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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 The object of this paper is the Grothendieck tame fundamental group \(\pi_ 1^ D(A)\) of an effective reduced divisor D on an abelian variety A over an algebraically closed field of arbitrary characteristic. This group fits into the exact sequence \(0\to I\to \pi_ 1^ D(A)\to \pi_ 1(A)\to 0\) with subgroup I called the inertia subgroup. The main result (theorem 5.3) is the calculation of the abelianization of I (i.e. the quotient by the closure of the commutator subgroup). This abelianization is found together with the natural structure of a module over the Tate module T(A) of A which comes from the identification (due to Serre and Lang) of \(\pi_ 1(A)\) with T(A) and the exact sequence above. More precisely if \({\mathbb{Z}}^{\vee}\) is the completion of \({\mathbb{Z}}\) by subgroups of index prime to \(p\) then the abelianization of the inertia subgroup is isomorphic to the sum over all irreducible components \(D_ i\) of the divisor D of the modules \({\mathbb{Z}}^{\vee}[[T(A/stab^ 0_ AD_ i)/T_ i]]\) where \(stab^ 0_ AD_ i\) is the connected component of the identity stabilizer of the component \(D_ i\) and \(T_ i\) is a closed subgroup of the Tate module \(T(A/stab^ 0_ AD_ i)\). Moreover if the codimension of the stabilizer is one then the corresponding subgroup \(T_ i\) of the Tate module is trivial. If the codimension of the stabilizer is greater than one and \(D_ i\) is geometrically unibranched then \(T_ i=T(A/stab^ 0_ AD_ i)\). As part of the proof the author has a series of results on the abelianized inertia subgroup for divisors on arbitrary smooth projective schemes.
These results are used for clarification of interrelationship between the following two conjectures. Conjecture B [E. Bombieri and \textit{S. Lang}, Bull. Am. Math. Soc. 14, 159-205 (1986; Zbl 0602.14019)]: Let V be a projective variety over an algebraic number field k. Then there is a non empty open subscheme U of V for which \(U(k)=\emptyset\). - Conjecture A [\textit{S. Lang}, Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)]: Let us call a subset S of k-rational points on A a D- integral with respect to a domain R which is a finitely generated \({\mathbb{Z}}\)-algebra, if there is \(d\in R\) such that for a basis \(x_ i\) of \(\Gamma\) (X,\({\mathcal O}(nD))\) one has \(x_ i(P)\in (1/d)R\). If D is an ample divisor on an abelian variety A/k then every D-integral subset with respect to a domain R of the group of rational points A(k) is finite. The author shows that conjecture B implies conjecture A. For this he needs coverings of the abelian variety A branched over D which will be varieties of general type. Existence of these coverings is derived from the aforementioned main result of this work. Grothendieck tame fundamental group; effective reduced divisor; abelian variety; abelianization; Tate module; integral subset with respect to a domain; group of rational points Brown, M.L.: The tame fundamental group of an abelian variety and integral points. Compos. Math.72, 1--31 (1989) Homotopy theory and fundamental groups in algebraic geometry, Arithmetic ground fields for abelian varieties, Rational points, Coverings in algebraic geometry The tame fundamental group of an abelian variety and integral 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 The purpose of this paper is to extend \textit{Top}'s result to the case of the superelliptic curves. Let \(p\) be a prime number, \(\zeta_ p\) a primitive \(p\)-th root of unity, and set \(K=\mathbb{Q} (\zeta_ p)\). Denote by \({\mathfrak O}_ K\) the ring of integers of \(K\). Let \(f \in {\mathfrak O}_ K[X]\) be a separable polynomial such that the degree of \(f\), denoted by \(n\), is prime to \(p\) and \({1 \over 2} (p-1)\) \((n-1) \geq 1\). Let \(C\) be a smooth projective model of the curve given by \(y^ p=f(x)\) and let \(J\) be the jacobian variety of \(C\). For every finite extension \(M\) over \(K\) we denote by \(J(M)\) the Mordell-Weil group of \(J\) over \(M\). Main theorem:
For every \(m \geq 1\) one can explicitly construct infinitely many extensions of \(K\) of the form \(L=K(\root p \of {d_ 1},\dots,\root p \of {d_ m})\) for which rank \((J(L)) \geq \text{rank} (J(K))+(p-1)m\). -- In the case of \(p=2\), this reduces to Top's theorem. Mordell-Weil rank of the jacobians; superelliptic curves; Mordell-Weil group Murabayashi, N.: Mordell -- Weil rank of the Jacobians of the curves defined by \(yp=f(x)\). Acta arith. 64, No. 4, 297-302 (1993) Rational points, Jacobians, Prym varieties, Elliptic curves, Arithmetic ground fields for surfaces or higher-dimensional varieties Mordell-Weil rank of the jacobians of the curves defined by \(y^ p=f(x)\) | 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 contains a number of results involving Cartier points on curves, foremost among them a strengthening of Ekedahl's theorem that a superspecial curve in characteristic \(p>0\) has genus at most \(p(p-1)/2\) if it is non-hyperelliptic, and at most \((p-1)/2\) if it is hyperelliptic.
Suppose \(C\) is a curve of positive genus over an algebraically closed field \(k\) of characteristic \(p\). A point \(P\) on \(C\) is a ``Cartier point'' of \(C\) if the Cartier operator takes the vector space of meromorphic differentials on \(C\) vanishing at \(P\) to itself. The main theorem of this paper states that
(1) if \(C\) has Cartier points \(P_1,\ldots, P_p\) such that the divisor \(p(P_i - P_j)\) is non-principal for every \(i\neq j\), then the genus of \(C\) is at most \(p(p-1)/2\); and
(2) if \(p>2\) and \(C\) is hyperelliptic, and if a hyperelliptic branch point of \(C\) is a Cartier point, then \(C\) has genus at most \((p-1)/2\).
Ekedahl's result follows easily from this, because if a curve \(C\) is superspecial then the Cartier operator acts as \(0\) on the meromorphic differentials, so that every point is a Cartier point. The author notes that his proof of Ekedahl's theorem does not rely on Nygaard's characterization of the Jacobians of superspecial curves.
The author also provides an explicit bound (in terms of \(g\) and \(p\)) on the number of Cartier points on a non-superspecial genus-\(g\) curve over a field of characteristic \(p\), and gives several examples of curves for which the Cartier points can be computed explicitly. For example, he shows that in characteristic \(3\) the modular curves \(X_0(43)\) and \(X_0(61)\) have no Cartier points. Cartier operator; superspecial curve; characteristic \(p\); genus; meromorphic differentials; number of Cartier points Baker, M.: Cartier points on curves. Internat. math. Res. notices 7, 353-370 (2000) Special algebraic curves and curves of low genus, Finite ground fields in algebraic geometry, Commutative rings of differential operators and their modules, Arithmetic ground fields for curves Cartier points on 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 aim of the paper is to establish comparison results between étale cohomology of schemes and rigid analytic varieties (in the sense of Fujiwara and Huber).
After the introduction, the second section of the paper deals with \textit{K. Fujiwara} spaces [Duke Math. J. 80, No. 1, 15--57 (1995; Zbl 0872.14014)]. Let \(S\) be a scheme, \(S_{0}\) a closed subscheme of it and~\(S^0\) its complement. To an \(S^0\)-morphism \(f: X \to Y\) between \(S_{0}\)-schemes of finite type and an object \(\mathcal{F}\) in \(D^b(X,\mathbb{Z}/\ell^n\mathbb{Z})\) is naturally associated a comparison morphism \(\varepsilon^*Rf_{*}\mathcal{F} \to Rf_{*}^{\text{an}} \varepsilon^* \mathcal{F}\), where \(\varepsilon\) is the natural morphism between the étale sites. In [loc. cit.], Fujiwara proved that it is an isomorphism when \(f\) is proper. The first main result of the paper asserts that it is still an isomorphism when \(S\) is of finite type over of field \(k\) of characteristic different from \(\ell\) and \(\mathcal{F}\) is in \(D_{c}^+(X,\mathbb{Z}/\ell^n\mathbb{Z})\). The author begins by studying how the comparison morphism behaves with respect to base-change and uses this to reduce the statement to known cases.
The third section deals with Huber's adic spaces. Let \(V\) be a complete discrete valuation ring with equal characteristic. Let \(S\) be a scheme of finite type over \(V\) and \(S_{0}\) be a closed subscheme of is closed fiber. Then, for every \(S\)-morphism \(f: X \to Y\) between \(S\)-schemes of finite type and every \(\mathcal{F}\) in \(D_{c}^+(X,\mathbb{Z}/\ell^n\mathbb{Z})\), with \(\ell\) invertible in \(V\), the comparison morphism is an isomorphim. Note that the case where \(f\) is proper (with no assumption of \(S\)) was proved by \textit{R. Huber} [Étale cohomology of rigid analytic varieties and adic spaces. Wiesbaden: Vieweg (1996; Zbl 0868.14010)]. Mieda's proof uses the same arguments as in the previous section together with results of Huber to handle open immersions and de Jong alterations.
In the fourth section, the author defines and studies nearby cycles for adic spaces. As a corollary of his previous results, he proves a comparison isomorphism \(\varepsilon^*R\psi_{X}\mathcal{F} \to R\Psi_{t(\mathcal{X})}\varepsilon^* \mathcal{F}\), where \(X\) is a scheme of finite type over \(V\), \(\mathcal{X}\) its completion along a closed subscheme \(Y\) and \(t(\mathcal{X})\) the adic space associated to it. At the very end of the paper, the author explains that the nearby cycles he constructed have a richer structure than \textit{V. G. Berkovich}'s (see [Invent. Math. 125, No. 2, 367--390 (1996; Zbl 0852.14002)]) and this additional information will be used in a forth-coming paper with Ito to study the \(\ell\)-adic cohomology of the Rapoport-Zink tower for \(\mathrm{GSp}(4)\). étale cohomology; comparison theorem; Fujiwara spaces; adic spaces; nearby cycles Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies Comparison results for étale cohomology in rigid 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 \(G\) be a reductive group scheme over a base scheme \(S\), \(T\) an \(S\)-torus and \(\Psi\) a twisted root datum associated to \(T\). The paper under review studies closed embeddings of \(T\) into \(G\) that respect the given root datum. The author defines the embedding functor \(\mathfrak{E}(G,\,\Psi)\) whose points parametrize these embeddings. She proves that this functor is a left homogeneous space under the automorphism group of \(G\) and is represented by an \(S\)-scheme. It is also shown that \(\mathfrak{E}(G,\,\Psi)\) is a torsor over the scheme \(\mathcal{T}\) of maximal tori of \(G\) under the right action of the automorphism group of \(\Psi\).
After having defined orientations of root data, the author introduces an oriented embedding functor, and proves a similar representability theorem for this functor. This functor has a structure of left homogeneous space under the adjoint action of \(G\) and a structure of torsor over \(\mathcal{T}\) under the action of the Weyl group of \(\Psi\).
Also explained in the paper are the relations between points of the embedding functors and certain embedding problems of Azumaya algebras with involution. Over global fields, various local-global principles for those embedding problems are obtained by using \textit{M. Borovoi}'s results on local-global principles for homogeneous spaces [Math. Ann. 314, No. 3, 491--504 (1999; Zbl 0966.14017)]. In particular, some results of [\textit{G. Prasad} and \textit{A. S. Rapinchuk}, Comment. Math. Helv. 85, No. 3, 583--645 (2010; Zbl 1223.11047)] are improved. torus; reductive group; root datum; embedding problem; central simple algebra; local-global principle Lee, T.-Y., Embedding functors and their arithmetic properties, Comment. Math. Helv., 89, 671-717, (2014) Classical groups, Group schemes, Group actions on varieties or schemes (quotients), Classical groups (algebro-geometric aspects), Linear algebraic groups over global fields and their integers Embedding functors and their arithmetic properties | 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 result of the interesting paper is the proof of Broué's conjecture concerning Deligne-Lusztig curves. Let \(G\) be a reductive algebraic group over an algebraic closure \(k_{p}\) of a prime finite field \({\mathbb F}_{p}\), \(p \neq l, F:G \rightarrow G\) the Frobenius endomorphism. Let \(T\) be a maximum \(F\)-stable torus, \(B\) a Borel subgroup containing \(T\) and \(U\) the unipotent radical of \(B.\) For the Lang isogeny \( {\mathcal L}: G \rightarrow G, g \mapsto g^{-1}F(g)\) the Deligne-Lusztig variety is defined by \(Y = Y_{G}(U) = ({\mathcal L}^{-1}(F(U))/(U \cap F(U)).\) The author of the paper under review shows that if
\[
\Hom_{\text{Ho}({\mathbb F}_{l}G)}({\widetilde R}\Gamma(Y,{\mathbb F}_{l}), \widetilde{R}\Gamma(Y,{\mathbb F}_{l})[i]) = 0
\]
for \(i \neq 0\) then Broué's conjecture is true (Theorem 4.5). The conjecture is formulated in the article by \textit{M. Broué} and \textit{G. Malle} [Astérisque 212, 119-189 (1993; Zbl 0835.20064))] and is defined more precisely in the article by \textit{M. Broué} and \textit{J. Michel} [in: Finite reductive groups, Prog. Math. 141, 73-139 (1997; Zbl 1029.20500)].
The organization of the paper under review is as follows: 1. Introduction, 2. Lifting, 3. Profinite groups and coverings of affine line, 4. Deligne-Lusztig curves. In the second section the author lifts construction of functors between derived categories of étale sheaves over schemes with a sheaf of algebras to pure derived categories. Theorem 2.29 of the section generalizes theorem 4.2 by \textit{J. Rickard} [Publ. Math., IHÉS 80, 81-94 (1994; Zbl 0838.14013)]. Section 3 contains results about some profinite groups and cohomology of étale covers of the affine line \({\mathbb A}^{1}(k_{p}).\) étale chain complexes; reductive algebraic group; Deligne-Lusztig variety; splendid equivalence; proof of Broué's conjecture; étale sheaves; pure derived categories Raphaël Rouquier, Complexes de chaînes étales et courbes de Deligne-Lusztig, J. Algebra 257 (2002), no. 2, 482 -- 508 (French, with English summary). Chain complexes (category-theoretic aspects), dg categories, Group actions on varieties or schemes (quotients), Étale and other Grothendieck topologies and (co)homologies, Derived categories, triangulated categories Étale chain complexes and Deligne-Lusztig 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 well known Hasse--Weil bound, equivalent to the Riemann Hypothesis, improved by J. P. Serre, establishes that the number \(N(C)\) of rational points on a curve \(C\) of genus \(g\) defined over \({\mathbb F}_q\) satisfies \(|N(C) -q-1|\leq g[2\sqrt{q}]\). Here a curve over \({\mathbb F}_q\) is an absolutely irreducible nonsingular projective algebraic variety of dimension \(1\) over \({\mathbb F}_q\). The \textit{discriminant} \(d({\mathbb F}_q)\) is defined by \(m^2-4q\), where \(m=[2\sqrt{q}]\). A curve \(C\) of genus \(g\) over a finite field \({\mathbb F}_q\) is called a \textit{maximal} (resp. \textit{minimal}) curve if \(N(C)\) attains the upper (resp. lower) Hasse--Weil--Serre bound: \(q+1\pm g[2\sqrt{q}]\). The main object of this paper is the study of maximal and minimal curves of low genera over fields with discriminant \(d({\mathbb F}_q)\in \{-3,-4,-7,-8\}\). It turns out that such curves are ordinary.
The main result of this paper is the following:
Theorem. If \(C\) is a curve of genus \(g\) over \({\mathbb F}_q\), then \(|N(C)-q-1|\leq g[2\sqrt{q}]-2\) if \(q\) and \(g\) satisfy:
\noindent \(\bullet\;d({\mathbb F}_q)=-3, q\neq 3, 3\leq g \leq 10\) (Theorem 3.2);
\noindent \(\bullet\;d({\mathbb F}_q)=-4, q\neq 2, 3\leq g \leq 10\) (Theorem 3.4);
\noindent \(\bullet\;d({\mathbb F}_q)=-7, 4\leq g \leq 8\) (Theorem 3.7);
\noindent \(\bullet\;d({\mathbb F}_q)=-8, \roman{char}({\mathbb F}_q)\neq 3, 3\leq g \leq 7\) (Theorem 3.9).
The method employed by the author is the use of the explicit classification of Hermite lattices by \textit{A. Schiemann} [J. Symb. Comput. 26, No. 4, 487--508 (1998; Zbl 0936.68129)]. curves over finite fields; the Hasse--Weil--Serre bound; maximal and minimal curves Arithmetic ground fields for curves, Arithmetic theory of algebraic function fields An improvement of the Hasse-Weil-Serre bound for curves over some finite 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 We give a moduli-theoretic proof of the classical theorem of \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323--448 (1962; Zbl 0201.35602)], stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statements for categories of sheaves twisted by a Gm-gerbe. Our methods allow us to treat even gerbes not coming from a Brauer class. As a pleasant consequence, we deduce a Morita theory for sheaves of abelian categories. reconstruction theorem Calabrese, J; Groechenig, M, Moduli problems in abelian categories and the reconstruction theorem, Algebra. Geom., 2, 1-18, (2015) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grothendieck categories, Generalizations (algebraic spaces, stacks), Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles Moduli problems in abelian categories and the reconstruction 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 A well known theorem due to \textit{D. O. Orlov} [J. Math. Sci., New York 84, 1361--1381 (1997; Zbl 0938.14019)] asserts that any equivalence \(F\) of derived categories of sheaves on smooth projective varieties \(X\) and \(Y\) is representable, i.e. there exists an object \(e\) of \(D(X \times Y)\) such that \(F\) is the integral functor \(\Phi^e\) associated to \(e\). In the paper under review the author extends this result to derived categories of sheaves on smooth stacks. More precisely he proves that, given stacks \(\mathcal{X}\) and \(\mathcal{Y}\) associated to normal projective varieties with quotient singularities \(X\) and \(Y\), any exact fully faithful functor \(F\) admitting a left adjoint is representable by a unique object (up to isomorphism).
