that the resulting Weierstrass points have desirable properties. Weierstrass points for higher dimensional schemes; wronskian determinant; sheaves of principal parts; osculating spaces Laksov, D. andThorup, A., Weierstrass points on schemes,J. Reine Angew. Math. 460 (1995), 127--164. Riemann surfaces; Weierstrass points; gap sequences, Schemes and morphisms Weierstrass points on schemes	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a somewhat expanded version of the first edition that appeared in 1982 [Springer Graduate texts in mathematics 84, (1982; Zbl 0482.10001). The first edition of 1982 consisted of 18 chapters which have been included in the second edition without alterations. The second edition contains two new chapters, one on the Mordell-Weil theorem for elliptic curves and one on recent developments in arithmetic geometry. The authors wrote the first edition with the purpose to give insight into modern developments in number theory by showing their close relationship with classical, 19th century number theory. The authors wrote the new chapters 19 and 20 in the same spirit. In chapter 19, they give a proof of the Mordell-Weil theorem for elliptic curves over \({\mathbb{Q}}\) without using Kummer theory or algebraic geometry: they first give Cassels' proof of the weak Mordell-Weil theorem which uses only a weaker version of Dirichlet's unit theorem for number fields; and then they derive the Mordell-Weil theorem using the standard descent argument which is worked out by elementary arithmetic. Chapter 19 is meant as a preparation for chapter 20, in which the authors give a very interesting overview of the important developments in arithmetic geometry after the appearance of the first edition of their book. Among other things, they discuss the Mordell conjecture proved by Faltings, the Taniyama-Weil conjecture and the result of Frey, Serre and Ribet that this implies Fermat's last theorem, recent progress on the Birch-Swinnerton-Dyer conjecture by Coates-Wiles, Gross-Zagier, Rubin, and Kolyvagin, and the derivation of Gauss' class number conjecture from the results of Gross-Zagier. In chapter 20, the authors do not give proofs but they give sufficient background to understand and appreciate the results. Chapter 20 is an excellent introduction for those who want to study the subject more thoroughly. unique factorization; congruence; quadratic reciprocity; quadratic Gauss sums; Jacobi sums; cubic and biquadratic reciprocity; equations over finite fields; zeta functions; quadratic and cyclotomic fields; Stickelberger relation; Eisenstein reciprocity law; Bernoulli numbers; Dirichlet L-functions; Mordell-Weil theorem for elliptic curves; Mordell conjecture; Taniyama-Weil conjecture; Fermat's last theorem; Birch- Swinnerton-Dyer conjecture; Gauss' class number conjecture K. Ireland and M. Rosen, \textit{A Classical Introduction to Modern Number Theory}, (2nd ed.), Springer, 1990. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Elementary number theory, Arithmetic algebraic geometry (Diophantine geometry), Multiplicative number theory, Algebraic number theory: global fields, Finite fields and commutative rings (number-theoretic aspects), Diophantine equations, Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Higher degree equations; Fermat's equation, Elliptic curves A classical introduction to modern number theory.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper gives a survey on the recent studies on fundamental groups of curves in positive characteristic. Let \(X\) be a smooth projective curve of genus \(g\) over an algebraically closed field of characteristic \(p>0\), \(U\) an affine open subscheme of \(X\) with complement \(S=X \backslash U\), and let \(\pi_A(U)\) be the collection of finite groups appearing in quotients of the fundamental group \(\pi_1(U)\). (A strong form of) Abhyankar's conjecture states that a finite group \(G\) is in \(\pi_A(U)\) if and only if every prime-to-\(p\) quotient of \(G\) has \(2g+\|S\|-1\) generators, and that such \(G\in \pi_A(U)\) can be realized by a Galois étale cover of \(U\) tamely ramified over \(X\) with at most one exceptional wild-ramification point in \(S\). In the paper under review, the author describes contributions to the solution of this conjecture, mainly focusing on those by \textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 311, No. 6, 341-346 (1990; Zbl 0726.14021)], \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)], and the author himself [\textit{D. Harbater}, Invent. Math. 117, No. 1, 1-25 (1994; Zbl 0805.14014)]. Expository explanations on the main ingredients of formal/rigid-analytic patching methods are also included. In the last section, there are described several related topics and stimulating open problems on structures of \(\pi_1(U)\), \(\pi_1(X)\) and embedding of fields. Among others, questions raised here on Grothendieck's anabelian geometry over finite fields have recently been settled by a remarkable paper by \textit{A. Tamagawa} [Compos. Math. J. (to appear)]. fundamental groups of curves; positive characteristic; quotients of the fundamental group; Abhyankar's conjecture; formal/rigid-analytic patching -, Fundamental groups of curves in characteristic \(p\), in Proceedings of the International Congress of Mathematicians, 1, 2 (Zürich, 1994), Birkhäuser, 1995, pp. 656-666. Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Inverse Galois theory, Finite ground fields in algebraic geometry, Local ground fields in algebraic geometry Fundamental groups of curves in characteristic \(p\)	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For part I see J. Math. Kyoto Univ. 28, No.1, 91-110 (1988; Zbl 0654.14021).] Kodaira introduced the notion of logarithmic transformations and showed that a non-Kähler elliptic surface and a Kähler one can be transformed to each other via logarithmic transformations. The main purpose of this paper is to prove the counterpart of Kodaira's theorem for elliptic bundles. Here by an elliptic bundle f: \(Y\to X\) over X, we mean that Y is a principal fiber bundle over a complex manifold X whose typical fiber and structure groups are non-singular elliptic curves. For an elliptic bundle f: \(Y\to X\) over X, we shall define generalized logarithmic transformations along an arbitrary \(Cartier\quad divisor\quad D\) on X, which is not necessarily assumed to be reduced, irreducible or effective. We state our main theorem: Theorem (A). Let f: \(Y\to X\) be an elliptic bundle over a projective manifold X, which satisfies either of the following conditions: (1) Y is Kähler; (2) \(h^{2,0}(X):=\dim (H^ 0(X,\Omega^ 2_ X))=0.\) Then Y can be obtained from the trivial elliptic bundle over X by a succession of generalized logarithmic transformations. However, theorem (A) does not necessarily hold if we drop the assumption that Y is Kähler. We can give the characterization of elliptic bundles which are obtained from the trivial bundle by generalized logarithmic transformations. logarithmic transformations; elliptic bundle Fujimoto, Y.: Logarithmic transformations on elliptic fibre spaces II, Math. Ann.288, 589--570 (1990) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Elliptic surfaces, elliptic or Calabi-Yau fibrations, Rational and birational maps Logarithmic transformations on elliptic fiber spaces. II: Elliptic bundle case	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It has been proven by \textit{J.-P. Serre} [``Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981'', in: Œuvres. Collected papers. Vol. IV: 1985--1998. Berlin: Springer. 1--12 (2000)], \textit{M. Larsen} and \textit{R. Pink} [Math. Ann. 302, No. 3, 561--579 (1995; Zbl 0867.14019)] and \textit{C. Chin} [J. Am. Math. Soc. 17, No. 3, 723--747 (2004; Zbl 1079.14029)], that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have ``the same'' \(\pi_0\) and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent \(F\)-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent \(F\)-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent \(F\)-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects. \(p\)-adic cohomology, crystalline cohomology, Galois representations The monodromy groups of lisse sheaves and overconvergent \(F\)-isocrystals	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this book the author studies properties of germs of plane curves, i.e., power series in \({\mathbb C}[[x, y]]\), the ring of formal power series in two indeterminates over the complex numbers, or power series in \({\mathbb C}\{x, y\}\), the ring of convergent power series. After some introductory remarks, the first chapter deals with Puiseux's theorem and the Newton-Puiseux algorithm, and the second chapter introduces branches of a germ and the notion of intersection multiplicity of two curves. Blowing up and infinitely near points are the subject of the third chapter; Enrique's definition of infinitely near points is also mentioned. This chapter deals also with the notion of proximate points, free and satellite points, and the resolution of singularities for germs of curves. There are also algebraic descriptions of the rings in the successive neighborhoods of the ring of a plane germ (cf. section 3.10 and 3.11). The notion of virtual multiplicity of clusters is defined in chapter 4; this chapter contains also a proof of Noether's \(Af + B\varphi\) theorem. Chapter 5 deals with characteristic exponents, the position of infinitely near points and the semigroup of values of an irreducible germ. In particular, it is shown that two irreducible germs are equisingular (i.e., have the same characteristic exponents) iff they have the same semigroup. (The well-known necessary and sufficient condition for a numerical semigroup to be the semigroup of a plane irreducible germ is given as exercises 5.10 and 5.11.) Polar germs are dealt with in chapter 6. Here the author gives Pham's example (showing that equisingular curves may have non-equisingular