The first step is the construction of a (possibly infinite) Beilinson-type resolution of the diagonal over \(X \times X\). It is pointed out here that this construction agrees with \textit{A. Canonaco}'s resolution for weighted projective spaces [J. Algebra 225, 28--46 (2000; Zbl 0963.14007)]. After proving some boundedness results, the author uses this resolution to define the representing object \(e\). The isomorphism between the integral functor \(\Phi^e\) and the original functor \(F\) is first constructed on a spanning class of locally free sheaves.
Some applications are also considered. The first one is a comparison of numerical invariants for varieties with quotient singularities having equivalent derived categories. The second one is an extension to orbifolds of \textit{A. Bondal} and \textit{D. Orlov}'s reconstruction theorem for varieties with ample (anti)canonical bundle [Compos. Math. 125, 327--344 (2001; Zbl 0994.18007)]. Fourier-Mukai transform; integral functors; representable functors; sheaves on orbifolds; weighted projective spaces Kawamata, Y., \textit{equivalences of derived categories of sheaves on smooth stacks}, Amer. J. Math., 126, 1057-1083, (2004) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Generalizations (algebraic spaces, stacks), Derived categories, triangulated categories Equivalences of derived categories of sheaves on smooth 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 authors study \({\mathbb F}_{q^2}\)-maximal curves when \(q\) is even, that is non-singular geometrically irreducible projective algebraic curves \(\mathcal X\) over \({\mathbb F}_{q^2},\) whose number of \({\mathbb F}_{q^2}\)-rational points attains the Hasse-Weil bound \(q^2+1+2qg\), where \(g\) is the genus of \(\mathcal X.\)
For general \(q\) it is known that either \(g=q(q-1)/2,\) in which case up isomorphism \(\mathcal X\) is the Hermitian curve, or \(g\leq g_2:=\lfloor (q-1)^2/4\rfloor.\) For \(q\) odd it was recently shown in joint work by the second author, that up to isomorphism there is a unique \({\mathbb F}_{q^2}\)-maximal curve of genus \(g,\) with \((q-1)(q-2)/4<g\leq g_2.\) This curve is the non-singular model determined by \(y^q+y=x^{(q+1)/2}\) and has \(g=g_2.\)
In this paper the authors extend this result to even characteristic, provided there exists a point \(P\in {\mathcal X}\) such that \(q/2\) is a Weierstrass non-gap at \(P.\) Making this assumption they show that for \(q=2^t,\) if \((q-1)(q-2)/4<g\leq g_2=q(q-2)/4,\) then \(\mathcal X\) is \({\mathbb F}_{q^2}\)-isomorphic to the non-singular model determined by \(\sum_{i=1}^t y^{q/2^i} = x^{q+1},\) and here again \(g=g_2.\)
The paper is well written, using special properties of maximal curves and their linear systems, in particular properties of Frobenius orders and the theory of Weierstrass points. maximal curves in characteristic two; projective algebraic curves; Frobenius orders; Weierstrass points M. Abdón; F. Torres, Maximal curves in characteristic two, Manuscripta Math., 99, 39, (1999) Curves over finite and local fields, Riemann surfaces; Weierstrass points; gap sequences, Arithmetic ground fields for curves, Arithmetic theory of algebraic function fields On maximal curves in characteristic 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 book is the second, revised and enlarged edition of the authors' book [Local cohomology. An algebraic introduction with geometric applications, Cambridge University Press (1998; Zbl 0903.13006)]. As it was mentioned in the review on the first edition (see Zbl 1133.13017) the intention of the authors was twofold: (1) There was a challenge for an algebraic introduction to Grothendieck's local cohomology theory originally invented by the aid of scheme theory (see \textit{A. Grothendieck} [Séminaire de géométrie algébrique par Alexander Grothendieck 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Fasc. I. Exposés I à VIII; Fasc. II. Exposés IX à XIII. 3ieme édition, corrigée. Bures-Sur-Yvette (Essonne): Institut des Hautes Études Scientifiques (1962; Zbl 0159.50402)]). (2) To present an introduction designed primarly to graduate students with some basic knowledge on homological and commutative algebra. -- Because of the authors' masterpiece for clear, complete and selfcontained writing the first edition became a well accepted and well studied textbook on the subject. The appearance of the book filled a large gap for textbooks as one might see that it is quoted in more than 320 research papers as a basic reference. For young researchers it opened the way for serious research on the vital field of local cohomology. (Note that during the last decades there are more than 500 research articles with the phrase ``local cohomology'' in the title.) Fifteen years after the first edition the authors decided to reflect on how they could change and extend the material in order to enhance its usefulness to the interested readers.
As a reaction of the recent developments the authors decided to include a new chapter devoted to the study the canonical module of a local ring \((R,\mathfrak{m})\). The canonical module is defined in the case of a factor ring of a Gorenstein ring. Its completion is the Matlis Dual of \(H^d_{\mathfrak{m}}(R), d = \dim R,\) for the \(d\)-th local cohomology of \(R\) with respect to the maximal ideal \(\mathfrak{m}\). Note that \(H^d_{\mathfrak{m}}(R)\) is (by Grothendieck's Theorems) the last non-vanishing local cohomology module of the family \(\{H^i_{\mathfrak{m}}(R)\}_{i\in \mathbb{Z}}\). It covers several local information of \(R\). In the first edition the authors were mainly interested in the Artinian \(R\)-module \(H^d_{\mathfrak{m}}(R)\) in particular when \(R\) is a Cohen-Macaulay ring. Here they decided to study the canonical module, which is finitely generated for \(R\). The authors study also the \(S_2\)-fication as the endomorphism ring of the canonical module. A second change, in contrast to the first edition concerns the chapters on gradings and graded local cohomology. Because of the usefulness of \(\mathbb{Z}^n\)-gradings in several applications the authors extend most of their results on \(\mathbb{Z}\)-gradings to the more general case of \(n > 1\). The authors illustrate their new perspective in particular to Stanley-Reisner rings. By the work of Hochster and Huneke characteristic \(p\)-methods in commutative algebra and local cohomology became an important tool during the last decades. As the key point of these methods there is an additional subsection on the so-called Frobenius action on the local cohomology modules \(H^i_I(R)\) for an ideal \(I\) in a ring \(R\) of prime characteristic \(p\). This is used in another additional subsection in order to include Hochster's proof of the Monomial Conjecture in the case of prime characteristic. A further new subsection is on locally free sheaves, where Serre's Cohomological Criterion for Local Freeness, Horrock's splitting Criterion and Grothendieck's Splitting Theorem are proved. The authors omitted also some items that they now consider no longer command sufficiently compelling reasons for inclusion (among them applications of local duality, a priori bounds of diagonal type on Castelnuovo-Mumford regularity). Minor changes concern the treatment of Faltings' Annihilator Theorem, the graded Delingne Isomorphism). The Chapters ``Links with projective varieties'', ``Castelnuovo regularity'', ``Connectivity in algebraic geometry'', ``Links with sheaf cohomology'' built a bridge to algebraic geometry. They are intended to graduate students as a starting point for geometric applications as anounced in the title of the book.
For an interested reader the book opens the view towards the beauty of local cohomology not as an isolated subject but as a tool helpful in commutative algebra and algebraic geometry.
Reviewer's remark: As a source for further applications not covered in the book under review like Gröbner bases, algorithmic aspects, \(D\)-modules, DeRham cohomology a.o. one might also see [\textit{S. B. Iyengar} et al., Twenty-four hours of local cohomology. Providence, RI: American Mathematical Society (AMS) (2007; Zbl 1129.13001)]). local cohomology; ideal transforms; vanishing theorems; canonical modules; connectivity; finiteness theorems; Castelnuovo-Mumford regularity; sheaf cohomology; locally free sheave Brodmann, M. P.; Sharp, R. Y., Local cohomology, \textrm{An algebraic introduction with geometric applications}, Cambridge Studies in Advanced Mathematics 136, xxii+491 pp., (2013), Cambridge University Press, Cambridge Local cohomology and commutative rings, Local cohomology and algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Homological methods in commutative ring theory Local cohomology. An algebraic introduction with geometric 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 Let \(X\) be a curve over the finite field \(\mathbb{F}_ q\) \((q = p^ m)\) with \(\infty \in X\) a fixed closed point; let \(A\) be the affine ring of \(X - \infty\). In his fundamental paper ``Elliptic modules'' [Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014), \textit{V. G. Drinfel'd} introduced elegant analogs (now called ``Drinfeld modules'') of elliptic curves and abelian varieties for \(A\). Although various instances of this theory go back to L. Carlitz in the 1930's, Drinfeld's seminal paper marked the beginning of the modern theory of function fields over finite fields. In particular, Drinfeld was able to give a moduli theoretic construction of the maximal abelian extension of \(k\) (= fraction field of \(A)\) which is split totally at \(\infty\), as well as to give a Jacquet-Langlands style two-dimensional reciprocity law where the infinite component is a Steinberg representation. The two-dimensional reciprocity law is established by decomposing the étale cohomology of the compactified rank 2 moduli scheme. The impediment to implementing this procedure for arbitrary \(d>2\) is the difficulty of obtaining a good compactification in general.
The operator \(\tau : x \mapsto x^ q\) satisfies many analogies with the classical differentiation operator \(D : = {d \over dx}\) and these analogies go surprisingly deep. A prime example of this was given in 1976 when Drinfeld found an interpretation of a Drinfeld module \(\varphi\) in terms of a locally free sheaf \({\mathcal F}\) on \(X\) (of the same rank as \(\varphi)\) with maps relating \({\mathcal F}\) and its twist by the Frobenius map; this sheaf-theoretic interpretation being analogous to results of I. Krichever on differential operators. An excellent reference for all this is a paper by \textit{D. Mumford} [in Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 115-153 (1977; Zbl 0423.14007)]. Such locally free sheaves are examples of ``shtuka,'' ``\(F\)-sheaves,'' or ``elliptic sheaves'' -- this last being the notation used in the paper being reviewed. More generally, shtuka appear when the axioms in the ``elliptic modules \(\leftrightarrow\) elliptic sheaves'' dictionary are weakened a bit.
The paper under review is a very nice summary of the important work of \textit{G. Laumon}, \textit{M. Rapoport} and \textit{U. Stuhler} on ``\({\mathcal D}\)- elliptic sheaves and the Langlands correspondence'' [cf. Invent. Math. 113, No. 2, 217-338 (1993)] where \({\mathcal D}\) is a central simple algebra over \(k\). These can be viewed, at least to first order, as ``elliptic sheaves with complex multipliction by \({\mathcal D}\)'' (and, among other axioms the dimension of \({\mathcal D}\) must also be the rank of the elliptic sheaf). A notion of ``level structure'' can be given generalizing that for Drinfeld modules. The point is that such \({\mathcal D}\)-elliptic sheaves have good, smooth, moduli spaces, and, when \({\mathcal D}\) is a division algebra, this moduli is projective (think of the theory of Shimura curves). Thus and importantly, the problems of noncompact moduli spaces are avoided. In particular, using the cohomology of such moduli spaces, G. Laumon, M. Rapoport and U. Stuhler prove a reciprocity law generalizing the one given in Drinfeld's original paper. As a consequence, they find enough representations to establish the basic local Langlands conjecture for \(\text{GL}_ d\) \((d\) arbitrary) of a local field of equal characteristic; this completes a program that was first begun by P. Deligne in the 1970's using Drinfeld's original work for \(d = 2\). Drinfeld modules; level structure; \({\mathcal D}\)-elliptic sheaves; curve over a finite field; function fields over finite fields; reciprocity law Carayol H.: Variétés de Drinfeld compactes, d'après Laumon, Rapoport et Stuhler. Astérisque 206, 369--409 (1992) Finite ground fields in algebraic geometry, Drinfel'd modules; higher-dimensional motives, etc., Algebraic functions and function fields in algebraic geometry, Formal groups, \(p\)-divisible groups, Langlands-Weil conjectures, nonabelian class field theory Compact Drinfeld varieties [after Laumon, Rapoport and Stuhler] | 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 prove the following:
Theorem. Let \(X\) be a smooth projective complex curve, \(A\) an abelian variety, and \(Y\) an effective divisor on \(A\). Assume that \(Y\) contains no translate of a non zero abelian subvariety. Then there exists a real constant \(C >0\), depending only on \(X\), \(A\), and \(Y\) with the property that for any morphism \(f:X\to A\) with \(f(X)\not\subset Y\) all points of the divisor \(f^*Y\) have multiplicity at most \(C\).
A very special case of this theorem is covered by a ``classical'' theorem (proved by Accola by extending an argument of Segre) saying that the multiplicities of the higher Weierstrass points on a curve are bounded [\textit{R. D. M. Accola}, ``Topics in the theory of Riemann surfaces'', Lect. Notes Math. 1595 (1994; Zbl 0820.30002)]. Let us note that the Segre-Accola argument breaks down when applied to our context; our proof will be based on a ``non-classical'' technique which we introduced earlier [\textit{A. Buium}, Ann. Math., II. Ser. 136, No. 3, 557-567 (1992; Zbl 0817.14021) and Int. Math. Res. Not. 1994, No. 5, 219-233 (1994; Zbl 0836.14025)], based on ``differential algebra plus Big Picard''. abelian variety; effective divisor; intersection multiplicity Buium, A., Intersection multiplicities on abelian varieties, Math. Ann., 1998, 310: 653--659. Algebraic theory of abelian varieties, Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Intersection multiplicities on abelian 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 method of descent [same authors, in Journées de géométrie algébrique Angers/France 1979, 223-237 (1980; Zbl 0451.14018)] reduces most arithmetic problems on a given rational variety X to a few basic problems on some auxiliary varieties, the ''universal torsors'' over X. These are defined in a quite abstract manner.
This note firstly gives a general description of the universal torsors (and others) as concrete fibre products. - Secondly, when X is a rational surface over a field k equipped with a relatively minimal k-morphism \(p:\quad X\to {\mathbb{P}}^ 1_ k\) whose generic fibre is a conic, under mild assumptions, it is shown that the universal torsors over X are, up to trivial factors, complete intersections of \((r-2)\quad quadrics\) in \({\mathbb{P}}_ k^{2r-1}\), where r denotes the number of geometric degenerate fibres of p. The crux of the proof is a variant of Abhyankar's lemma (ramification eats up ramification), which accounts for a magic change of variables [see \textit{A. Beauville}, the authors and \textit{P. Swinnerton-Dyer}, Ann. Math., II. Ser. 121, 283-318 (1985; Zbl 0589.14042) and the authors and \textit{P. Swinnerton-Dyer}, J. Reine Angew. Math. 373, 37-107 and 374, 72-168 (1987)]. - Thirdly, for \(X/{\mathbb{P}}^ 1_ k\) as above and \(r=4\), some arithmetic consequences are drawn. In particular, if k is a number field, the set of k-rational points on X decomposes into finitely many R-equivalence classes, as defined by \textit{Yu. I. Manin} [''Cubic Forms'' (1974 and 1986; Zbl 0255.14002 and Zbl 0582.14010)]. A K-theoretical but closely related approach to conic bundles, with arithmetical consequences for \(r=4\), is due to \textit{P. Salberger} [C. R. Acad. Sci., Paris, Sér. I 303, 273-276 (1986); Sémin. Théorie Nombres, Paris 1985-1986 (to appear)] and it inspired the present more geometric note, which itself has led to more results for \(r=4\) [\textit{J.-L. Colliot-Thélène} and \textit{A. N. Skorobogatov}, ''R- equivalence on conic bundles of degree 4'' (preprint)] and \(r=5\) [\textit{A. N. Skorobogatov} (in preparation)]. descent; universal torsors; complete intersections; ramification; rational points; conic bundles J. Colliot-Thélène and J. Sansuc, ''La descente sur les surfaces rationnelles fibrées en coniques,'' C. R. Acad. Sci. Paris Sér. I Math., vol. 303, iss. 7, pp. 303-306, 1986. Global ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Rational and unirational varieties, Complete intersections, Special surfaces, Applications of methods of algebraic \(K\)-theory in algebraic geometry La descente sur les surfaces rationnelles fibrées en coniques. (Descent on rational surfaces fibered in conics) | 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 course of the recent ten years, a rapid progress in applying methods from algebraic geometry to arithmetic problems has been achieved. In the early 1970's the Russian mathematician \textit{S. Yu. Arakelov} established an intersection theory for divisors on arithmetic surfaces, essentially by compactifying those surfaces with respect to the archimedean places and then introducing suitable hermitean structures on arithmetic line bundles. A decisive break through was made around 1982, when the author of the present book proved a Riemann-Roch theorem for arithmetic surfaces by amplifying Arakelov's theory. Faltings' innovation was based upon the construction of volume forms on the cohomology of hermitean bundles, and this allowed the extension of various techniques and theorems for complex surfaces to arithmetic surfaces. The main application of this discovery was the author's spectacular proof of the Mordell conjecture [cf. the author, Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)]. At about the same time, \textit{D. Quillen} gave a similar construction, using Cauchy-Riemann operators on Riemann surfaces and the Ray-Singer theory of analytic torsion [cf. \textit{D. Quillen}, Funct. Anal. Appl. 19, 31-34 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 37-41 (1985; Zbl 0603.32016)].
These achievements lead to a growing interest in this area, in particular with regard to significant applications in mathematical physics (string theory), and in the course of the recent five years the theory of arithmetic varieties underwent an enormous development. After P. Deligne's generalization of Faltings' volume forms to a wider class of arithmetic varieties [cf. \textit{P. Deligne} in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)], H. Gillet and C. Soulé succeeded in establishing both an arithmetic intersection theory for general arithmetic varieties and a hermitean \(K\)-theory for them [cf. \textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst Hautes Étud. Sci. 72, 93-174 (1990)]. Together with \textit{J. M.Bismut} they constructed, in the sequel, the determinant of cohomology of an arbitrary arithmetic variety. Finally, \textit{J. M. Bismut} and \textit{G. Lebeau} [cf. C. R. Acad. Sci., Paris, Sér. I 309, No. 7, 487-491 (1989; Zbl 0681.53034)], using probability-theoretic methods (stochastic integration), proved a Riemann-Roch theorem for closed immersions of arithmetic varieties, i.e., a Riemann-Roch-type result for the determinant of cohomology with respect to closed immersions.
In the spring of 1990 the author gave a course at Princeton University on these very recent results by Gillet-Soulé and Bismut-Lebeau. The present book grew out of these lectures, which were originally intended to explain the various techniques involved. However, the present text appears as an overworked and subsequently improved version of this lecture course. The aim of the book is now to present the arithmetic Riemann-Roch theory in a more coherent and technically simplified way. The author has managed, in an ingenious manner, to widely replace all the probability-theoretic arguments in the original approach --- which are barely familiar to algebraic geometers and number theorists --- by algebraic, complex-analytic and geometric considerations. This might help to make the topic more accessible to a wider class of interested readers, although the material still remains highly demanding and complicated.
Chapter I provides an introduction to the classical Riemann-Roch theorem for smooth morphisms of regular schemes over. The author explains \(K\)- groups and Chow groups, Chern classes, deformations of regular embeddings to normal cones, the reduction of the Riemann-roch problem to the case of projective bundles, and proves then the Riemann-Roch theorem (via flag schemes) in this situation. Precisely this strategy is used, later on, to derive the general Riemann-Roch theorem for arithmetic schemes.
In Chapter II the author develops the theory of arithmetic Chern classes (due to Gillet-Soulé) by a new method which is closer to the classical approach of Grothendieck. After a brief survey on Kähler manifolds and their Hodge theory, the construction of arithmetic \(K\)-groups, arithmetic Chow groups, and arithmetic Chern classes is carried out in a remarkably simplified way.
Chapter III contains the analytical framework that the author developed for the purpose of replacing the stochastic arguments by Bismut-Lebeau in proving the arithmetic Riemann-Roch theorem. The new ingredient consists in computing the asymptotic behavior of the diagonal values of the heat kernel of the Laplacian on certain Riemann manifolds. The presentation is based on a simplified ad-hoc procedure which avoids fancy technicalities, and suffices for all relevant cases later on.
This is used, in chapter IV, to prove a local index theorem for Dirac operators on compact Kähler manifolds. This chapter begins with a survey on Clifford algebras and Dirac operators, and closes, after the proof of the local index theorem for them, with the construction of super-Dirac operators as limits of ordinary Dirac operators in the sense of Clifford algebras.
Chapter V is devoted to the construction of direct images in arithmetic \(K\)-theory, i.e. of direct images with respect to smooth proper maps of arithmetic varieties. This is done by using new so-called number operators and various estimates for Laplacians, and provides an approach different from the original ones by Bismut-Gillet-Soulé and Quillen.
Chapter VI puts then everything together and culminates in presenting the author's proof of the arithmetical Riemann-Roch theorem. Along the line which was followed in proving the classical Riemann-Roch theorem in chapter I, the problem is reduced to some deformation argument with respect to normal cones for closed embeddings, and to the Riemann-Roch problem for projective bundles. Both reduced problems are proved, again in a new way, by constructiong a modified Todd class and using subtle estimates for heat kernels of Laplacians.
The concluding chapter VII is of completing nature. It discusses the very recent theorem of Bismut-Vasserot [cf. \textit{J.-M. Bismut} and \textit{E. Vasserot}, Commun. Math. Phys. 125, No. 2, 355-367 (1989; Zbl 0687.32023)] on the behavior of the analytic torsion of ample line bundles on complex algebraic manifolds, and that from the point of view of the author's approach developed in the present book. It turns out that this methods are nicely applicable to this kind of problems, too.
Altogether, this treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it. The text combines introductory material with a detailed presentation of new methods and results in arithmetical algebraic geometry and hermitean geometry. Although the subject is rather complicated, the author succeeded in providing a comprehensible, fairly self-contained and methodically coherent account of it. This book will certainly become one of the most important standard references for both specialists and interested non-specialists in arithmetic geometry and its applications. index theorem; analytic torsion; heat kernel of the Laplacian on Riemann manifolds; Arakelov's theory; hermitean bundles; Mordell conjecture; arithmetic intersection theory for general arithmetic varieties; arithmetic Riemann-Roch theory; arithmetic Chern classes; arithmetic \(K\)- groups; arithmetic Chow groups; Dirac operators on compact Kähler manifolds; super-Dirac operators [15] Faltings (G.).-- Lectures on the arithmetic Riemann-Roch theorem, Annals of Math. Studies, vol. 127, Princeton University Press, 1992. &MR~11 | &Zbl~0744. Arithmetic varieties and schemes; Arakelov theory; heights, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Riemann-Roch theorems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes Lectures on the arithmetic Riemann-Roch theorem. Notes taken by Shouwu Zhang | 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 presents the construction of a moduli space of stable objects in a triangulated category. In the paper of the same author [J. Math. Kyoto Univ. 42, No. 2, 317--329 (2002; Zbl 1063.14013)] it is proved that if \(f:X\longrightarrow S\) is a projective, flat morphism of noetherian schemes, then the functor Splcpx\(^{\text{ét}}_{X/S}\) is an algebraic space over \(S\), where Splcpx\(^{\text{ét}}_{X/S}\) is the étale sheafification of the functor Splcpx\(_{X/S}\) which associates to any \(S-\)scheme \(T\) the set of equivalence classes of complexes \(E\in D^{b}(\mathrm{Coh}(X\times_{S} T))\) such that \(E(t):=E\otimes^{L}k(t)\) is bounded and simple. This result was generalized by \textit{M. Lieblich} [J. Algebr. Geom. 15, No. 1, 175--206 (2006; Zbl 1085.14015)] to any proper, flat morphism of algebraic spaces.
The aim of the paper is to construct a projective moduli space as a Zariski open subset of Splcpx\(^{\text{ét}}_{X/S}\). The construction is presented in the general setting of fibered triangulated categories \(p:\mathcal{D}\longrightarrow (Sch/S)\) with base change property (see Definitions 2.1 and 2.2). An important example is the following: let \(f:X\longrightarrow S\) be a flat, projective morphism, and for every \(U\in(Sch/S)\) let \(X_{U}:=X\times_{S} U\). The category \(\mathcal{D}=\{D^{b}(\mathrm{Coh}(X_{U}/U))\}_{U\in Sch/S}\) is a fibered triangulated category with base change property, where \(D^{b}(\mathrm{Coh}(X_{U}/U))\) is the full subcategory of \(D^{b}(\mathrm{Coh}(X_{U}))\) of complexes of finite Tor-dimension over \(U\).
In this general setting, the author introduces the notion of strict ample sequence, which is a family \(\mathcal{L}=\{L_{n}\}_{n\geq 0}\) such that \(L_{n}\) is an object in the fiber \(\mathcal{D}_{S}\) of \(\mathcal{D}\) over \(S\), verifying a list of axioms (see Definition 3.1) generalizing the notion of ample sequence introduced by \textit{A. Bondal} and \textit{D. Orlov} in [Compos. Math. 125, No.3, 327--344 (2001; Zbl 0994.18007)]. If \(\mathcal{L}\) is a strict ample sequence, the author defines a notion of \(\mathcal{L}-\)stability of objects in \(\mathcal{D}_{k}\) for every geometric point \(\mathrm{Spec}(k)\longrightarrow S\). This notion of \(\mathcal{L}-\)stabilty is different from the notion of stability condition introduced by \textit{T. Bridgeland} in [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], and the relation between these two notions is still an open problem.
Fixing a numerical polynomial \(P\in\mathbb{Q}[t]\) one has then the moduli functor \(\mathcal{M}_{\mathcal{D}}^{P,\mathcal{L}}:(Sch/S)\longrightarrow(Sets)\) which associates to any \(S-\)scheme \(T\) the set of equivalence classes of objects \(E\in\mathcal{D}_{T}\) such that for every \(s\in T\) we have that \(E_{s}\) is \(\mathcal{L}-\)stable and \(\Hom((L_{n})_{s},E_{s})\) has dimension \(P(n)\). Similarily, one defines the moduli functor \(\overline{\mathcal{M}_{\mathcal{D}}^{P,\mathcal{L}}}\) of \(\mathcal{L}-\)semistable objects. In section 4, the author proves that \(\mathcal{M}_{\mathcal{D}}^{P,\mathcal{L}}\) and \(\overline{\mathcal{M}_{\mathcal{D}}^{P,\mathcal{L}}}\) admit a coarse moduli space \(M_{\mathcal{D}}^{P,\mathcal{L}}\) and \(\overline{M_{\mathcal{D}}^{P,\mathcal{L}}}\). Moreover, if \(S\) is of finite type over a universally Japanese ring, then \(\overline{M_{\mathcal{D}}^{P,\mathcal{L}}}\) is projective over \(S\).
In section 5 several examples are presented. A first example shows that the classical moduli space of semistable sheaves with fixed Hilbert polynomial on a smooth projective scheme \(X\) (ore, more generally, the relative moduli space) is a particular case of this construction. More general examples are described as particular cases of this construction, like the moduli space of \(G-\)twisted semistable sheaves (where \(G\) is any vector bundle), or the moduli space of \(G-\)twisted semistable \(\alpha-\)twisted sheaves (where \(G\) is any vector bundle and \(\alpha\) is an element in the cohomological Brauer group of \(X\)). moduli; triangulated category Inaba-2 M.~Inaba, Moduli of stable objects in a triangulated category, J.\ Math.\ Soc.\ Japan, 62 (2010), 395--429. Algebraic moduli problems, moduli of vector bundles, Derived categories, triangulated categories Moduli of stable objects in a triangulated category | 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 0695.00009.]
This paper reviews results and sketches some techniques concerning the author's work on Mordell-Weil groups of universal abelian varieties. Throughout she works over \(\mathbb C\); roughly speaking the results are that the generic abelian variety with a certain level structure has (over the function field of the moduli space corresponding to the level structure) no more rational points than the ones forced upon it by the level structure.
The method can be divided into two parts. First one shows that the number of rational points is finite. E.g., the author manages to prove this in examples by specializing to all possible abelian varieties isogenous to a \(g\)-fold product of an elliptic curve. To make this work one needs similar results valid in the elliptic modular case. One also needs an argument which implies that the orders of such specialized sections cannot vary with the degree of the used isogeny; the author obtains this last result by studying the (dense) set of complex multiplication fibres and using the knowledge of fields of definition of torsion points in that theory. Once finiteness has been proven one can identify the Mordell-Weil group as a frequently easy cohomology group. As noted by the author, in the elliptic modular case such results are not true in characteristic \(p\) in general. For instance, in certain examples the universal family is a \(K3\)-surface with maximal Picard number, and one obtains additional sections in all characteristics where this surface becomes supersingular. Shioda gave some explicit examples of this. It would be very interesting to see whether similar phenomena exist for higher dimensional abelian varieties.
Hopefully the place where this very nice paper appeared will not prevent too many mathematicians interested in Mordell-Weil groups and/or moduli of abelian varieties from finding it. Mordell-Weil groups of universal abelian varieties; level structure; number of rational points; set of complex multiplication fibres; fields of definition of torsion points; K3-surface with maximal Picard number Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties, Rational points Universal families of abelian 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 conjectural analytic properties of Hasse-Weil \(L\)-functions of curves over number fields are a consequence of the widely believed expectation that such functions arise from the world of automorphic representations. This paper deals with zeta functions of arithmetic schemes, which are quotients of Hasse-Weil \(L\)-functions up to finitely many bad factors. Such quotients do not inherit automorphic properties and this work attempts to replace the lost automorphicity with a mean-periodicity condition. On the other hand, it is stated as an expectation that mean-periodicity can be proved independently of automorphicity. This claim was motivated by the appearance of mean-periodicity in the first author's works on two-dimensional adelic analysis [Doc. Math., J. DMV Extra Vol., 261--284 (2003; Zbl 1130.11335); Mosc. Math. J. 8, No. 2, 273--317 (2008; Zbl 1158.14023); J. K-Theory 5, No. 3, 437--557 (2010; Zbl 1225.14019)], though the document in question does not require knowledge of this as a prerequisite. Further evidence is discussed at intervals throughout the paper. The need for approaches to meromorphic continuation and functional equation which do not depend on automorphicity was mentioned by \textit{R. P. Langlands} [in: Representation theory and automorphic forms. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 61, 457--471 (1997; Zbl 0901.11032)].
As remarked in the introduction, mean-periodicity is not widely used in number theory and thus deserves some further discussion -- much care is taken to introduce three formulations of the notion, each of which is valid in only certain classes of functional spaces, with large overlaps between these classes. The first states that the set of translates of a function is not dense in the functional space, the second is existence of a non-trivial homogeneous convolution equation, and the third is the existence of a form of generalized Fourier expansion.
The main results of the paper state that, up to technical conditions, the meromorphic continuation and functional equation of zeta functions of arithmetic schemes is equivalent to the mean-periodicity of a related function, referred to in this paper as the ``boundary function''. The boundary function is given as a transform of the product of finitely many Riemann zeta functions and the zeta function of the arithmetic scheme. It is so called on account of its interpretation as an integral over a semi-global adelic boundary in dimension two -- this idea is not needed for understanding of the paper. Aside from meromorphic continuation and functional equation this paper also considers zeros and poles of zeta functions. In particular, it is studied a ``single sign property'', which is again motivated by the ideas originating in the first author's adelic analysis.
The paper concludes with interesting examples of mean-periodic functions arising from Dedekind zeta functions, cuspidal automorphic forms and Eisenstein series. zeta functions of arithmetic schemes; Hasse-Weil \(L\)-functions of curves over global fields; zeta functions of elliptic curves over number fields; zeta functions of arithmetic schemes; mean-periodicity; boundary terms of zeta integrals; higher adelic analysis Ivan Fesenko, Guillaume Ricotta, and Masatoshi Suzuki, Mean-periodicity and zeta functions, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 5, 1819 -- 1887 (English, with English and French summaries). Other Dirichlet series and zeta functions, Elliptic curves over global fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Classical almost periodic functions, mean periodic functions Mean-periodicity and zeta 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 absolutely irreducible, smooth projective curve over a field \(F\); let \(K:=F(X)\) be its function field and \({\mathcal O}_ X\) its structural sheaf. The authors define (for all \(r\geq 1)\) an \(r\)-divisor, \(D\), as a rank \(r\) \({\mathcal O}_ X\)-submodule of the constant sheaf \(K^ r\); a rank \(r\) vector bundle on \(X\) is associated to each such \(D\) and the set of all \(r\)-divisors is partitioned according to the isomorphism classes of the corresponding vector bundles. Counting all the cardinalities related to the partition classes, in this very well written paper the authors give a new proof of the Siegel formula.
In an appendix they sketch the original proof [due independently to the teams \textit{U. V. Desale} and \textit{S. Ramanan}, Invent. Math. 38, 161-185 (1976; Zbl 0323.14012) and \textit{G. Harder} and \textit{M. S. Narasimhan}, Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] of this formula (shown to be equivalent to the classical (i.e. of A. Weil) fact that the Tamagawa number of \(SL(r,K)\) is 1) and to its original application to the computation, via the just proven Weil conjectures, of the Betti numbers of the moduli spaces of stable vector bundles on \(X\). divisors of higher rank on a curve; \(r\)-divisor; Tamagawa number; Betti numbers of the moduli spaces of stable vector bundles Ghione F and Letizia M, Effective divisors of higher rank on a curve and the Siegel formula,Composite Math. 83 (1992) 147--159 Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli Effective divisors of higher rank on a curve and the Siegel formula | 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{Y. Laszlo} and \textit{M. Olsson} [Publ. Math., Inst. Hautes Étud. Sci. 107, 109--168 (2008; Zbl 1191.14002); Publ. Math., Inst. Hautes Étud. Sci. 107, 169--210 (2008; Zbl 1191.14003)] constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. We give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed by \textit{W. Zheng} [``Gluing pseudo functors via \(n\)-fold categories'', Preprint, \url{arXiv:1211.1877}]. As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and \(\ell\)-adic coefficients. Deligne-Mumford stack; étale cohomology; six operations; pseudofunctor; Lefschetz-Verdier formula Zheng, W, Six operations and Lefschetz-verdier formula for Deligne-Mumford stacks, Sci. China Math., 58, 565-632, (2015) Étale and other Grothendieck topologies and (co)homologies, Generalizations (algebraic spaces, stacks), Double categories, \(2\)-categories, bicategories, hypercategories Six operations and Lefschetz-Verdier formula 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 Let \(C\) be a smooth projective curve of genus \(2\) over an algebraically closed field \(k\) of characteristic \(>2\). Denote by \(M_2(C)\) the moduli space of vector bundles of rank \(2\) with trivial determinant on \(C\). Pulling back by the Frobenius defines the generalized Verschiebung map, a rational map \(V_2\) from \(M_2(C^p)\) to \(M_2(C)\). The map is not defined precisely on the locus \(B\) corresponding to Frobenius-unstable vector bundles. The author assumes that \(B\) consists of \(\delta\) reduced points. Note that such an assumption holds if \(C\) is a general curve.
Theorem. (1) By a single blow up at each \(b\in B\), the rational map \(V_2\) can be extended to a morphism of degree \(p^3 - \delta\).
(2) The exceptional divisor \(E_b\) at \(b\in B\) maps bijectively to \({\mathbb P}(\) Ext\(^1(L,L^{-1}))\subset M_2(C), \, L\) being a theta characteristic destabilizing the Frobenius pull back of the vector bundle corresponding to \(b\).