polars), proves Merle's result on the polar quotients, introduces the Milnor number, and shows [using result of his two papers: Math. Ann. 287, 429-454 (1990; Zbl 0675.14009); Comp. Math. 89, 339-359 (1993; Zbl 0806.14021)] that in the case of a single characteristic exponent, the polars of a germ may be used for unveiling properties of a germ that depend on its isomorphism class and not only on its equisingularity class. In section 6.10 and 6.11 the author treats the polar invariants of a reduced germ \(\xi\) and shows how to compute them from an Enrique diagram of \(\xi\). Linear families of germs are the subject of chapter 7. In section 7.2 the author introduces the weighted clusters of base points of pencils and proves their main properties. Section 7.5 and 7.6 deals with the notion of \(E\)-sufficiency (or \(C^0\)-sufficiency): A positive integer \(n\) is said to \(E\)-sufficient for a reduced \(f \in {\mathbb C} [[x,y]]\) iff all \(h \in f + m^n\) are non-zero, reduced and equisingular to \(f\) (\(m\) is the maximal ideal of \({\mathbb C} [[x, y]]\)). ({}The letter \(E\) stands for equisingularity: there is a similar notion with regard to being analytically isomorphic which is dealt with in section 7.7.) The last chapter presents -- in the context of this book -- a classification of valuations of \({\mathbb C}\{x,y\}\) and a proof of Zariski's factorization theorem for complete ideals in \({\mathbb C}\{x, y\}\). This book covers a lot of classical material in a modern fashion; in particular, it clarifies some obscure definitions, remarks and statements in older papers from the end of the 19th century and the beginning of the 20th century. As prerequisites for reading this book a course in algebra should be enough. The clear presentation of the proofs makes life for a reader very easy; exercises at the end of each chapter are either examples to give the reader the possibility of proving for himself that he has mastered the text or provide new theorems not dealt with in the main text. Taken all together: A book making nice reading; it belongs to the shelf of every mathematician interested in curves and their singularities. Puiseux's theorem; Newton-Puiseux algorithm; infinitely near points; proximate points; satellite and free points; resolution of singularities; equisingularity; semigroup of a curve; characteristic exponents; linear families of germs; clusters; polar germ; formal power series; convergent power series E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser. 276, Cambridge University Press, Cambridge 2000. Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Equisingularity (topological and analytic), Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Germs of analytic sets, local parametrization Singularities of plane curves	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with Grothendieck's standard conjectures on algebraic cycles, introduced independently by Grothendieck and Bombieri to explain the Weil conjectures on the \(\zeta\)-function of algebraic varieties. We prove that the semisimplicity of the algebra of algebraic correspondences \({\mathcal A}^({\mathcal X}\times {\mathcal X})\) of a projective irreducible smooth variety \({\mathcal X}\) implies the standard conjecture of Lefschetz type for \({\mathcal X}\). It was proved by U. Jannsen that the algebra \({\mathcal A}^({\mathcal X}\times {\mathcal X})\) is semisimple when the numerical and homological equivalences of algebraic cycles on \({\mathcal X}\) are the same. Thus, with Jannsen's theorem our result asserts that the standard conjecture of Lefschetz type follows from Grothendieck's conjecture about the equality of the numerical and homological equivalences. This was known before only in the presence of the standard conjecture of Hodge type. algebraic cycles; Weil conjectures; algebraic correspondences; standard conjecture of Lefschetz type doi:10.1007/s002220050140 Algebraic cycles, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Graded rings and modules (associative rings and algebras) Graded associative algebras and Grothendieck standard conjectures	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This booklet contains the author's doctoral dissertation accomplished at the University of Köln, Germany under the guidance by U. Jannsen as academic adviser. Thematically, the research conducted here is devoted to a number of problems linking algebraic geometry over local fields, and that in its motivic-cohomological aspects, and the accompanying theory of local Galois representations. More precisely, the author investigates more closely two related conjectures due to \textit{S. Bloch} and \textit{K. Kato} [in: The Grothendieck Festschrift, Collect. Artic. Honor A. Grothendieck, Vol. I, Prog. Math. 86, 333--400 (1990; Zbl 0768.14001)], on the one hand, and to \textit{U. Jannsen} on the other, which predict certain good properties of the so-called motivic pairs arising from the étale cohomology of smooth and proper