As an application, the author deduces that if \(C\) is general, then \(V_2\) has degree \((p^3+2p)/3\). Recently, H. Lange and C. Pauly have obtained the result by different methods. In some cases in characteristic \(2\) and \(3\), \textit{Y. Laszlo} and \textit{C. Pauly} have given explicit polynomials giving \(V_2\), note that \(M_2(C) \cong {\mathbb P}^3\) [J. Algebr. Geom. 11, No. 2, 219--243 (2002; Zbl 1080.14527); Adv. Math. 185, No. 2, 246--269 (2004; Zbl 1055.14038)]. Osserman B., The generalized Verschiebung map for curves of genus 2, Math. Ann., 2006, 336(4), 963--986 Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The generalized Verschiebung map for curves of genus 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 The book under review is based on the minicourse given by the author in 2008 at IMPA (a part of the material reflects his earlier courses given in 2006 in Göttingen and in 2008 in Strasbourg). Chapter 1 is a brief introduction to the Brauer groups of schemes and Galois cohomology aiming to state the period-index problem and Serre's Conjectures I and II. Chapter 2 introduces the reader to diophantine problems over finite fields and over function fields of curves by discussion of the classical Chevalley--Warning and Tsen--Lang theorems and their application to Brauer groups. Chapter 3 is devoted to rationally connected varieties focusing on the conjecture by \textit{J.~Kollár, Y.~Miyaoka}, and \textit{S.~Mori} [J. Algebraic Geom. 1, No. 3, 429--448 (1992; Zbl 0780.14026)] stating that every rationally connected fibration over a curve has a section. A sketch of proof is given which is slightly different from the proof published in \textit{T.~Graber, J.~Harris}, and \textit{J.~Starr} [J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)] (in particular, it takes into account some major simplifications due to A.~J.~de Jong). Chapter 4 contains a review of the period-index theorem of \textit{A.~J.~de Jong} [Duke Math. J. 123, No. 1, 71--94 (2004; Zbl 1060.14025)]. In particular, the second proof of this theorem is given (due to de Jong and the author). Note that the third proof of the same theorem was obtained by \textit{M.~Lieblich} [Compos. Math. 144, 1--31 (2008; Zbl 1133.14018)]. All three proofs are based on different ideas and techniques. Finally, Chapter 5 contains a description of known results about Serre's Conjecture II and a sketch of the proof of its split case over function fields of surfaces using the notion of rational simply connectedness (based on a recent work of \textit{A.~J.~de~Jong, X.~He, and J.~M.~Starr} [\url{arXiv:0809.5224}]). Brauer group; Galois cohomology; rationally connected variety; function field Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fibrations, degenerations in algebraic geometry, Formal methods and deformations in algebraic geometry, Ramification problems in algebraic geometry, Brauer groups of schemes, Rational points, Other nonalgebraically closed ground fields in algebraic geometry, Linear algebraic groups over arbitrary fields Brauer groups and Galois cohomology of function fields of 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 Contents: -- Introduction -- 1. Galois theory of fields 2. Fundamental groups in topology 3. Riemann surfaces 4. Fundamental groups of algebraic curves 5. Fundamental groups of schemes 6. Tannakian fundamental groups -- Bibliography and index.
Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at the graduate level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory.
The first three chapters of the book cover basic algebraic, topological and analytical concepts, presented from the modern viewpoint advocated by Grothendieck. The first chapter consists of a concise introduction to classical Galois theory. The second chapter focusses on topology, discussing Galois covers, local systems and the monodromy action. The third chapter unites the algebraic and the topological picture by introducing Riemann surfaces and establishing the correspondence between finite Galois branched covers of a Riemann surface \(X\) and finite Galois extensions of the field of meromorphic functions on \(X\).
This background enables then a systematic and very accessible development of the theory of fundamental groups of algebraic curves in a first step, and fundamental groups of schemes in a second step. At the heart of the book under review lies Grothendieck's construction of the étale fundamental group \(\pi_1^{\mathrm{et}}(S,s)\) of a pointed connected scheme \(S\), and the correspondence between finite étale covers of \(S\) and finite continuous left \(\pi_1^{\mathrm{et}}(S,s)\)-sets. Further topics are the homotopy exact sequence splitting the étale fundamental group in its arithmetic and its geometric part, structure results for étale fundamental groups, comparison with topological fundamental groups and the abelianised fundamental group.
Input from algebraic geometry is kept at a minimum and is always explained in a honest manner, with appropriate references, and illustrated by examples.
The last chapter of the book is about Tannakian fundamental groups. Large parts of it are logically independent from the rest of the book, except for the last two section which discuss differential fundamental groups, Nori's fundamental group scheme and its relation with the étale fundamental group.
The book is consistently written in a clear and rigorous style, with a healthy balance between the abstract and the concrete. Much of the covered material appears for the first time in book form. The references given in the text satisfy those readers who seek further up-to-date information as well as those who are interested in the historical development of the theory. The typesetting is flawless.
The book suits well both, self study and class work. Each chapter is followed by an ample collection of instructive exercises. Key applications, for example on the inverse Galois problem, and remarks on recent results and open problems, for example on Grothendieck's section conjecture are given throughout. Thus, also the more advanced reader who already knows about étale fundamental groups will enjoy this reading matter. Galois groups; fundamental groups; Tannakian categories T. \textsc{Szamuely}, \textit{Galois Groups and Fundamental Groups}, Cambridge Studies in Advanced Mathematics, vol.~117, Cambridge Univ. Press, Cambridge, 2009. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Separable extensions, Galois theory, Monoidal categories (= multiplicative categories) [See also 19D23] Galois groups and fundamental 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 This paper is a gentle introduction to \textit{P. Deligne} [``Les constantes des équations fonctionnelles des fonctions \(L\)'', in: Modular Functions of One Variable, II, Proc. Int. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 501-597 (1993; Zbl 0271.14011)]. The first part is devoted to the general formalism. After defining the Weil-Deligne group of a non-archimedean local field \(K\) and considering its (complex) representations, the author discusses in turn the \(L\)-factor, conductor, \(\varepsilon\)-factor, and root number. He also indicates the connection between such representations and the \(\ell\)-adic Galois representations arising from the étale cohomology of varieties over \(K\).
In the second part, this connection is pursued for the simplest non-trival example, that of elliptic curves. In this case, it is shown that the representation of the Weil-Deligne group arising from \(\ell\)-adic cohomology is independent of \(\ell\). Moreover, the various invariants (\(L\)-factor, conductor, etc.) are computed explicitly in terms of the reduction of \(E\). The author concludes with a brief discussion of the archimedean case and the global setting. Weil-Deligne group; non-archimedean local field; \(L\)-factor; conductor; \(\varepsilon\)-factor; root number; elliptic curves Rohrlich, D. E., \textit{elliptic curves and the Weil-Deligne group}, Elliptic curves and related topics, 125-157, (1994), American Mathematical Society, Providence, RI Elliptic curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Elliptic curves over local fields, Local ground fields in algebraic geometry Elliptic curves and the Weil-Deligne 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 main purpose of this book is to develop an arithmetic analogue of the theory of ordinary differential equations. In arithmetic theory, replace ``time variable'' \(t\) by a fixed prime number \(p\), smooth real functions \(x\to x(t)\) by integers \(a\in{\mathbb{Z}}\) (or more generally by algebraic integers), the derivative \(x(t)\to {dx\over {dt}}(t)\) by a ``Fermat quotient operator'' \(\delta: {\mathbb{Z}}\to{\mathbb{Z}}: a\mapsto \delta\,a:={{a-a^p}\over{p}}\). Smooth manifolds are replaced by algebraic varieties over number fields. Jet spaces are replaced by algebraic jet spaces constructed using \(\delta\). Differential equations are replaced by ``arithmetic differential equations''. In particular, differential equations that are invariant under certain group actions are replaced by arithmetic differential equations which are invariant under various correspondences on the varieties in question. The main goal of the book is to construct new quotient spaces which have no counterparts in the classical algebraic geometry. This is done by introducing a new geometry called \(\delta\)-geometry, where the categorical quotients are non-trivial (contrary to the situation in classical algebraic geometry). Three classes of examples are discussed in detail illustrating the new \(\delta\)-geometry, namely, (1) Spherical case: quotients of the \({\mathbb{P}}^1\) by actions of \(\text{SL}_2({\mathbb{Z}})\), (2) Flat case: quotients of \({\mathbb{P}}^1\) by actions of postcritically finite maps \({\mathbb{P}}^1\to{\mathbb{P}}^1\) with (orbifold) Euler characteristic zero, and (3) Hyperbolic case: quotients of modular or Shimura curves by actions of Hecke correspondences.
A conjecture is formulated which characterizes the existence of non-trivial categorical quotients.
Conjecture. The quotient of a curve over a number field by a correspondence is non-trivial for almost all primes \(p\) if and only if the correspondence has a complex ``analytic uniformization''.
The main results of the book assert that the ``if''part of the conjecture holds under some mild assumptions in the three cases (1),(2) and (3). The ``only if'' part of the conjecture is proved in the ``dynamical system case''.
Let \(\mathcal{C}\) be an arbitrary category. A correspondence in \(\mathcal{C}\) is a tuple \({\mathbb{X}}=(X, \check{X}, \sigma_1,\sigma_2)\) where \(X\) and \(\check{X}\) are objects of \(\mathcal{C}\) and \(\sigma_1,\,\sigma_2\,:\, \check{X}\to X\) are morphisms of \(\mathcal{C}\). A categorical quotient for \({\mathbb{X}}\) is defined to be a pair \((Y,\pi)\) where \(Y\) is an object of \(\mathcal{C}\) and \(\pi:X\to Y\) is a morphism in \(\mathcal{C}\) satisfying the following properties: (1) \(\pi\circ \sigma_1=\pi\circ\sigma_2\); (2) For any pair \((Y^{\prime},\pi^{\prime})\) where \(Y^{\prime}\) is an object of \(\mathcal{C}\) and \(\pi^{\prime}: X\to Y^{\prime}\) is a morphism such that \(\pi^{\prime}\circ\sigma_1=\pi^{\prime}\circ \sigma_2\) there exists a\ unique morphism \(\gamma: Y\to Y^{\prime}\) such that \(\gamma\circ \pi=\pi^{\prime}\). If a categorical quotient exists, it is unique up to isomorphism, and denoted by \(Y=X/\sigma\). In the classical algebraic geometry, there are interesting correspondences but in many cases their categorical quotients turn out to be trivial. This is the ``basic pathology''. The motivation of the book stems from the attempt to remedy this pathological situation. The author's approach is to enlarge algebraic geometry so that the categorical quotients would become non-trivial. For this, there are at least two approaches, that is, invariant theoretic method, and groupoid theoretic method. The author takes the viewpoint of invariant theoretic approach, and tries to enlarge algebraic geometry by adjoining some new functions \(\delta\), satisfying certain ``polynomial compatibility conditions'' with respect to addition and multiplication. Locally, there are four types of such \(\delta\): derivation operators, difference operators, \(p\)-derivation operators and \(p\)-difference operators. When \(\delta\) are \(p\)-derivation operators attached to various prime numbers \(p\) (via ``Fermat quotient operators''), this leads to the arithmetic differential algebra, arithmetic differential equation and its corresponding \(\delta\)-geometry, and the book develop some of the basic elements of \(\delta\)-geometry and then construct and study interesting categorical quotients whose categorical quotients in the usual algebraic geometry are trivial.
The theory of arithmetic differential algebras, arithmetic differential equations and the corresponding \(\delta\)-geometry is then compared to other theories. When \(\delta\) is derivation, this leads to differential algebra of \textit{J. F. Ritt} [``Differential Algebra.'' Colloquium Publications, 33. New York: American Mathematical Society (AMS). (1950; Zbl 0037.18402)] and \textit{E. R. Kolchin} [``Differential algebra and algebraic groups.'' Pure and Applied Mathematics, 54. New York-London: Academic Press. (1973; Zbl 0264.12102)]. If \(\delta\) is a difference operator, this gives rise to the difference algebra of \textit{R. Cohn} [``Difference Algebra.'' New York-London-Sydney: Interscience Publishers. (1965; Zbl 0127.26402)]. The case when \(\delta\) is \(p\)-difference operator seems to lead to a less interesting theory. Only when \(\delta\) is \(p\)-derivation, it gives rise to an arithmetically futile theory of arithmetic differential algebra and its \(\delta\)-geometry. Comparisons to Connes noncommutative geometry, Dwork's theory, Drinfeld modules, Mochizuki's \(p\)-adic Teichmüller theory, Ihara's congruence relations, among others, are also briefly discussed. arithmetic analogue of ordinary differential equations; \(\delta\)-geometry; quotient spaces; spherical correspondences; flat correspondences; hyperbolic correspondences Buium, Alexandru: Arithmetic differential equations, (2005) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local ground fields in algebraic geometry, Modular and Shimura varieties, Geometric invariant theory Arithmetic differential equations | 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 is a very good resource for the theory of algebraic geometry codes. It includes more extensive treatments of the topics in the lectures of \textit{J. H. van Lint} and \textit{G. van der Geer} [Introduction to coding theory and algebraic geometry. DMV Seminar 12, Basel, Birkhäuser (1988; Zbl 0639.00048)], and it also includes the arithmetic theory found in the text by \textit{C. Moreno} [Algebraic curves over finite fields, Cambridge Tracts in Mathematics, 97. Cambridge, Cambridge University Press (1991; Zbl 0733.14025)].
The book is divided into four parts. The first part contains basic definitions, bounds, and constructions of coding theory. The projective systems of Tsfasman and Vladut are treated in some exercises. The second part is an introduction to algebraic curves, with special attention to curves over finite fields. Several proofs for foundational results in algebraic geometry are omitted here, with references given to standard sources. The Riemann-Roch Theorem is proved using repartitions to get the needed duality statements. The definition of Weierstrass point (p. 97) is only appropriate for classical curves. The Riemann hypothesis for curves over finite fields is proved using the argument developed by the author and \textit{E. Bombieri} [Sém. Bourbaki 1972/73, Exp. No. 430, Lect. Notes Math. 383, 234-241 (1974; Zbl 0307.14011)]. The author additionally proves Serre's improvement of the Hasse-Weil bound. The theory of function fields is also used in this part, with references given to the texts by \textit{M. Deuring} [Lectures on the theory of algebraic functions of one variable, Lect. Notes Math. 314 (1973; Zbl 0249.14008)] and \textit{H. Stichtenoth} [Algebraic function fields and codes. Universitext. Berlin, Springer (1993; Zbl 0816.14011)].
The third part deals with elliptic and modular curves. The author describes classical modular curves and the Galois theory of elliptic function fields. This part culminates in a statement of the Eichler-Selberg trace formula, which is used, as in C. Moreno's book, to give a proof of the Tsfasman-Vladut-Zink theorem. The final part is devoted to geometric Goppa (or algebraic geometry) codes. The author considers standard examples and then discusses asymptotically good geometric codes. Here, the author not only considers codes from modular curves, as done by Tsfasman-Vladut-Zink, but also the more recent work by \textit{A. Garcia} and \textit{H. Stichtenoth} [Invent. Math. 121, 211-222 (1995; Zbl 0822.11078)] that uses towers of Artin-Schreier extensions to produce asymptotically good sequences of codes. In addition, the author includes his work [Discrete Math. Appl. 7, No. 1, 77-88 (1997; Zbl 0904.94026)] on codes from fiber products of hyperelliptic curves. There is also a chapter on decoding in this part that treats the Skorobogatov-Vladut basic algorithm for decoding algebraic geometric codes, as well as decoding algorithms due to Porter, Erhard, and Duursma.
There are exercises at the end of each chapter, with hints given for the more difficult ones. There is an extensive bibliography of over 200 papers and books. A few topics, such as generalized Hamming weights and applications of Gröbner bases to codes, do not appear in this book, but it is nonetheless one of the most comprehensive introductions to geometric codes to appear to date. geometric Goppa code; algebraic curve; zeta function; modular curve; asymptotically good sequence Stepanov, S.A.: Codes on Algebraic Curves. Springer, Berlin (1999) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry, Research exposition (monographs, survey articles) pertaining to information and communication theory, Arithmetic algebraic geometry (Diophantine geometry) Codes on algebraic 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 \(C\) be a smooth projective curve of genus \(g \geq 3\) with Jacobian \(J = \text{Pic}^0 C\) and \(\alpha: C \hookrightarrow J\) an Abel-Jacobi map. The image of \(\alpha\) gives a corresponding point in \(\text{Hilb}_J\) which lies on a unique irreducible component \(\text{Hilb}_{C,J} \subset \text{Hilb}_J\). [\textit{H. Lange} and \textit{E. Sernesi}, Ann. Mat. Pura Appl. (4) 183, No. 3, 375--386 (2004; Zbl 1204.14012)]; see also [\textit{P. A. Griffiths}, J. Math. Mech. 16, 789--802 (1967; Zbl 0188.39302)] proved that if \(C\) is nonhyperelliptic, then \(\text{Hilb}_{C/J}\) is smooth of dimension \(g\),
while if \(C\) is hyperelliptic, then \(\text{Hilb}_{C/J}\) is irreducible of dimension \(g\) and everywhere nonreduced with Zariski tangent space of dimension \(2g-2\): in both cases, the only deformations of \(C\) in \(J\) are by translation. In the hyperelliptic case, the author clarifies the non-reduced scheme structure by proving that \(\text{Hilb}_{C/J} \cong J \times R_g\) where \(R_g = \text{Spec}[s_1,\dots,s_{g-2}]/\mathfrak m^2\) and \(\mathfrak m = (s_1, \dots, s_{g-2})\) is the maximal ideal. He uses this result to describe the scheme theoretic fibers of the Torelli morphism \(\tau_g: \mathcal M_g \to \mathcal A_g\) along the non-hyperelliptic locus, where \(\mathcal M_g\) is the moduli stack of genus \(g\) curves and \(\mathcal A_g\) is the moduli stack of principally polarized abelian varieties of dimension \(g\).