arithmetic schemes over \(\mathbb{Q}\). In order to tackle these conjectures, at least so in manageable special cases, the author introduces what is called adelic pairs, describes then the underlying algebraic framework of monodromy representations and Galois representations, and applies the general results obtained here to the geometric case of arithmetic abelian varieties and arithmetic surfaces. The main results of this subtle analysis enable the author to verify the above-mentioned conjectures in those particular cases, which certainly represents a remarkable progress in this current field of research within arithmetic algebraic geometry. Diophantine geometry; local ground fields; étale cohomology; Galois representations; monodromy representations; Abelian varieties; arithmetic surfaces; algebraic geometry over local fields; motivic-cohomological aspects; adelic pairs; arithmetic algebraic geometry Local ground fields in algebraic geometry, Varieties over finite and local fields, Arithmetic ground fields for surfaces or higher-dimensional varieties, Arithmetic ground fields for abelian varieties, Brauer groups of schemes, Arithmetic varieties and schemes; Arakelov theory; heights Finiteness theorems for local Galois representations	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{P. Berthelot} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185--272 (1996; Zbl 0886.14004)] generalized the notion of PD structure in crystalline cohomology theory to that of ``\(m\)-PD structure (PD structure of level \(m\))'' for each natural number \(m\). From this, one can generalize the usual crystalline cohomology theory to \(m\)-crystalline cohomology using a level-\(m\) version of the crystalline site. In this paper, the author proves the boundedness, the base change, and the finiteness of the crystalline cohomology of level \(m\). \textit{B. Le Stum} and \textit{A. Quirós} [J. Algebra 240, 559--588 (2001; Zbl 1064.14015)] proved the exact crystalline Poicaré lemma, namely the ``jet complex of order \(p^m\)'' calculates the crystalline cohomology of level \(m\). Unfortunately, this complex is not bounded and the proof to its local freeness contains a gap. So the finiteness of crystalline cohomology of level \(m\) does not follow from the exact crystalline Poincaré lemma. The main contribution of this paper is to introduce an auxiliary ``de Rham-like'' complex which locally resolves a \textit{direct sum} of finitely many copies of the structure sheaf (as opposed to the structure sheaf itself). This allows the author to prove cohomological boundedness and the base change theorem. Together with the exact crystalline Poincaré lemma of Le Stum and Quirós, the author concludes the finiteness of crystalline cohomology of level \(m\). crystalline cohomology of higher level; Poincaré lemma; PD structure of higher level Miyatani, K.: Finiteness of crystalline cohomology of higher level. Ann. inst. Fourier 65, No. 3, 975-1004 (2015) \(p\)-adic cohomology, crystalline cohomology Finiteness of crystalline cohomology of higher level	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author establishes an analogy between a modern proof of the classical theorem of N. H. Abel giving the rational equivalence classes of divisors \(P\) and \(Q\) of degree \(d\) on a Riemann surface from equality of their images in the Jacobian variety \(J(X)\), on the one hand, and a construction for certain cohomologous curves \(P\) and \(Q\) on a threefold \(Y\) for which the image \(Q- P\) in the intermediate Jacobian \(IJ(Y)\) is zero, on the other hand. After recasting the classical proof of Abel's theorem in the language of differential forms with values in distributions (or currents), the author turns to smooth projective threefolds and constructs, for two suitable effective algebraic one-cycles \(P\) and \(Q\) therein, a rank-2 vector bundle \(E_\infty\) whose first Chern class is trivial and whose second Chern class is represented by \(P\) or by \(Q\). Then, using Chern-Simons theory and following the distribution-theoretic proof of Abel's theorem for divisors on curves, he produces a certain three-form on the underlying threefold, which depends on a fixed quaternionic structure on the bundle \(E_\infty\). If \(P\) and \(Q\) are algebraically equivalent curves and \(P- Q\) is zero in the intermediate Jacobian of the threefold, then this differential three-form ``almost'' gives the rational equivalence of \(P\) and \(Q\). As the author points out, this paper was motivated by some work of R. Thomas [cf.: \textit{S. K. Donaldson} and \textit{R. Thomas}, in: The geometric universe: Science, geometry, and the work of Roger Penrose, 31--47 (1998; Zbl 0926.58003)]. Further details can be found in the author's recent paper [``Cohomology and obstructions II: Curves on \(K\)-trivial threefolds'', Preprint, \texttt{http://arxiv.org/abs/math.AG/0206219}]. intermediate Jacobians; rational equivalence of cycles; vector bundles; connections; currents H. Clemens, An analogue of Abel's theorem (English summary), The legacy of Niels Henrik Abel, Springer-Verlag, Berlin, 2004, pp. 511-530. \(3\)-folds, Picard schemes, higher Jacobians, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Transcendental methods, Hodge theory (algebro-geometric aspects), Currents in global analysis An analogue of Abel's theorem	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with some aspects of the theory of algebraic systems of effective divisors on a non singular curve and of one dimensional systems of curves on a smooth surface. In the case of curves, defined over an algebraically closed field, a short proof is given of a remarkable result due to Allen, which extends to non-linear systems of effective divisors the well known formula of De Jonquières. As an application of Allen's theorem a criterion of linear equivalence by Castelnuovo and Torelli is derived; this, in turn, implies a well known inequality by Castelnuovo and Severi concerning correspondences on a curve. Turning to surfaces and assuming the base field of characteristic zero, a formula is proved, giving a bound for the number of curves in a one dimensional system of generically smooth curves having a singular point. This bound was already given, but with incomplete proof, by Torelli. From this result a criterion of linear equivalence, analogous to the aforementioned one of Castelnuovo and Torelli for curves, is deduced. algebraic systems of effective divisors; correspondences on a curve C. Ciliberto--F. Ghione,Serie algebriche di divisori su una curva e su una superficie, Ann. Mat. Pura Appl.,136 (1984), pp. 329--353. Divisors, linear systems, invertible sheaves, Rational and birational maps Algebraic series of divisors on a curve and on a surface	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) denote a smooth integral complex algebraic curve of genus \(g \geq 2\). Fix points \(x_1, \dots, x_m\) of \(X\) and integers \(n_i\) attached to each \(x_i\). For a semisimple simply connected algebraic group \(G\), parahoric bundles or parahoric \(G\)-torsors on \(X\) are pairs \((E, \theta)\) where \(E\) is a torsor (i.e. a principal homogeneous space) on \(X\) under a parahoric Bruhat-Tits group scheme \(\mathcal{G}\) and weights \(\theta \in Y(T)\otimes {\mathbb Q}\). Here \(Y(T)\) denotes the group of \(1\)-parameter subgroups for a (fixed) maximal torus \(T\) of \(G\). To a set \(\tau\) of conjugacy classes, one associates weights \(\theta_{\tau} = \{ \theta_i \} \in (Y(T)\otimes \mathbb{Q})^m\) and a parahoric group scheme \(G_{\theta_{\tau},X}\) with only ramification points \(x_i\). The authors introduce notions of stability and semistability for parahoric \(G\)-torsors, construct their moduli spaces \(M_X(G_{\theta_{\tau},X})\) and show that they are irreducible, normal, projective varieties. Let \(K\) be a maximal compact subgroup of \(G\). The authors show that there is a Fuchsian group \(\pi\) and a homeomorphism of the space of conjugacy classes of representations of \(\pi\) in \(K\) of local type \(\tau\) onto the moduli space \(M_X(G_{\theta_{\tau},X})\) which identifies the subset corresponding to irreducible representations with the subset of \(M_X(G_{\theta_{\tau},X})\) corresponding to isomorphism classes of stable \(G_{\theta_{\tau},X}\)-torsors. This generalises the work of \textit{V. B. Mehta} and \textit{C. S. Seshadri} on parabolic vector bundles [Math. Ann. 248, 205--239 (1980; Zbl 0454.14006)] to parahoric \(G\)-torsors. Parahoric torsors (without weights) were studied by \textit{G. Pappas} and \textit{M. Rapoport} [Adv. Math. 219, No. 1, 118--198 (2008; Zbl 1159.22010); Adv. Stud. Pure Math. 58, 159--171 (2010; Zbl 1213.14028)], many results on moduli stacks of these torsors were proved by \textit{J. Heinloth} [Math. Ann. 347, No. 3, 499--528 (2010; Zbl 1193.14014)] over arbitrary ground fields. \(G\)-torsors; parahoric bundles; representations of Fuchsian groups; Riemann surface Balaji, V. and Seshadri, C. S. Moduli of parahoric G-torsors on a compact Riemann surface J.~Algebraic Geom.24 (2015) 1--49 Math Reviews MR3275653 Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moduli of parahoric \(\mathcal G\)-torsors on a compact Riemann surface	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first volume of \textit{G. Harder's} ``Lectures on algebraic geometry I. Sheaves, cohomology of sheaves, and applications to Riemann surfaces'' appeared three years ago [Aspects of Mathematics E 35. Wiesbaden: Vieweg (2008; Zbl 1129.14001)] and was predominantly devoted to those fundamental concepts, methods, and techniques which are absolutely indispensable for formulating and treating algebraic geometry in its scheme-theoretic and cohomological language à la A. Grothendieck and his successors. Actually, two thirds of this first volume dealt with the basic prerequisites from general category theory, abstract homological algebra, sheaf theory, and cohomology theory of sheaves, nevertheless so with a