There is an application to the moduli space of Picard sheaves on the Jacobian. Fix a smooth curve \(C\) of genus \(g \geq 2\) with Jacobian \(J\) and dual \(\hat J\). For \(p \in C\) and \(1 \leq d \leq g-1\), view the line bundle \(\xi = {\mathcal O}_C (dp)\) as a sheaf on \(\hat J\) by first pushing it forward along an Abel-Jacobi map \(\alpha: C \hookrightarrow J\) and then using the identification of \(J\) with \(\hat J\): applying his Fourier transform to this sheaf on \(\hat J\), \textit{Mukai} constructs an associated \textit{Picard sheaf} \(F\) on \(J\). Letting \(M(F)\) be the connected component containing \(F\) in the moduli space \(\text{Spl}_J\) of simple coherent sheaves on \(J\), \textit{Mukai} proves that if \(g=2\) or \(C\) is non-hyperelliptic, then the natural morphism \(\hat J \times J \to M(F)\) is an isomorphism [\textit{S. Mukai}, Nagoya Math. J. 81, 153--175 (1981; Zbl 0417.14036)] and when \(C\) is hyperelliptic it is an isomorphism onto \(M(F)_{\text{red}}\) [\textit{S. Mukai}, Adv. Stud. Pure Math. 10, 515--550 (1987; Zbl 0672.14025)]. Analogous to the main theorem, the author describes the non-reduced scheme structure by proving that \(M(F) \cong \hat J \times J \times R_g\). Jacobian; Torelli morphism; Hilbert schemes; Picard sheaves; Fourier-Mukai transform Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard 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 Let $S$ be a Noetherian scheme and $f:X\rightarrow S$ a proper morphism. By SGA 4 XIV [Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Springer, Cham (1973; Zbl 0245.00002)], for any constructible sheaf $\mathscr{F}$ of $\mathbb{Z}/n\mathbb{Z}$-modules on $X$, the sheaves of $\mathbb{Z}/n\mathbb{Z}$-modules $\mathtt{R}^{i}f_{\star}\mathscr{F}$ obtained by direct image (for the étale topology) are themselves constructible, that is, there is a stratification $\mathfrak{S}$ of $S$ on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which $S$ is generically of characteristic zero or the sheaves $\mathscr{F}$ are constant (with invertible torsion on $S$), here we study the dependency of $\mathfrak{S}$ on $\mathscr{F}$. We show that a natural 'uniform' tameness and constructibility condition satisfied by constant sheaves, which was introduced by O. Gabber, is stable under the functors $\mathtt{R}^{i}f_{\star}$. If $f$ is not proper, this result still holds assuming tameness at infinity, relative to $S$. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the sheaves $\mathtt{R}^{i}f_{\star}\mathbb{F}_{\ell}$, where $\ell$ ranges through all prime numbers invertible on $S$. étale cohomology; constructible sheaf; stratification; tameness; cohomological descent; alteration; ultraproduct Étale and other Grothendieck topologies and (co)homologies, Ramification problems in algebraic geometry, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Ultraproducts (number-theoretic aspects), Algebraic moduli problems, moduli of vector bundles Uniform constructability and tameness 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 We construct a well-behaved Weil-étale complex for a large class of \(\mathbb{Z}\)-constructible sheaves on a regular irreducible scheme \(U\) of finite type over \(\mathbb{Z}\) and of dimension 1. We then give a formula for the special value at \(s = 0\) of the \(L\)-function associated to any \(\mathbb{Z}\)-constructible sheaf on \(U\) in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of [GS20]. We deduce a special value formula for Artin \(L\)-functions of integral representations twisted by a singular irreducible scheme \(X\) of finite type over \(\mathbb{Z}\) and of dimension 1. This generalizes and improves all results in [Tra16]; as a special case, we obtain a special value formula for the arithmetic zeta function of \(X\). \(L\)-functions; special values; Weil-étale cohomology Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Étale and other Grothendieck topologies and (co)homologies, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Special values of \(L\)-functions on regular arithmetic schemes of dimension 1 | 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 sufficient conditions for the existence of the torsion component of the Picard functor of an algebraic stack, and for the finite generation of the Néron-Severi groups or of the Picard group itself. For this he proves a stacky version of the relative representability theorem from [\textit{P. Berthelot, A. Grothendieck} and \textit{L. Illusie} (eds.), Séminaire de géométrie algébrique du Bois Marie 1966/67. Lect. Notes Math. 225 (1971; Zbl 0218.14001), exp.XII]. A special care was taken to avoid superfluous tameness assumptions. Because of the fact that an algebraic stack might have infinite cohomological dimension, it is often convenient to assume that the algebraic stacks under consideration are tame. In Appendix A, the author shows that the cohomology of an arbitrary algebraic stack is tractable as soon as the base scheme is regular and noetherian. As a byproduct he get the semicontinuity theorem for algebraic stacks. algebraic stacks; Picard functor; Néron-Severi groups; torsion component; semicontinuity theorem Brochard, S, Finiteness theorems for the Picard objects of an algebraic stack, Adv. Math., 229, 1555-1585, (2012) Generalizations (algebraic spaces, stacks), Picard groups, (Equivariant) Chow groups and rings; motives Finiteness theorems for the Picard objects of an algebraic stack | 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 earlier work [Manuscr. Math. 111, No. 1, 105--139 (2003; Zbl 1089.11031)] the author defined two abelian varieties \(A\) and \(B\) over a field \(K\) to be isokummerian if there is a nonzero integer \(N\) such that for every positive integer \(n\) that is coprime to \(N\), the extension field of \(K\) generated by the \(n\)-torsion points of \(A\) is isomorphic to the extension field generated by the \(n\)-torsion points of \(B\). When \(K\) is finite, the author showed that two varieties are isokummerian if and only if the multiplicative subgroup \(\Phi_A\) of the complex numbers generated by the Frobenius eigenvalues of \(A\) is equal to the corresponding subgroup \(\Phi_B\) for \(B\).
Motivated by these earlier results, in the present paper the author investigates the question of which abelian varieties \(A\) over \(\mathbb F_q\) have the property that the set of Weil \(q\)-numbers in \(\Phi_A\) is equal to the set of Frobenius eigenvalues for \(A\). He shows that if \(A\) is simple and ordinary, and if the Galois group of the number field generated by the Weil polynomial for \(A\) is the Weyl group \(W_{2g}\) of the symplectic group \(\text{ Sp}(2g)\), then the only Weil \(q\)-numbers in \(\Phi_A\) are the Weil numbers of \(A\).
Using this result and work of \textit{N. Chavdarov} [Duke Math. J. 87, No. 1, 151--180 (1997; Zbl 0941.14006)], the author proves the following: For every prime power \(q\) and positive integer \(g\), let \(A_g(q)\) denote the set of isomorphism classes of principally-polarized \(g\)-dimensional abelian varieties over \(\mathbb F_q\), and let \(I_g(q)\) denote the subset consisting of those \(A\) such that any \(B\) iso\-kummerian with \(A\) is isogenous to a power of \(A\). Then for each \(g\), as \(q\) tends to infinity over the powers of a fixed prime, the ratio \(\#I_g(q) / \#A_g(q)\) tends to \(1\).
Using the large sieve and some results of \textit{S. A. DiPippo} and the reviewer [J. Number Theory 73, No. 2, 426--450 (1998); corrigendum ibid. 83, 182 (2000); Zbl 0931.11023], the author proves a similar result for isogeny classes of abelian varieties. abelian variety; Weil number; torsion point; monodromy; large sieve; isogeny class Kowalski, E., Weil numbers generated by other Weil numbers and torsion field of abelian varieties, J. Lond. Math. Soc. (2), 74, 2, 273-288, (2006) Abelian varieties of dimension \(> 1\), Isogeny, Curves over finite and local fields, Varieties over finite and local fields, Applications of sieve methods Weil numbers generated by other Weil numbers and torsion fields of abelian 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 This paper is concerned with the study of algebraic cycles on higher-dimensional algebraic varieties, by introducing and studying certain sets called \textit{base loci}. Such base loci are subsets \(B \subset X\) attached to \(k\)-cycles on a projective variety \(X\) over an algebraically closed field, where \(1 \leq k \leq \dim(X)-1\). They are generalizations of previously defined notions of stable, augmented and restricted base loci of Cartier divisors \(D\) on \(X\). Indeed, restricted and augmented basi loci of Cartier divisors were introduced in Section 1 of the paper [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. The goal of the author is to extend these definitions to arbitrary \(k\)-cycles, and to prove a number of interesting results about them.
The study of ample line bundles has a long history. Similarly, nef, base-point-free, big and semi-ample line bundles are important objects that arise in many areas of algebraic geometry. To ask whether a line bundle satisfies one of these properties is to ask about its \textit{positivity}. The cone of divisors defined by such and other positivity conditions, on a smooth projective variety \(X\), have been the subject of a great deal of work. The same holds for cones of curves defined by similar positivity conditions. The study of positivity for cycles of higher codimension and dimension have started to come into focus only recently. One direction of such a generalization is the notion of `ampleness' of subschemes \(Y \subset X\) of a projective variety \(X\), see [\textit{R. Hartshorne}, Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0208.48901)] and [\textit{J. C. Ottem}, Adv. Math. 229, No. 5, 2868--2887 (2012; Zbl 1285.14006)].
This paper takes another direction, focusing on positivity properties of \(k\)-cycles \(\alpha = \sum_{i = 1}^m n_i [Z_i]\) on a smooth projective variety \(X\) over an algebraically closed field, and their relation with certain base loci \(B \subset X\) defined by these cycles. Recall that the \textit{augmented base locus} \(\textbf B_+(D)\) and the \textit{restricted base locus} \(\textbf B_-(D)\) of a \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisor \(D\) on a normal complex projective variety \(X\) were defined and studied in [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. For every \(\mathbb Q\)-divisor, one has \(\textbf B_-(D) \subset \textbf B+(D)\), and \(D\) is called \textit{stable} if equality holds (see [loc. cit.]).
The first goal of this paper is to generalize these definitions to \(k\)-cycles on a projective variety of dimension \(n\) over any algebraically closed field, for arbitrary \(k\) with \(1 \leq k \leq n-1\). To explain this, let \(\mathcal Z_k(X)_{\mathbb R}\) be the vector space of real \(k\)-cycles, and let \(N_k(X)\) be its quotient by the numerical equivalence relation. For \(\alpha \in \mathcal Z_k(X)_{\mathbb R}\), let \([\alpha] \in N_k(X)\) be the numerical equivalence class of \(\alpha\). The key notion of this paper is:
Definition 1.1. Let \(\alpha \in N_k(X)\). Set \(|\alpha|_{\mathrm{num}} = \{e \in \mathcal Z_k(X)_{\mathbb R} \colon e \text{ is effective and } [e] = \alpha\}\). The \textit{numerical stable base locus} \(\textbf B_{\mathrm{num}}(\alpha) \subset X\) of \(\alpha\) is defined as follows. We take \(\textbf B_{\mathrm{num}}(\alpha) = X\) if \(|\alpha|_{\mathrm{num}} = \emptyset\) and \(\textbf B_{\mathrm{num}}(\alpha) = \cap_e \mathrm{Supp}(e)\) otherwise, where \(e\) ranges over the elements in \(|\alpha|_{\mathrm{num}}\). The \textit{augmented} and \textit{restricted base loci} of \(\alpha\) are the respective loci
\[
\textbf B_+(\alpha) = \bigcap_{A_1, \dotsc, A_{n-k}} \textbf B_{\mathrm{num}}(\alpha - [A_1 \cdots A_{n-k}])\text{ and }
\textbf B_-(\alpha) = \bigcap_{A_1, \dotsc, A_{n-k}} \textbf B_{\mathrm{num}}(\alpha + [A_1 \cdots A_{n-k}]),
\]
where \(A_1, \dotsc, A_{n-k}\) run through all ample \(\mathbb R\)-Cartier \(\mathbb R\)-divisors on \(X\).
This generalizes the definitions \(\textbf B_+(D)\) and \(\textbf B_-(D)\) for divisors \(D\) on \(X\) introduced in [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. In Sections 3 and 4, Lopez proves a number of properties of these base loci, analogues of similar properties for divisors (see [loc. cit.]).
The second goal of the author is to answer the following question: What are the positivity properties of \(\alpha \in N_k(X)\) when \(\textbf B_+(\alpha)\) or \(\textbf B_-(\alpha)\) are empty or properly contained in \(X\)? He proves (Theorem 1) that
\[
P_k(X) = \{[A_1 \cdots A_{n-k}] + \beta \in N_k(X) \mid A_1, \dotsc, A_k \text{ ample }\mathbb R\text{-Cartier }\mathbb R\text{-divisors}, \beta \in N_k(X) \colon \textbf B_{\mathrm{num}}(\beta) = \emptyset \}
\]
is a convex cone in \(N_k(X)\), open and full-dimensional if \(X\) is smooth. In addition, the author proves that, for \(\alpha \in N_k(X)\), the following assertions are true (see Theorems 2 and 3):
\begin{itemize}
\item[1.] \(\textbf B_+(\alpha) \subsetneq X\) if and only if \(\alpha\) is big;
\item[2.] \(\textbf B_+(\alpha) = \emptyset\) if and only if \(\alpha \in P_k(X)\);
\item[3.] \(\textbf B_-(\alpha) \subsetneq X\) implies that \(\alpha\) is pseudo-effective; the converse is true if the base field is uncountable;
\item[4.] \(\textbf B_-(\alpha) = \emptyset\) implies that \(\alpha \in \overline{P_k(X)}\), and the converse is true if \(X\) is smooth.
\end{itemize}
Finally, in analogy with the divisor case, the author defines a cycle \(\alpha \in N_k(X)\) to be \textit{stable} if \(\textbf B_-(\alpha) = \textbf B_+(\alpha)\) (see Definition 10.1), and proves several results on stable cycles generalizing some results of [loc. cit.] base loci; algebraic cycles; positivity of cycles Algebraic cycles, Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)) Augmented and restricted base loci of cycles | 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 the ring of integers of an algebraic number field \(K\), and denote by \(X\) a separated noetherian scheme which is projective, flat, and with one-dimensional fibers over the arithmetic curve \(S = \text{Spec} (A)\). Such a datum, with \(X\) being a two-dimensional arithmetic variety, is called a relative arithmetic curve over \(S\), and \(X\) is referred to as an arithmetic surface over \(A\). If the generic fibre of the structure morphism \(f : X \to S\) is smooth, then \(X\) is called a regular arithmetic surface over \(A\). For regular arithmetic surfaces, there is an intersection theory for divisors and line bundles, which has been established by S. Arakelov in the early 1970s [cf. \textit{S. Yu. Arakelov}, Math. USSR, Izv. 8(1974), 1167-1180 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002)]. Arakelov's intersection theory on arithmetic surfaces is based upon ``completing'' \(X\) by adding fibers (Riemann surfaces) over the archimedean places of the number field \(K\), and by enlarging both the divisor class group and the Picard group of \(X\) in a suitable way. The so-called Arakelov-Picard group (of isomorphism classes of ``admissibly metrized line bundles'') of \(X\) is equipped with an intersection pairing \(\langle , \rangle\), whose significance culminates in a Riemann-Roch-type theorem for regular arithmetic surfaces. In his celebrated paper ``Calculus on arithmetic surfaces'' [Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)], \textit{G. Faltings} constructed certain volume forms on the cohomology of a metrized line bundle \(L\) restricted to the fibres over the points at infinity, and these allowed to define the notion of an ``Euler characteristic'' \(\chi (L)\) for an admissibly metrized line bundle \(L\) on \(X\). Falting's Riemann-Roch theorem for regular arithmetic surfaces implies the formula \(\chi (L) = {1 \over 2} \cdot \langle L,L - K_{X/S} \rangle + \chi ({\mathcal O}_X)\), where \(K_{X/S}\) denotes the relative dualizing sheaf on \(X\) over \(S\). Actually, Faltings's Riemann-Roch theorem has a formulation not only in terms of numerical invariants but in terms of a certain metric isomorphism between certain sheaves on \(X\). In the meantime, the arithmetic Riemann-Roch theorem has been generalized to arbitrary projective and flat morphisms \(f : X \to Y\) between regular arithmetic varieties \(X\) and \(Y\). The ``arithmetic Riemann-Roch theorem'', in this general setting, is due to \textit{J. M. Bismut}, \textit{G. Lebeau}, \textit{M. Gillet} and \textit{C. Soulé}, and was obtained around 1990. Detailed accounts on these recent developments are given in the lecture notes by \textit{G. Faltings}: ``Lectures on the arithmetic Riemann-Roch theorem'', Ann. Math. Stud. 127 (1992; Zbl 0744.14016), and in the book of \textit{C. Soulé}, \textit{D. Abramovich}, \textit{J.-F. Burnol} and \textit{J. Kramer}: ``Lectures on Arakelov geometry'', Camb. Stud. Adv. Math. 33 (1992; Zbl 0812.14015).
In the present work, the author turns to the (so far still open) case of non-regular arithmetic varieties. His objects of study are singular arithmetic surfaces \(X\) over \(S\), whose fibers are Cohen-Macaulay curves, and whose relative dualizing sheaf \( K_{X/S}\) is invertible. The text gives a development of a modified Arakelov theory for arithmetic surfaces, which is general enough to handle also a large class of singular surfaces, and which culminates in a Riemann-Roch-type theorem for such arithmetic surfaces.
Basically, the work is divided into two parts.
The first part, chapters 1 through 3, provides a very detailed and self-contained discussion of Deligne's functorial intersection theory for invertible sheaves on families of curves. The author's approach, however, is more direct and concrete, especially tailored to the specific arithmetic situation to come. Bypassing the abstract framework, as developed by \textit{P. Deligne} [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)], and \textit{D. Mumford} and \textit{F. Knudsen} [Math. Scand. 39, 19-55 (1976; Zbl 0343.14008)], he gives a merely ad-hoc treatment which is closer to the construction of admissible metrics on arithmetic line bundles due to \textit{L. Moret-Bailly} [in: Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell, Astérisque 127, 29-87 (1985; Zbl 0588.14028)]. This approach also yields much of the needed duality theory, including the geometry of the dualizing sheaf, the associated duality isomorphisms, and the respective Riemann-Roch isomorphism.
The second part, chapters 4 and 5, is devoted to developing a class of intersection functions on complex curves, as they occur as fibers over the points at infinity in (possibly singular) arithmetic surfaces, and to extending the relative Riemann-Roch isomorphism (of the first part) to the ``completed'' object. The intersection functions constructed here behave analogously to the canonical Green's functions used in the smooth case, but they also show major differences to them. For example, Arakelov's canonical Green's function for regular arithmetic surfaces is unique, whereas the author's functions are parametrized by a finite-dimensional vector space. Also, Arakelov's function is bounded from below, whereas the author's functions are not bounded but, nevertheless, asymptotically nicely behaved at the singularities. Using his intersection functions in combination with his adapted functorial intersection theory on singular relative curves, the author obtains metrized invertible sheaves on particular (see above) singular arithmetic surfaces, which have a well-defined degree and a meaningful Euler characteristic. The general Riemann-Roch isomorphism extended to the ``completed'' (singular) arithmetic surfaces yields then a Riemann-Roch formula, just by taking degrees on both sides of it.
Altogether, the present work is a comprehensive and self-contained treatise which provides not only a generalization of Faltings's Riemann-Roch theorem to certain singular arithmetic surfaces, but also an alternative approach to two-dimensional Arakelov theory from a functorial point of view. arithmetic curve; Arakelov-Picard group; arithmetic Riemann-Roch theorem; non-regular arithmetic varieties; Green's functions; Riemann-Roch isomorphism; Arakelov theory Arithmetic varieties and schemes; Arakelov theory; heights, Singularities of surfaces or higher-dimensional varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields An arithmetic Riemann-Roch theorem for singular 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 article studies the existence of Néron models of abelian varieties over a regular base scheme \(S\) of any dimension. Usually, the theory of Néron models is considered over Dedekind schemes. The definition in this more general setting is the following: Let \(A\rightarrow U\) be an abelian scheme over an open dense subscheme \(U\) of \(S\). A Néron model for \(A\) is a smooth separated algebraic space \(N\rightarrow S\) together with an isomorphism \(A\rightarrow U_U:=N\times_SU\) satisfying the universal property: If \(T\to S\) is a smooth morphism of algebraic spaces and \(f:T_U\rightarrow N_U\) is a smooth \(U\)-morphism of algebraic spaces, then there exists a unique \(S\)-morphism \(F:T\rightarrow N\) such that \(F|_U=f\).
If the base scheme is of dimension greater 1, such Néron models do not exist, in general. In the case of a Jacobian of a relative semi-stable curve with smooth generic fiber, the author gives a precise criterion for the existence of a Néron model; cf. Theorem 1.2. This shows in particular that even in this special situation a Néron model does not exist without further restriction on the relative curve. The criterion in Theorem 1.2 is expressed by a condition on the local structure of the singularities of the fibres of the relative curve; cf. Definition 2.11 and 2.2. Due to the extension property for invertible sheaves from \(T\times_SU\) to \(T\times_SS\), the relative Picardfunctor of invertible sheaves of total degree zero is a helpful means for constructing a Néron model. Jacobians, Prym varieties, Families, moduli of curves (algebraic) Néron models of Jacobians over base schemes of dimension greater than 1 | 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 There are several possible extensions of the usual definition of the (degree \(d\)) Jacobian of a smooth curve to singular ones. The paper under review compares two of them in the case of a single stable curve \(X\) over an algebraically closed field.
On the one hand, one can consider \(\bar P^d_X\), the coarse moduli scheme of of equivalence classes of degree \(d\) balanced line bundles on quasistable curves having \(X\) as stable model; equivalently, it is the fiber of \({\overline{Pic}}_{g,d} \to \bar M_g\) over \(X\), where \(\bar {Pic}_{g,d}\) is the compactification of the universal Picard stack constructed by the author in [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)]. The scheme \(\bar P^d_X\) is proper, and its smooth locus is a group scheme.
On the other hand, one can choose a smoothing \(f : \mathcal X \to S\) over a DVR, where the generic fiber \(\mathcal X_\eta\) is smooth and \(\mathcal X_s \cong X\). Then the closed fiber of the Néron model of \({Pic}^d \mathcal X_\eta\), denoted \(N^d_X\), provides another generalization of the degree \(d\) Jacobian of \(X\).
The main theorem of the paper is a characterization of the curves for which \(N^d_X\) coincides with the smooth locus of \(\bar P^d_X\) (in which case one says that \(\bar P^d_X\) is of Néron type). It turns out that this depends only on the dual graph of \(X\). Moreover, this property is invariant under smoothing a non-separating node. One can therefore assume that all nodes are separating. The theorem is that for a curve \(X\) all of whose nodes are separating, \(\bar P^d_X\) is of Néron type if and only \(X\) is \(d\)-general. stable curve; Picard scheme; Néron model; compactification; balanced line bundle Caporaso, L., Compactified Jacobians of Néron type, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 23, 2, 213-227, (2012), MR 2924900 Picard schemes, higher Jacobians, Families, moduli of curves (algebraic), Jacobians, Prym varieties Compactified Jacobians of Néron type | 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 complete integral curve of arithmetic genus \(g \geq 2\) over an algebraically closed field of arbitrary characteristic. The canonical model \(C'\) of \(C\) was defined by \textit{M. Rosenlicht} [Ann. Math. (2) 56, 169--191 (1952 (1952; Zbl 0047.14503)], where he introduced the dualizing sheaf \(\omega\).
In this paper the authors give refined statements and modern proofs of Rosenlicht's results about \(C'\):
They give a version of Clifford's Theorem more general with respect to Rosenlicht's one and they characterize the case in which the canonical model \(C'\) is equal to the rational normal curve \(N_{g-1}\).
Then they show that, if \(C\) is nonhyperelliptic, the canonical map induces an open embedding of its Gorenstein locus into its canonical model \(C'\); the proof involves the blowup of \(C\) with respect to \(\omega\).
Also, using Castelnuovo Theory and some results due to \textit{V. Barucci} and \textit{R. Fröberg} [J. Algebra 188, No. 2, 418--442 (1997; Zbl 0874.13018)], they give necessary and sufficient conditions for the canonical model \(C'\) to be arithmetically normal.
Finally, they prove Rosenlicht's Main Theorem which essentially asserts that, if \(C\) is nonhyperelliptic, the canonical map between the blowup of \(C\) with respect to \(\omega\) and \(C'\) is an isomorphism; they apply this result to characterize the non-Gorenstein curves \(C\) whose canonical model is projectively normal. canonical model; singular curve; non-Gorenstein curve Kleiman, S.L., Martins, R.V.: The canonical model of a singular curve. Geom. Dedic. 139, 139--166 (2009) Singularities of curves, local rings, Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory) The canonical model of a singular 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 \(f:\mathcal X\to\mathcal C\) be a morphism of degree \(k\) between smooth connected projective curves of genus \(g\) and \(q\) respectively. In this paper the study of cohomological properties of the sheaf \(E:={f_*}(\mathcal O_\mathcal X)/\mathcal O_\mathcal C\) is considered. If \(q=0\), \(E\) is a direct sum of \(k-1\) line bundles, and the rank 1 summands of \(E\) uniquely determine the scrollar invariant of \(f\) [see \textit{F. O. Schreyer}, Math. Ann. 275, 105--137 (1986; Zbl 0578.14002)].
In this paper, the author considers the analogous problem for \(q>0\). He also introduces the notion of a (total) Weierstrass point of \(f\) at \(P\in\mathcal X\) as follows. Let \(N(f,P)\) and \(n(f,P)\) denote respectively the sequence of integers \(h^0(C,f_*(\mathcal O_\mathcal X)(tP))\) and \(h^0(C,E(tP))\), \(t\in \mathbb N_0\). The point \(P\) is called a total Weierstrass point (Weierstrass point) if \(N(f,P)\neq N(f,Q)\) (\(n(f,P)\neq n(f,Q)\)) for a generic point \(Q\in \mathcal X\). The author presents several examples to illustrate his results. semistable vector bundle; Weierstrass point Coverings of curves, fundamental group, Vector bundles on curves and their moduli, Special algebraic curves and curves of low genus, Riemann surfaces; Weierstrass points; gap sequences Cohomological properties of multiple coverings of smooth projective 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 article considers the problem of evaluation of the quotient of a Jacobian variety \(J\) of a curve \(C\) by a maximal isotropic subgroup \(V\) in its \(\ell\)-torsion for \(\ell\) an odd prime integer different from the characteristic of \(K\). Firstly, the problem is considered generally. It has been considered how to quickly design and compute standard functions on \(J/V\). It has been shown that if the dimension \(g\) of \(J\) equals two, the quotient is the Jacobian of another curve \(D\). The complexity of evaluating standard functions on Jacobians including Weil functions and algebraic theta functions has been bounded. A formula for the divisor of certain functions on J defined using determinants has been given. An expression for eta functions as combinations of these determinants has been deduced. The resulting algorithm for evaluating Eta function has been detailed. One has deduced a bound for the complexity of computing a basis of sections for the bundle associated with a multiple of the natural polarization of \(J\). The algebraic definition of canonical theta functions has been recalled and the evaluating canonical theta functions has been given. One has been bounded the complexity evaluating functions on the quotient of \(J\) by a maximal isotropic subgroup \(V\) in \(J [\ell]\) when \(\ell\) is and odd prime different from the charscteristic of \(\mathbf K\). Assuming that the characteristic p of \(\mathbf K\) is odd one has been bounded the complexity of computing an isogeny \(J_{C} \to J_{D}\) between two Jacobians of dimension two, the expected form of such isogeny has been given. A specific algorithm for genus 2 curves have been presented. A complete example with \(\mathbf K\) being the field with 1009 elements has been given. Jacobian variety; theta functions; eta function Couveignes, J-M; Ezome, T, Computing functions on Jacobians and their quotients, LMS J. Comput. Math., 18, 555-577, (2015) Isogeny, Computational aspects of algebraic curves, Effectivity, complexity and computational aspects of algebraic geometry, Theta functions and abelian varieties Computing functions on Jacobians and their quotients | 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 normal algebraic variety over a perfect field and \(\Delta\) be a \(\mathbb{Q}\)-divisor such that \(K_X+\Delta\) is \(\mathbb{Q}\)-Cartier. In this paper, the authors define an ideal, in artibrary equal characteristic as follows:
\[
J:=\cap_Y\text{Image}(\pi_*O_Y(\lceil K_Y-\pi^*(K_X+\Delta)\rceil)\to K(X))
\]
where the intersection runs over all generically finite proper separable maps \(\pi\): \(Y\to X\) where \(Y\) is regular and the map to the function field \(K(X)\) is induced by the Grothendieck trace map \(\text{Tr}_{\pi}\): \(\pi_*\omega_Y\to\omega_X\). The main results are: (1) If \(X\) has equal characteristic \(0\), then \(J=J(X,\Delta)\) is the multiplier ideal of \((X,\Delta)\) and (2) If \(X\) has equal characteristic \(p>0\), then \(J=J(X,\Delta)\) is the test ideal of \((X,\Delta)\). This result in characteristic \(0\) can be viewed as a generalization of the transformation rule for multiplier ideals under alterations, and in positive characteristic gives a new characterization of test ideals and \(F\)-rational singularities under alterations.
The authors find many applications of the main results. One of them is a Nadel-type vanishing theorem in characteristic \(p>0\). Let \((X,\Delta)\) be as above and let \(L\) be a Cartier divisor such that \(L-(K_X+\Delta)\) is big and semi-ample \(\mathbb{Q}\)-divisor. Then there exists a finite surjective map \(f\): \(Y\to X\) such that the natural map
\[
f_*O_Y(\lceil K_Y+f^*(L-K_X-\Delta)\rceil)\to O_X(L)
\]
induced by the trace map has image \(\tau(X,\Delta)\otimes O_X(L)\), and the induced map on cohomology
\[
H^i(Y, O_Y(\lceil K_Y+f^*(L-K_X-\Delta)\rceil))\to H^i(X, \tau(X,\Delta)\otimes O_X(L))
\]
vanishes for every \(i>0\). Using this vanishing theorem, the authors obtain results on extending sections. test ideals; multiplier ideals; alterations; \(F\)-rationality; vanishing theorems Blickle, Manuel; Schwede, Karl; Tucker, Kevin, \textit{F}-singularities via alterations, Amer. J. Math., 137, 1, 61-109, (2015) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Multiplier ideals \(F\)-singularities via alterations | 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 studies base points of the generalized theta divisor on the moduli space of vector bundles on a smooth algebraic curve of genus \(g > 0\).
Let \(X\) be a smooth algebraic curve of genus \(g\). Consider its Picard group Pic\(^0(X)\) and its Jacobian variety Jac\((X)\), which are each other dual, and the Poincaré bundle \(\mathcal P\) on their product. Consider the Fourier-Mukai transform with kernel \(\mathcal P\) and apply it to the line bundle on Pic\(^0(X)\) associated to \(-m \Theta\). The vector bundles \(P_m\) defined in \textit{M. Raynaud} [Bull. Soc. Math. France 110, 103--125 (1982; Zbl 0505.14011)] are obtained here by applying to the image of such a line bundle the natural involution on the Jacobian and by pulling it back to \(X\).
Vector bundles \(E\) on \(X\) such that \(H^0(E \otimes L) \neq 0\) for any line bundle \(L\) form the base locus of the generalized theta divisor. The first example of such vector bundles is given by \(P_m\) for any integer \(m\). The author then shows that a vector bundle \(E\) on \(X\) has such a property if and only if there exists a nontrivial morphism \(P_m \to E\) for \(m\gg 0\). A more accurate analysis fixes an upper bound for \(m\) to be \(r g\), where \(r\) is the rank of the vector bundle.
In the last part of the paper, the author studies minimal bundles, that is bundles in the base locus which does not admit any subbundle in the base locus, and base points for higher rank bundles. base points of the theta divisor; generalized theta divisor; Fourier-Mukai transform Hein G.: Raynaud's vector bundles and base points of the generalized Theta divisor. Mathematische Zeitschrift 257, 597--611 (2007) Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard schemes, higher Jacobians Raynaud's vector bundles and base points 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 Elementary transformations along a divisor \(D\) introduced by \textit{M. Maruyama} [in algebraic geometry, Proc. int. Conf., La Rabida 1981, Lect. Notes Math. 961, 241-266 (1982; Zbl 0505.14009)] have been widely used. Here the author shows that under suitable assumptions one can find elementary transformations which transform a vector bundle to a vector bundle less unbalanced (more stable) than the original bundle. This is a useful technique. The case \(D=\) a rational smooth curve on a smooth surface is studied in detail. One example with \(D\) a smooth elliptic curve is also given. The technique is applied to the study of stable rank 2 vector bundles \(F\) on \(\mathbb{P}^ 2\) in positive characteristics. Normalise \(c_ 1(F)=0\), \(-1\). For a general line \(l\) if \(F|_ l={\mathcal O}(t)\oplus{\mathcal O}(-t+c_ 1(F))\), \(t>0\) then \(t\) is called the splitting order of \(F\). Starting with such a bundle, other bundles with splitting order \(t-1\) (with same or different Chern classes) are produced by elementary transformations. It is proved that every rank 2 stable bundle on \(\mathbb{P}^ 2\) with splitting order \(t\) is the limit of an irreducible, unirational, flat family of stable vector bundles of splitting order \(t-1\). elementary transformations along a divisor; splitting order of vector bundles Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Strange bundles on \(\mathbb{P}^ 2\) and elementary transformations | 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 0683.00009.]
This is a very nice down-to-earth introduction to Grothendieck's theory of cotangent complex applied to a concrete situation of deformations of a hyperplane section. It is applied to the problem of explicit description of graded canonical (or subcanonical) rings of surfaces and curves in terms of generators and relations.
The most interesting part of this article is a collection of concrete examples. For instance the author works out in detail the deformation theory in degree \(\leq 0\) of the half-canonical ring of a hyperelliptic curve of genus \( 6.\) This can be applied to some Horikawa surfaces (numerical quintics) whose canonical system contains hyperelliptic curves of genus 6. This example serves a warm-up example to the real job: to compute the canonical ring of surfaces with \(p_ g=0\) with torsion \({\mathbb{Z}}/2\). In this example the ring needs 8 generators, 20 relations, 64 syzygies and 90 second syzygies. An attempt to implement the computations using the Maple program has been made; its success could eventually decide the irreducibility of the moduli space of such surfaces.
Among concrete results proved in this paper we cite only one: Let X be a canonical surface of general type with \(q=0\), \(p_ g\geq 2\) and \(K^ 2\geq 3\). Assume X has an irreducible curve in its canonical linear system. Then the canonical ring is generated in degrees \(\leq 3\) and related in degrees \(\leq 6\). cotangent complex; deformations of a hyperplane section; half-canonical ring of a hyperelliptic curve; Horikawa surfaces; canonical ring of surfaces; Maple program; canonical surface of general type Miles Reid, Infinitesimal view of extending a hyperplane section --- deformation theory and computer algebra, Algebraic geometry (L'Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 214 -- 286. Formal methods and deformations in algebraic geometry, Computational aspects of algebraic curves, Computational aspects of algebraic surfaces, Surfaces of general type, Local deformation theory, Artin approximation, etc. Infinitesimal view of extending a hyperplane section - deformation theory and computer algebra | 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 general introduction the authors briefly discuss the classical approach to computing the cohomology of a manifold by means of differential forms in the algebraic, analytic and smooth settings originated with E. Cartan, G. de Rham, A. Grothendieck, P. Deligne and others. In fact, the main idea is to set up an isomorphism between hypercohomologies of appropriate De Rham complexes or, in other words, to prove the so-called comparison theorem. In the book under review the authors give a purely algebraic proof of some known comparison theorems based on the modern theory of differential modules. This enables them to avoid Hironaka's resolution of singularities, monodromy arguments and some other standard constructions of differential-analytic character. As a result they obtain a comparison theorem in the non-archimedean case and they describe some useful properties of connections in the \(p\)-adic setting.
The book is divided into 4 chapters and 5 appendixes. The first chapter provides a self-contained exposition of the algebraic theory of regularity in several variables. Let \(X\) be an algebraic variety over an algebraically closed field of characteristic zero, let \(Z\) be a divisor with strict normal crossings in \(X,\) and let \(\nabla\) be an algebraic integrable connection on \(X\setminus Z.\) There are known at least \(4\) different conditions under which the connection \(\nabla\) is regular singular along the divisor \(Z.\) The classical analysis of regularity conditions is based essentially on transcendental arguments and Hironaka's resolution of singularities [see \textit{P. Deligne}, ``Equations différentielles à points singuliers réguliers'', Lect. Notes Math. 163 (1970; Zbl 0244.14004)]. In contrast with earlier works the authors prove the equivalence of these conditions making use of the theory of \(D\)-modules and of simple geometric and algebraic considerations.
The second chapter deals with the notion of irregularity studied in a similar manner. First the authors remark that one of the characterization of regularity along a divisor can be formulated as follows: Logarithmic differential operators of increasing order act with poles of bounded order at the generic point of \(Z.\) Thus a natural notion of a generalized Poincaré-Katz rank of irregularity of \(\nabla\) at \(Z\) arises. Then the stratification of the singular divisor \(Z\) by Newton polygons is introduced. The authors study the variation of Newton polygons, prove a semicontinuity theorem and obtain a formal decomposition of an integrable connection at a singular divisor. The last section of this chapter contains a very useful and less known material concerning properties of the indicial polynomial and Turrittin exponents of a connection.