steady prevailing view toward their later applications to algebraic geometry, algebraic topology, and complex-analytic geometry. The book under review is the long-announced second volume within the author's respective publication program, which still is intended to culminate in a future third volume devoted to the current active research topic of the cohomology of arithmetic groups. While the first volume presented the first five chapters of the whole treatise, this second volume completes the introduction to modern algebraic geometry by the subsequent five chapters. Each chapter is divided into several sections and subsections, which certainly enhances the overall lucidity of the entire exposition. As for the more precise contents, Chapter 6, the first chapter of the current second volume, is titled ``Basic Concepts of the Theory of Schemes''. The author develops here the language of schemes in its general and abstract setting. This includes affine schemes and general schemes, quasi-coherent sheaves of modules, algebraic vector bundles on schemes, affine and flat morphisms, relative schemes and base change, the various concepts of points in a scheme, representable functors in algebraic geometry, and an introduction to the theory of descend for later use. Chapter 7 provides some basic commutative algebra and its use in algebraic geometry. Among the selected topics discussed here are those which are especially related to geometry and algebraic number theory: finitely generated algebras, low-dimensional rings and basic results from arithmetic, flat morphisms, formal schemes, infinitesimal schemes, regular rings and smoothness, relative differentials, vector fields, derivations, infinitesimal automorphisms, group schemes and basic examples, and actions of group schemes on relative schemes. Chapter 8 gives an introduction to projective geometry, that is, to projective schemes, their special morphisms, their sheaf cohomology, and their specific geometric properties. Among other topics, the reader gets here acquainted with locally free sheaves on \(\mathbb{P}^n_A\), valuative criteria for morphisms, ample and very ample sheaves, the cohomology of quasi-coherent and coherent sheaves, the fundamental finiteness theorems for the latter, the constructions of blowing-up and contracting subschemes, base change properties, basic intersection theory, and Bertini's theorem. Chapter 9 is devoted to algebraic curves and the Riemann-Roch theorem. The main objects of study are here smooth projective curves over an arbitrary ground field, together with their function fields, divisors, and line bundles. The main theme in the first part of this chapter is the classical Riemann-Roch-theorem, together with the Serre duality theorem, which are treated in a very interesting, highly enlightening, and somewhat non-standard manner. In fact, the author's approach uses classical ideas of \textit{R. Dedekind} and \textit{H. Weber} [cf.: ``Theorie der algebraischen Funktionen einer Veränderlichen'', Kronecker J. XCII. 181--291 (1882; JFM 14.0352.01)] and combines them effectively with the modern sheaf-theoretic and cohomological framework. The second part of this chapter discusses modern generalizations and applications of the Riemann-Roch theorem, ranging from the coarse moduli space of elliptic curves to the general idea of moduli spaces for arbitrary smooth projective curves. At the end, the general Riemann-Roch-Grothendieck theorem is explained in a more informal way, with the emphasis on some important special cases. However, the final highlight is the application to curves over finite fields, where the relationship between the Riemann-Roch theorem and the Zeta-function of such a curve is analyzed, on the one hand, and an analogue of the Riemann hypothesis (à la Mattuck-Tate and Grothendieck, 1958) is expounded on the other hand. Chapter 10, the most advanced part of the book, describes another moduli problem in algebraic geometry, namely the representability of the Picard functor for algebraic curves and their Jacobians. Referring to the construction of the Jacobian of a compact Riemann surface as described in Chapter 5 of Volume I, the author first defines the Picard functor on the category of schemes of finite type over a base scheme B, establishes its local representability and the construction of the Picard scheme of a curve, and obtains the Jacobian of a curve as a particular Picard scheme, which turns out to be a connected projective variety equipped with the structure of a commutative group scheme, that is, an abelian variety. In the sequel, the theory of abelian varieties and their Picard schemes is developed, with Jacobians of curves as prototype, and endomorphism rings of Jacobians, the Weil pairing, Néron-Severi groups, and the ring of correspondences are treated just as well as the \(\ell\)-adic Tate modules. As the author points out in the preface to the present volume, the final goal of this book is to bring the reader to the