The next chapter develops a new approach to the study of direct images of connections with respect to a smooth morphism \(X \rightarrow S\) between smooth algebraic varieties in characteristic zero or, in other words, to the study of direct images of the relative de Rham complex over \(X/S\) with coefficients, endowed with the Gauss-Manin connection [\textit{N. M. Katz}, Publ. Math., Inst. Hautes Étud. Sci. 39, 175--232 (1970; Zbl 0221.14007)]. The authors prove the generic and fundamental finiteness, regularity, monodromy and base change theorems for direct images of regular algebraic connections with using neither resolution of singularities, nor the holonomy theory. The main tool of their proofs is the dévissage method from Artin's theory of elementary fibrations [cf. \textit{Y. André} and \textit{F. Baldassarri} in: Arithmetic geometry, Proc. Symp., Cortona 1994, Symp. Math. 37, 1--22 (1997; Zbl 0936.14014)]. As a result the problem reduces to the simplest case of an ordinary differential operator in one variable.
In the last chapter an effective proof of the classical Grothendieck-Deligne comparison theorem is presented. Since this proof does not rely on resolution of singularities, does not make use of moderate growth conditions and properties of the monodromy, the authors' arguments remain valid also in the \(p\)-adic setting. This leads to a simple proof of the Kiehl-Baldassarri theorem [\textit{F. Baldassarri}, Math. Ann. 280, No.~3, 417--439 (1988; Zbl 0651.14012)]. Among other things the authors explain how the comparison theorem extends to the case of an irregular connection in the non-archimedean setting.
Five appendixes contain some additional interesting material. In particular, a non-published original proof of Berthelot's comparison theorem between two different notions of the dual of a differential module is reproduced and a natural interpretation of Dwork's algebraic dual theory in terms of the relative algebraic de Rham cohomology with compact supports is given.
The book is written in a clear and concise style, almost all key topics are followed by examples, non-formal remarks and comments. The authors underline that they use neither the language of derived categories nor the general theory of holonomic modules. Thus, a major part of the book is accessible to non-specialists and graduate students while the main results should be of interest to specialists of \(D\)-modules and differential equations as well as to algebraic and arithmetic-algebraic geometers. differential modules; comparison theorems; integrable connections; regular singular connections; direct images; Gauss-Manin connections; Gauss maps; irregularity of connections; Turrittin exponents; Fuchs exponents; indicial polynomials; relative De Rham complex; fibrations; logarithmic differential operators; Riemann existence theorem; rigid-analytic cohomology; \(p\)-adic de Rham cohomology; Dwork's dual theory André, Y., Baldassarri, F.: De Rham Cohomology of Differential Modules on Algebraic Varieties. Progress in Mathematics, vol.\ 189. Birkhäuser, Basel (2001) de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Modules of differentials, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) De Rham cohomology of differential modules on 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 Let \(R\) be a discrete valuation ring with field of fractions \(K\). Let \(A_K\) be an abelian variety over \(K\) with dual variety \(A_K'\). Then, with regard to their respective Néron models \(A\) and \(A'\), there is a bilinear pairing of the component groups of these Néron models
\[
\langle,\rangle: \Phi_A \times \Phi_{A'} \to\mathbb{Q}/ \mathbb{Z},
\]
which has been introduced by \textit{A. Grothendieck} [SGA 7, I, Lect. Notes Math., 288, Springer Verlag (1972; Zbl 0237.00013)] thirty years ago. Grothendieck himself conjectured that this pairing is perfect. In the meantime, Grothendieck's conjecture has been proved in various special cases, but also series of counter-examples in other cases have been found. In the paper under review, the authors investigate the case of the Jacobian \(J_K\) of a smooth proper curve \(X_K\) admitting a \(K\)-rational point. Their main result is an explicit formula for the pairing \(\langle,\rangle\), which is then used to prove Grothendieck's conjecture for certain types of such Jacobians, on the one hand, and disprove it for particular Jacobians in the case where the residue field of the ground ring \(R\) is imperfect. The method of proof is purely geometric and based on the invention of a certain pairing attached to a symmetric matrix, which leads to a more practical description of Grothendieck's pairing. Many explicit examples, at the end of the paper, demonstrate the power of the authors' approach. abelian varieties over arithmetic ground fields; Néron models; Jacobians over arithmetic ground fields Bosch, S., Lorenzini, D.: Grothendieck's pairing on component groups of Jacobians. Invent. Math. \textbf{148}(2), 353-396 (2002) Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties Grothendieck's pairing on component groups of 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 Let \(C\) be a smooth projective curve of genus \(g \geq 3\) over \({\mathbb C}\), and let \(E\) be a vector bundle of rank \(r\) and slope \(2g-1\) over \(C\). If \(E\) is general in moduli, then the set
\[
\{ \ell \in \mathrm{Pic}^{-g} (C) : h^0 ( C, E \otimes \ell ) \geq 1 \}
\]
is the support of a divisor \(\Theta_E\), the \textit{theta divisor of \(E\)} (see Beauville-Narasimhan-Ramanan [\textit{A. Beauville} et al., J. Reine Angew. Math. 398, 169--179 (1989; Zbl 0666.14015)]). Such generalised theta divisors have been the subject of much interest; see for example [\textit{A. Beauville}, Adv. Stud. Pure Math. 45, 145--156 (2006; Zbl 1115.14025)].
In this paper, the author studies an interesting connection between \(\Theta_E\) and the geometry of the \textit{tautological model} of \(E\). The latter is the image \(P_E\) of the natural map
\[
{\mathbb P}(E^*) \dashrightarrow {\mathbb P} ( H^0 ( C, E )^* ) \cong | {\mathcal O}_{{\mathbb P}(E^*)} (1) |^* .
\]
Let \({\mathcal V}_{g, g-2} ( P_E )\) be the variety of \((g-2)\)-planes which are \(g\)-secant to \(P_E\). The author shows that if \(C\) is a Petri curve and \(r > g-1\), then for \(E\) stable and general in moduli, \(\Theta_E\) is birational to an irreducible component of \({\mathcal V}_{g, g-2} ( P_E )\). Conversely, with some assumptions on \(E\), the existence of a suitable component of \({\mathcal V}_{g, g-2} ( P_E )\) implies that \(E\) admits a theta divisor. Another result which may have further applications is Proposition 8.3 (1), which gives a sufficient condition for \(E\) \textit{not} to admit a theta divisor. curves; vector bundles; theta divisors; moduli spaces; tautological map Vector bundles on curves and their moduli Theta divisors and the geometry of tautological model | 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 theory of \(\mathbb Q\)-Cartier divisors on the space of \(n\)-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of \(\mathbb Q\)-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in \(\mathbb P^r\) is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree \(d\) plane curves is explicitly evaluated. enumerative geometry; Cartier divisors; \(n\)-pointed, genus 0, stable maps; intersection products of \(\mathbb{Q}\)-divisors; characteristic numbers of rational curves; 1-cuspidal rational locus R. Pandharipande, Intersections of \(\({ Q}\)\)-divisors on Kontsevich's moduli space \(\({\overline{M}}_{0, n}({ P}^{r}, d)\)\) and enumerative geometry. Trans. Am. Math. Soc. 351(4), 1481-1505 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Birational geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves, Picard groups Intersections of \(\mathbb{Q}\)-divisors on Kontsevich's moduli space \(\overline M_{0,n}(\mathbb P^r,d)\) and enumerative 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 A main motivation for this paper is the Abelianisation program iniated by \textit{N. Hitchin} [``Stable bundles and integrable systems'', Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)].
Let \(W\) be a finite group, \(T\) an abelian group and \(N\) an extension of \(W\) by \(T\). Let \(\pi : Z \to X\) be an étale Galois cover of smooth projective curves of Galois group \(W\). Let \({\mathcal M}_X(N)\) (respectively \({\mathcal M}_X(W)\)) be the set of isomorphism classes of principal \(N\)-bundles (respectively \(W\)-bundles) on \(X\). The main result in this paper is the computation of the fiber of the quotient by \(T\) map
\[
q_T : {\mathcal M}_X(N) \to {\mathcal M}_X(W)
\]
as a precise torsor over the Prym-Donagi variety. So \({\mathcal M}_X(N)\) is a finite union of torsors over abelian varieties. Interesting applications to the Abelianisation program are are also given. For completeness, it should also be noted that a known result (but for which no proof is published) about Mumford groups is proved. abelian variety; algebraic groups; Abelianisation program; G-bundles Jacobians, Prym varieties, Vector bundles on curves and their moduli Principal \(N\)-bundles for \(N\) an extension of a finite group by an abelian 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 purpose of this comprehensive book is twofold. On the one hand, it gives a systematic exposition of the general theory of algebraic curves, their Jacobians, and their uniformizations over non-Archimedean fields. On the other, it describes recent results of the author on abeloid varieties, which are the rigid-analytic equivalent of compact Lie groups over the complex numbers.
Both themes are thoroughly explained and explored by the author. The work is written in a lucid writing style, and details of proofs are always provided. It contains many important contributions to the literature on rigid geometry, both from the point of view of research and from that of exposition. The new results on the construction of Jacobians of general rigid-analytic curves and the theorems on abeloid varieties in the second half of the book are highly satisfying, and these results form the theoretical highlight of the book. The work additionally fills a gap in the literature by the detail that it provides in its consideration of rigid-analytic curves, also when treating the more classical subjects of Mumford curves, their automorphic functions and their Jacobians.
The book is therefore highly useful both as a standard reference and as a main resource for an advanced graduate course on rigid geometry. It is to be hoped, however, that the large amount of typos, among which there are some rather glaring ones (``ground braking article'' in the preface to Chapter 2, just to name one example) will be removed in later editions.
Chapter 1 treats classical rigid geometry. It gives standard results on restricted power series rings over non-Archimedean fields before treating affinoid spaces, which are glued together via the Grothendieck topology to form rigid-analytic spaces. Kiehl's results on coherent sheaves on rigid-analytic spaces are described, after which line bundles and their rigidifications are considered in detail. The chapter concludes by proving the algebraizability of (smooth and geometrically connected) proper analytic curves, which leads to the Riemann-Roch theorem for these curves.
After first considering Tate's elliptic curves, Schottky groups and Mumford curves form the focus of Chapter 2. Mumford curves are rigid-analytic curves that allow a uniformization by an open subset of the rigid-analytic projective line, and Schottky groups are the discrete groups that intervene in this uniformization. The chapter contains a very careful exposition on Schottky groups, filling in many details left open in other references. It then considers the corresponding automorphic functions and Drinfeld's polarization. The latter is the rigid-analytic version of the Riemann form on the Jacobian on the corresponding Mumford curve. Rigid-analytic complex tori and a characterization of the algebraic ones among these are described; the results here are largely analogous to those over the complex numbers. After this, the chapter concludes with a thorough study of the Jacobians of Mumford curves, their Poincaré bundles, and the Riemann vanishing theorem in the rigid-analytic context.
Chapter 3 considers the relations between rigid geometry and formal geometry. It first discusses the classical result that for a non-Archimedean field \(K\) with valuation ring \(R\), there is an equivalence of categories between the category of quasi-compact and quasi-separated admissible formal \(R\)-schemes, localized by admissible blow-ups, and the category of quasi-compact and quasi-separated rigid-analytic \(K\)-spaces. This result is due to Raynaud, Bosch and the author, using crucial flattening techniques developed by Gruson and Raynaud. Finiteness theorems by Grauert-Remmert and Gruson are also discussed. The book then moves on to the question of the existence of \(R\)-models with a reduced special fiber. After treating methods of Grauert-Remmert, Gruson, and Epp, the author gives what he calls the ``natural'' approach to the reduced fiber theorem, which is explained in great detail.
In Chapter 4, the stable reduction theorem for smooth projective algebraic curves over non-Archimedean fields is proved. The proof does not require any desingularization result. It proceeds in two steps: first the results of the previous chapter are used to obtain a model with reduced special fiber, and after this the remainder of the proof rests on an analysis of the formal fiber of points on the special fiber. This leads to an analytic way to study the blow-ups and blow-downs required to obtain a model with stable reduction. After this, the universal cover of a rigid-analytic curve is constructed, and a criterion on the stable reduction is given for this universal cover to be an open subset of the projective line, and with it, for a geometrically connected smooth projective of genus \(g \geq 2\) with semi-stable reduction to be a Mumford curve: namely, the curve should have split rational reduction (or, equivalently, \(\text{rk} \; H^1 (X_K, \mathbb{Z}) = g\)).
Chapter 5 considers Jacobian varieties of general smooth projective curves over a non-Archimedean field and constructs their uniformization, thus vastly generalizing the result for Mumford curves in Chapter 2. This construction is intimately bound up with the analysis of the special fiber of a semi-stable model of the curve, whose generalized Jacobian is an extension of an abelian variety by a torus; such extensions are analyzed in great detail. By a pushout construction, this leads to the sought-for universal cover of \(J_K\) of a smooth projective curve over \(K\), and more precisely to the so-called \textit{Raynaud representation} \(J_K = \widetilde{J}_K / M\) of \(J_K\), where \(M\) is a lattice in the universal cover \(\widetilde{J}_K\). As a consequence of these results, one obtains the semi-abelian reduction theorem of Grothendieck.
Chapter 6 studies the general theory of Raynaud extensions; it shows that the universal cover \(\widetilde{J}_K\) is in fact an extension of an abelian variety (with good reduction) by a torus. This question is studied in the broader context of \textit{abeloid varieties}, that is, connected smooth proper rigid-analytic groups varieties. A Raynaud representation of an abeloid variety is a representation of said variety as a quotient \(E / M\), where \(E\) an extension of a proper rigid-analytic group with good reduction by a torus and where \(M\) is a lattice in \(E\). In this context, the author also studies duality, polarizations and degenerations of abelian varieties over \(K\).
Finally, Chapter 7 is devoted to abeloid varieties themselves. It proves the main structure theorem for these varieties, namely that they admit a Raynaud representation after a suitable extension of the base field. This results generalizes both the rigid-analytic uniformization of abelian varieties and Grothendieck's semi-abelian reduction theorem. Its proof is based on reducing considerations to the semi-stable reduction theorem for smooth curve fibrations, using compactifications to cover the given abeloid variety by a finite number of these and then applying suitable approximation theorems derived earlier on. As a consequence of these results, it is shown that every abeloid variety admits a dual.
The appendices give some needed background on graphs, torus extensions and group cohomology, making the book as self-contained as reasonably possible. rigid-analytic geometry; curves; Jacobians; abelian varieties; uniformization; Raynaud extensions; abeloid varieties; Schottky groups; Mumford curves Lütkebohmert, Werner, Rigid geometry of curves and their Jacobians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 61, xviii+386 pp., (2016), Springer, Cham Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rigid analytic geometry, Jacobians, Prym varieties, Group schemes, Abelian varieties and schemes Rigid geometry of 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 Let \(X\) be a variety over a finite field \(\mathbb{F}_q\). The category \(D^{\text{Weil}}(X)\) consisting of complexes having the property that the eigenvalues of the Frobenius action on the stalks have a certain form, plays an important role in Deligne's work on the Weil conjectures. It is known that this category is preserved by the usual sheaf operations. However, for some applications this category and its abelian subcategory \(P^{\text{Weil}}(X)\) of perverse sheaves are too large. If \(X\) is, for example, the flag variety of a reductive algebraic group, \(P^{\text{Weil}}(X)\) can be substituted by a category \(P^{\text{mix}}_\mathcal{S}(X)\) of perverse sheaves which are smooth along a certain stratification \(\mathcal{S}\) and such that the Frobenius action is semisimple and has integral eigenvalues. This latter category is then a so-called Koszul category.
The motivating questions of the article under review are whether there is a triangulated category \(D^{\text{mix}}_\mathcal{S}(X)\subset D^{\text{Weil}}_\mathcal{S}(X)\) analogous to \(P^{\text{mix}}_\mathcal{S}(X)\) and whether the usual sheaf operations preserve the stronger conditions on the Frobenius action mentioned above. The authors provide answers for so-called affable stratifications.
The paper is divided into three parts. In the first part the authors describe various notions from homological algebra needed later on. More precisely, Section 2 deals with mixed and Koszul categories, while in Section 3 infinitesimal extensions are introduced and studied. Here, if \(\mathcal{D}\) is a triangulated category, the infinitesimal extension \(\mathcal{ID}\) is the category having the same objects but with \(\text{Hom}_\mathcal{ID}(A,B)=\text{Hom}_\mathcal{D}(A,B)\oplus \text{Hom}_\mathcal{D}(A,B[-1])\). Note that \(\mathcal{ID}\) is not a triangulated category, but some facts from homological algebra still hold. Very roughly, one can establish results about \(\mathcal{ID}\) by using the obvious functors between \(\mathcal{D}\) and \(\mathcal{ID}\). The following section deals with Orlov categories, which are certain additive categories with finite-dimensional Hom-spaces equipped with a degree function from the set of isomorphism classes of indecomposable objects to the integers subject to some conditions, while in Section 5 the authors show that there is a one-to-one correspondence between equivalence classes of split Koszul abelian categories and equivalence classes of Koszulescent Orlov categories.
The second part is the core of the paper. In Section 6 the authors present some facts about mixed and Weil categories of perverse sheaves, before defining affable stratifications in Section 7. Very roughly, a stratification \(\mathcal{S}\) is affable if it admits a refinement where all strata are affine spaces and a further technical condition is satisfied. Now, if one simply writes a condition on objects which generalizes the definition of \(P^{\text{mix}}_\mathcal{S}(X)\), one gets the so-called miscible category \(D^{\text{misc}}_\mathcal{S}(X)\) which, however, is not triangulated. But, as proved in Section 7, if \(\mathcal{S}\) is affable, one can define a triangulated category \(D^{\text{mix}}_\mathcal{S}(X)\) and then \(D^{\text{misc}}_\mathcal{S}(X)\subset D^{\text{Weil}}_\mathcal{S}(X)\) is the infinitesimal extension of \(D^{\text{mix}}_\mathcal{S}(X)\). In particular, there is a faithful functor \(\iota_X\colon D^{\text{mix}}_\mathcal{S}(X)\to D^{\text{Weil}}_\mathcal{S}(X)\) whose essential image is precisely \(D^{\text{misc}}_\mathcal{S}(X)\). Furthermore, \(P^{\text{mix}}_\mathcal{S}(X)\) is the heart of a bounded t-stricture in \(D^{\text{mix}}_\mathcal{S}(X)\). This answers the first question posed above.