foothills of the mountain range of étale cohomology. This is done in the last section of this concluding chapter, where an outlook to this subject is given. After explaining the basic concepts and some of the fundamental theorems, this section touches upon the famous Weil conjectures for projective varieties over a finite field, Deligne's confirming theorem, and the special case of abelian varieties and of curves. Finally, the study of a degenerating family of elliptic curves serves as an illustrating, highly instructive example for the nature of the Weil conjectures and for compactifications of moduli spaces, respectively. All together, this second volume of [loc. cit.] stands out by the same features that already distinguished the foregoing Volume 1. Again, the very special and individual disposition of the subject matter, the purposeful arrangement of the material, the broad spectrum of topics and their interrelations, the great clarity of exposition, the topicality, the mathematical depth, and the reader-friendly style of writing make this text a particularly valuable enrichment of the existing course book literature in algebraic geometry and its arithmetic aspects. algebraic geometry (textbook); schemes and morphisms; projective schemes; algebraic curves; Jacobians; Picard functor; abelian varieties; étale cohomology Harder, G.: Algebraic Geometry. Vieweg. Manuscript (to appear) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Relevant commutative algebra, Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies, Jacobians, Prym varieties, Algebraic theory of abelian varieties Lectures on algebraic geometry II. Basic concepts, coherent cohomology, curves and their Jacobians	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a smooth algebraic curve of genus \(g\) with Brill-Noether number \(\varrho(g,r,d) = 0\), a finite number of linear series of degree \(d\) and dimension \(r\) over \(C\) exists. A classical result by Castelnuovo counts these series and a generalization by \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 90, 359--387 (1987; Zbl 0631.14023)] provides a formula for the number of linear series with a prescribed vanishing sequence \(a: 0 \leq a_0 \leq \cdot \leq a_r \leq d\) at a fixed general point~\(p\). In this article, the authors consider curves with adjusted Brill-Noether number \(\varrho(g,r,d,a) = -1\) and count the number of pairs of points \(p\) and linear series with prescribed vanishing sequence \(a\) at \(p\). Pointed curves with \(\varrho(g,r,d,a) = -1\) at the marked point form a divisor in the moduli stack \({\mathcal M}_{g,1}\) and the authors compute classes of closures in \(\overline{\mathcal M}_{g,1}\) of such divisors. Moreover, they show that classes of Brill-Noether loci of codimension two are generally not proportional. For Brill-Noether divisors, this is known to be true [\textit{D. Eisenbud} and \textit{J. Harris}, Invent. Math. 90, 359--387 (1987; Zbl 0631.14023)]. Brill-Noether theory; linear series; vanishing sequence; enumerative geometry; Castelnuovo numbers Farkas, G; Tarasca, N, Pointed Castelnuovo numbers, Math. Res. Lett., 23, 389-404, (2016) Special divisors on curves (gonality, Brill-Noether theory), Computational aspects of algebraic curves Pointed Castelnuovo numbers	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a field of characteristic different from 2 and \(K_a\) its algebraic closure. The main result proved in the paper is the following: Let \(f(x)\in K[x]\) be an irreducible and separable polynomial of degree \(n\geq 5\) whose Galois group is \(S_n\) or \(A_n\) and let \(J(C_f)\) be the Jacobian of the hyperelliptic curve of equation \(y^2=f(x)\) whose \(K_a\)-endomorphism ring is denoted by \(\text{End} (J(C_f))\). Then either \(\text{End} (J(C_f))= \mathbb{Z}\) or \(\text{char}(K)>0\) and \(J(C_f)\) is a supersingular abelian variety. The proof is based on the study of representation of \(\text{End} (J(C_f))\) in the \(\mathbb{F}_2\)-vector space of 2-torsion points of \(J(C_f)\). hyperelliptic Jacobians; complex multiplication; representations; endomorphism ring Yu. G. Zarhin, Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett. 7 (2000), 123--132. Jacobians, Prym varieties, Complex multiplication and abelian varieties, Abelian varieties of dimension \(> 1\) Hyperelliptic Jacobians without complex multiplication	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this work we consider the relation between the residues of a Weil differential \(\omega\) on a non-singular projective algebraic curve \(C\) over a perfect constant field. Let \(P\) is a plase of curve \(C\) of degree \(n, F_P\) is the residue class field, which is a finite Galois extension of constant field \(K, F=K(C)\) is the field of rational functions on \(C, F'=F \cdot F_P\) is the constant field \(F\) by means of \(F_P\). Then in \(F'\) there exist exactly \(n\) places \(Q_1, \dots, Q_n\) lying over \(P\). The degree of \(Q_i\) is one, for any \(i:1 \leq i \leq n\). Moreover \(\text{Re}s_{P} (\omega)=\sum \limits_{i=1}^{n} \text{Re}s_{Q_{i}}(\omega)\), and a local component \(\omega_P\) of the Weil differential \(\omega\) in \(P\) can be viewed as \(\omega_{P}(u)=\text{Re}s_{P}(u \cdot \omega)\) for any \(u \in F\), that gives a simple proof of the Residue Theorem. Algebraic functions and function fields in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials About residues of Weil differentials in a constant field extension of an algebraic curve.	0
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a survey article devoted to recent developments in constructing moduli spaces of geometric objects such as vector bundles or principal \(G\)-bundles on a fixed smooth projective variety defined over an algebraically closed field \(k\) of arbitrary chracteristic. In view of the tremendous recent progress achieved in this field of contemporary research, the author had to focus on a few selected aspects of moduli theory. Thus, according to both his taste and expertise, the author has chosen the following guiding principles for the current survey article: 1. Analytic techniques are throughout avoided, and only a purely algebraic part of the theory is discussed, with special emphasis on the case of positive characteristics. 2. The main objective is to present some recent constructions of certain moduli spaces, that is, to establish their very existence, whereas any comments concerning their (mostly still unknown) geometry have been largely neglected. 3. Methodologically, the framework of Geometric Invariant Theory (GIT) is generally adopted, and therefore the classes of geometric objects to be classified are chosen with respect to their feasibility within this framework. With these principles in mind, the author has divided this survey into two major parts. The first part (Sections 2 and 3) explains the theory underlying the construction of moduli spaces of semistable (torsion-free) sheaves on a smooth projective variety \(X\). This includes the concept of Gieseker semistability, slope semistability in characteristic zero, and strong slope semistability in positive characteristic likewise, together with some old and new results on the basic properties of semistable sheaves such as Bogomolov's inequality, restriction theorems, and dimension bounds for their cohomology groups. Along the way, the author explains his own recent contributions in the case of positive characteristic in greater detail [cf. \textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)]. These basic results allow to apply the standard GIT methods to construct the corresponding moduli spaces as projective schemes right away. The second part (Sections 4 and 5) turns to the construction of (semistable) principal \(G\)-bundles. Historically, this theory was initiated by \textit{A. Ramanathan} in his Ph.D.thesis at the Tata Institute, Mumbai, mainly in the case of compact Riemann surfaces, Math. Ann. 213, 129--152 (1975; Zbl 0284.32019). In the last few years, this theory (in characteristic zero) was extended to higher-dimensional projective varieties by \textit{T. Gómez} and \textit{I. Sols} [Moduli space of principal-sheaves over projective varieties, Ann. Math. (2) 161, No. 2, 1037--1092 (2005; Zbl 1079.14018)], on the one hand, and by \textit{A. H. W. Schmitt} [Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.14001)] on the other. But only very recently, the construction of compactified moduli spaces of semistable principal \(G\)-bundles on smooth projective varieties in arbitrary characteristic has been accomplished by \textit{T. L. Gómez}, \textit{A. Langer}, \textit{A. H. W. Schmitt} and \textit{I. Sols} [Moduli spaces for principal bundles in large characteristic, Adv. Math. 219, No. 4, 1177--1245 (2008; Zbl 1163.14008)], and a fairly detailed sketch of their novel approach is the main topic of the second part of the paper. Also, it is carefully explained how this final result is based upon the author's foregoing work in this direction [cf. \textit{A. Langer}, Semistable principal \(G\)-bundles in positive characteristic, Duke Math. J. 128, No. 3, 511-540 (2005; Zbl 1081.14018)]. Written in a very lucid and panoramic style, the survey under review effectively combines classical and utmost topical aspects of algebraic moduli theory, thereby providing an excellent source for further reading in a field at the forefront of current research in algebraic geometry. survey articles (algebraic geometry); algebraic moduli problems; moduli of semistable sheaves; principal \(G\)-bundles; geometric invariant theory; moduli of vector bundles A. Langer, Moduli spaces of sheaves and principal \textit{G}-bundles, Algebraic geometry. Part 1 (Seattle 2005), Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 273-308. Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vanishing theorems in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Geometric invariant theory Moduli spaces of sheaves and principal \(G\)-bundles	0