In Section 8 the authors establish several technical results on the so-called miscibility of objects and morphisms on an affine space, which lays the groundwork for more general results proved in the following section. To roughly state these, we need to introduce some notions. One says that a functor \(F\colon D^{\text{Weil}}_\mathcal{S}(X)\to D^{\text{Weil}}_\mathcal{S}(Y)\) is miscible if it maps \(D^{\text{misc}}_\mathcal{S}(X)\) into \(D^{\text{misc}}_\mathcal{S}(Y)\). If \(F\) is miscible, it is called genuine if there is a functor \(\tilde{F}\colon D^{\text{mix}}_\mathcal{S}(X) \to D^{\text{mix}}_\mathcal{S}(Y)\) such \(F\iota_X\cong \iota_Y\tilde{F}\). In Section 9 the authors show that a number of common functors, for example proper push-forwards, are miscible and some are even genuine, which addresses the second question posed above.
In the last part some applications of the above results are given. In particular, the authors prove some results about Andersen-Jantzen sheaves and about Wakimoto sheaves. Koszul duality; perverse sheaves; flag variety Achar, P.; Riche, S., \textit{Koszul duality and semisimplicity of Frobenius}, Ann. Inst. Fourier, 63, 1511-1612, (2013) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras Koszul duality and semisimplicity of Frobenius | 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 Summary of results in the author's doctoral thesis: ``Let \(f\) and \(g\) be two germs of (analytic) plane curves without common branches. We define the Jacobian quotients of \((g,f)\), which are `first-order invariants' of the discriminant curve of \((g,f)\), and we prove that they depend only on the topological type of \((g,f)\). We give a method for computing these invariants using the minimal resolution of \(f\cdot g\). If \(g\) is a linear form transverse to \(f\), we recover the results of \textit{Le Dung Trang}, \textit{F. Michel} and \textit{C. Weber} [Compos. Math. 72, No. 1, 87-113 (1989; Zbl 0705.32002); Ann. Sci. Ec. Norm. Supér., IV. Sér. 24, No. 2, 141-169 (1991; Zbl 0748.32018)] concerning polar curves''. plane curves; Jacobian quotients; polar curves Maugendre, H.: Topologie des germes jacobiens. Th` ese de Doctorat, Uni- versité de Nantes, 1995. Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities of curves, local rings Topology of Jacobian germs | 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 profound paper the author generalizes the Serre-Tate theory to the ordinary locus of good reduction of Shimura varieties of PEL-type. Let \(\mathcal D\) be a usual PEL datum and let \(\mathcal A_{\mathcal D}\) be the moduli space of abelian varieties with the prescribed additional structures. Assume that \(p>2\) and the moduli space \(\mathcal A_{\mathcal D}\) has good reduction at \(p\). On the reduction \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\), there are the Newton polygon stratification and the Ekedahl-Oort stratification. The first one comes from the classification of the associated \(p\)-divisible groups with additional structures up to isogeny, which is intensively studied by \textit{F. Oort} [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 417--440 (2001; Zbl 1086.14037)], \textit{R. E. Kottwitz} [Compos. Math. 56, 201--220 (1985; Zbl 0597.20038)], \textit{M. Rapoport} and \textit{M. Richartz} [Compos. Math. 103, 153--181 (1996; Zbl 0874.14008)], \textit{C.-L. Chai} [Am. J. Math. 122, 967--990 (2000; Zbl 1057.11506)], and many others. The latter stratification comes from the classification of the associated \(\text{ BT}_1\)'s with additional structures up to isomorphism. This is studied by \textit{F. Oort} [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 345--416 (2001; Zbl 1052.14047)], \textit{E. Z. Goren} and \textit{F. Oort} [J. Algebr. Geom. 9, 111--154 (2000; Zbl 0973.14010)], \textit{T. Wedhorn} [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 441--471 (2001; Zbl 1052.14026)], and the author [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 255--298 (2001; Zbl 1084.14523)].
A point in \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\) is said to be \textit{\(\mu\)-ordinary} if it lies in an open NP stratum. A point in \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\) is said to be \textit{\([p]\)-ordinary} if it lies an open EO stratum. The first main result the author proves is that these two notions of ordinary agree. Using another result of the author and \textit{T. Wedhorn} on the dimensions of the EO strata [math.Ag/0208161, to appear in Ann. Inst. Fourier(Grenoble)], [loc. cit. Zbl 1052.14026)], the author gives another proof of the density of the \(\mu\)-ordinary locus, which is proved by \textit{T. Wedhorn} [Ann. Sci. École Norm. Sp. (4) 32, No.~5, 575--618 (1999; Zbl 0983.14024)].
The Serre-Tate theory says that the formal deformation space of an ordinary abelian variety has a natural structure of formal torus. The main result of this paper under review is to generalize the Serre-Tate theory to the ordinary locus of the considered moduli space. The author proves that the formal deformation of an ordinary object has a ``group-like'' structure called \textit{cascade}, which is a generalization of the notion of a biextension. Roughly speaking, it is a sequence of tree-like fibration, which is the extension part of graded pieces of the slope filtration under forgetful maps, with fibers close to a B.-T.~group. Recently, instead of looking at the whole formal deformation, Chai considered subvarieties with a fixed isomorphism type of the associated \(p\)-divisible groups, called leaves. This fine geometric structure as introduced by Oort. Chai proved that any formal completion of a leaf in the Siegel moduli space has similar group-like fibration structure. This further investigation provides the satisfactory generalized Serre-Tate theory.
In the last section the author gives a very interesting application on congruence relations of the Frobenius correspondences on the ordinary locus. Serre-Tate theory; moduli space; PEL-type; stratifications; congruence relation Ben Moonen, ``Serre-Tate theory for moduli spaces of PEL type'', Ann. Sci. Éc. Norm. Supér.37 (2004) no. 2, p. 223-269 Arithmetic aspects of modular and Shimura varieties, Algebraic theory of abelian varieties Serre-Tate theory for moduli spaces of PEL type | 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 separably closed field of characteristic \(p>0\) and \(g \geq 2\) be an integer. By \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36(1969), 75-110 (1970; Zbl 0181.488)] there exists a universal stable curve \(Z_ g \to H_ g\), where \(H_ g\) is a \(k\)-subscheme of a convenient Hilbert scheme such that every stable curve over \(k\) of genus \(g\) is isomorphic to a fiber of \(Z_ g \to H_ g\) and \(H_ g\) is geometrically irreducible and smooth over \(k\), moreover the set of \(x \in H_ g\) whose fiber in \(Z_ g\) is smooth is an open dense subset of \(H_ g\). Let \(\eta\) be the generic point of \(H_ g\) and \(L\) the algebraic closure of \(k(\eta)\). The generic curve of genus \(g\) is denoted by \(X=Z_ g \times_{H_ g} \text{Spec} L\). It is a proper, smooth and connected curve over \(L\). Given a scheme \(S\) of characteristic \(p\) and \(f:Z \to S\) any morphism of schemes, we denote by \(Z^{(p)}=Z \times_ SS\) with respect to the absolute Frobenius morphism \(S \to S\). Given a semi-stable curve \(Z\) over a field \(K\) of characteristic \(p>0\) the relative Frobenius \(F:Z \to Z^{(p)}\) induces a map \(F^*:H^ 1(Z^{(p)}, {\mathcal O}_{Z^{(p)}}) \to H^ 1(Z, {\mathcal O}_ Z)\). We say that \(Z\) is ordinary if \(F^*\) is bijective. The author's main result states that given any étale connected Galois covering \(Y\) of \(X\) with Galois group of order prime to \(p\) then \(Y\) is ordinary. In particular, \(X\) is ordinary. Furthermore, this result together with a result of \textit{R. M. Crew} [cf. Compos. Math. 52, 31-45 (1984; Zbl 0558.14009); corollary 1.8.3] which says that if \(Y\) is a complete nonsingular connected curve defined over an algebraically closed field \(k\) of characteristic \(p>0\) and \(X \to Y\) is a finite étale Galois covering of degree a power of \(p\), then \(X\) is ordinary if and only if \(Y\) is ordinary, implies that every étale abelian covering of a generic curve is ordinary. characteristic \(p\); absolute Frobenius; ordinary curve; étale abelian covering of a generic curve Finite ground fields in algebraic geometry, Coverings of curves, fundamental group Abelian étale coverings of generic curves and ordinarity | 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 0504.00007.]
In this article the author verifies a generalization of the famous ''Brill-Noether problem'' for a smooth curve. The precise statement is as follows: Let X be a smooth curve of genus g and F a locally free sheaf of rank s and degree d. Define: \(W^ r_ m(F)=\{L\in Pic^ mX| \dim H^ 0(F\otimes L)\geq r+1\}. \rho =sm+d-(g-1)(s-1) \tau(F)=g-(f+1)(r-\rho +g).\)- Then \((a)\quad \dim W^ r_ m(F)\geq \tau(F).\) (b) If equality holds in (a), then \([W^ r_ m(F)]=\alpha(F)\cdot s^{g-\tau(F)}\cdot \theta^{r-\tau(F)}\) in the Chow ring of \(Pic^ m_ X\) where \(\theta =theta\) divisor and \(\alpha(F)=\prod^{r}_{i=0}i!/(r-\rho +g+i)!\). The author conjectures that like in the case of line bundles, equality must hold in (a) for a general vector bundle on a general curve. - The proofs use results on Quot-schemes, which the author has proved in another paper: ''Quot scheme over a smooth curve'', Prepr. Univ. Napoli (1982). Another ingredient, as one would expect, are the 'Porteous formulas'. Picard group; Brill-Noether problem; locally free sheaf; Chow ring Ghione, F. : '' Un probleme de type Brill-Noether pour les fibres vectioriels '', in: LNM 997, Springer-Verlag (1983). Singularities of curves, local rings, Picard groups, Enumerative problems (combinatorial problems) in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Riemann-Roch theorems, Divisors, linear systems, invertible sheaves Un problème du type Brill-Noether pour les fibres vectoriels | 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 complete non-archimedean valued field and $X$ be a compact quasi-smooth strictly $k$-analytic space. The article under review shows that, after performing a finite extension of the base field and a quasi-étale covering, one may always find a space that admits a strictly semistable formal model. (The word ``altered'' in the title of the article refers to the fact that one can conjecturally replace the quasi-étale morphism by an embedding of a disjoint union of affinoid domains.) This theorem is used by the author in [in: Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 -- April 6, 2013 and Puerto Rico, February 1--7, 2015. Cham: Springer. 195--285 (2016; Zbl 1360.32019)] where he investigated pluricanonical forms on quasi-smooth Berkovich spaces.
In the case where the base field $k$ is discretely valued, the theorem had formerly been proved by \textit{U. T. Hartl} [Manuscr. Math. 110, No. 3, 365--380 (2003; Zbl 1099.14010)], building on techniques introduced by \textit{A. J. de Jong} in [Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)] and involving moduli spaces of proper curves.
In order to go beyond the discretely valued case, the author uses the stable modification theorem from [J. Algebr. Geom. 19, No. 4, 603--677 (2010; Zbl 1211.14032)] that applies to arbitrary relative curves, with no properness assumption. This allows him to first prove an algebraic version of the result (Section 2), whereas Hartl's method needed to use analytic methods from the start in order to obtain relative compactifications. Formal and analytic versions of the result are then deduced in Sections 3 and 4. local uniformization; Berkovich spaces Rigid analytic geometry Altered local uniformization of Berkovich 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 For a scheme \(X\) over a field of characteristic \(p\), \textit{L. Illusie} [Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979; Zbl 0436.14007)] defined the so called de Rham Witt complex, a sheaf complex on \(X\), functorial in \(X\), which, if \(X\) is smooth and the base is perfect, computes the crystalline cohomology of \(X\) as its hypercohomology. Illusie's construction has subsequently been generalized into various directions; in particular, \textit{A. Langer} and \textit{T. Zink} [J. Inst. Math. Jussieu 3, No. 2, 231--314 (2004; Zbl 1100.14506)] constructed a relative de Rham Witt complex for morphisms \(X\to S\) where \(p\) is nilpotent on the \({\mathbb Z}_{(p)}\)-scheme \(S\).
In the present paper, this is generalized to fine log schemes \(X\) over fine log schemes \(S\) over \({\mathbb Z}_{(p)}\). Following the approach of Langer and Zink, the (log) de Rham Witt complex is constructed as the initial object of a certain category of log \(F-V\)-complexes. It is then shown that its hypercohomology computes the relative log crystalline cohomology of \(X\to S\) in cases where \(X/S\) is a relative semistable log scheme, or where \(X/S\) is a log scheme associated with a smooth scheme with a normal crossings divisor. Next, in these situations, generalizing constructions of \textit{A. Mokrane} [Duke Math. J. 72, No. 2, 301--337 (1993; Zbl 0834.14010)] and \textit{C. Nakayama} [Am. J. Math. 122, No. 4, 721--733 (2000; Zbl 1033.14012)], a weight spectral sequence for the crystalline cohomology is constructed. It is shown to degenerate modulo torsion at \(E_2\) when the base scheme is the spectrum of a (not necessarily perfect) field.
As already indicated, the approach taken here is inspired by the one suggested by Langer and Zink and thus differs from e.g. the one taken by \textit{O. Hyodo} and \textit{K. Kato} [in: Périodes \(p\)-adiques. Séminaire du Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France. 221--268 (1994; Zbl 0852.14004)] (resp. by Mokrane for the spectral sequence) in a more restricted log scheme setting. The key technical ingredient is the identification of a certain explicit bases of the (log) de Rham Witt complex in certain explicit cases; its elements are called log basic Witt differentials.
Finally, overconvergent de Rham Witt complexes for log schemes are introduced, generalizing the overconvergent de Rham Witt complexes for usual schemes introduced by Davis, Langer and Zink [\textit{C. Davis} et al., Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 197--262 (2011; Zbl 1236.14025)]. More precisely, for a log scheme \(X\) arising from a smooth scheme together with a normal crossings divisor \(D\), an overconvergent de Rham Witt complex is constructed. It is shown that its hypercohomology computes the rigid cohomology of the open complement \(X-D\). De Rham Witt complex; crystalline cohomology; log scheme; weight spectral sequence; log basic Witt differentials; overconvergent de Rham Witt complex; rigid cohomology \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry On relative and overconvergent de Rham-Witt cohomology for log 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 The original Bertini theorems [\textit{E. Bertini}, Rend. Lomb. (2) 15, 24-29 (1982; JFM 14.0433.02)] have been generalized in many directions. We are interested in two particular generalizations: the axiomatic theory proposed by Cumino, Greco and Manaresi and the transversality theory studied by Kleiman, Laksov and Speiser, both developed on algebraically closed ground fields.
We construct a general theory which generalizes both the axiomatic theory and the transversality theory extending the results to larger families of \(k\)-schemes and morphisms where \(k\) is an infinite ground field. -- In section 1 we give some preliminary remarks on the topology of the rational points of a scheme, and on the base change and we prove a useful Jacobian criterion. In section 2 we introduce the axioms for the local geometric properties and we verify that many properties satisfy such axioms. In section 3, which is the technical part of the paper, we prove the main theorem (axiomatic theorem 3.5) of our axiomatic theory.
In theorem 3.5 we propose a new axiomatic approach to the transversality theory that allows us to improve the known results. We generalize also the axiomatic theory introduced by \textit{C. Cumino, S. Greco} and \textit{M. Manaresi} [J. Algebra 98, 171-182 (1986; Zbl 0613.14006)] to arbitrary fields and we reobtain theorem 1 of this paper as a particular case of our result.
In the last two sections we apply our result to some significant particular cases. In section 4, we give explicit generalizations of the known Bertini type theorems in positive characteristic. -- In section 5, we compare our axiomatic theory with the transversality theory and we show that our results allow to obtain some Bertini type theorems without using group actions. families of morphisms; JFM 14.0433.02; transversality theory; families of \(k\)-schemes; Bertini type theorems Spreafico M.L.: Axiomatic theory for transversality and Bertini type theorems. Arch. Math. (Basel) 70(5), 407--424 (1998) Birational geometry, Schemes and morphisms Axiomatic theory for transversality and Bertini type theorems | 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 generalize results of P. Le Duff to genus \(n\) hyperelliptic curves. More precisely, let \(C/\mathbb Q\) be a hyperelliptic genus \(n\) curve, let \(J(C)\) be the associated Jacobian variety, and let \(\bar{\rho}_\ell:G_{\mathbb Q}\to\mathrm{GSp}(J(C)[\ell])\) be the Galois representation attached to the \(\ell\)-torsion of \(J(C)\). Assume that there exists a prime \(p\) such that \(J(C)\) has semistable reduction with toric dimension 1 at \(p\). We provide an algorithm to compute a list of primes \(\ell\) (if they exist) such that \(\bar{\rho}_\ell\) is surjective. In particular we realize \(\mathrm{GSp}_6(\mathbb F_\ell)\) as a Galois group over \(\mathbb Q\) for all primes \(\ell\in[11,500,000]\). Representation-theoretic methods; automorphic representations over local and global fields, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties, Rational points, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois representations and Galois groups over \(\mathbb Q\) | 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 \(\overline{{\mathcal M}}_{g,n}({\mathbb{P}}^r,d)\) be the moduli space of stable maps from \(n\)-pointed, genus \(g\) curves to \({\mathbb{P}}^r\) of degree \(d\). In this paper the author computes the Poincaré polynomial of \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\), using what are called Serre characteristics in [\textit{E. Getzler} and \textit{R. Pandharipande}, J. Algebr. Geom. 15, No. 4, 709--732 (2006; Zbl 1114.14032)]. Serre characteristics are defined for varieties \(X\) over \(\mathbb{C}\) via the mixed Hodge theory of Deligne as follows. For a mixed Hodge structure \((V,F,W)\) over \(\mathbb{C}\), set \(V^{p,q}= F^p\text{gr}^W_{p+q} V\cap\overline F^q\text{gr}^W_{p+q} V\) and let \({\mathcal X}(V)\) be the Euler characteristic of \(V\) as a graded vector space. Then the Serre characteristic \(\text{Serre}(X)\) of \(X\)is defined to be \(\text{Serre}(X)= \sum^\infty_{p,q=0} u^p v^q{\mathcal X}(H^\bullet_c(X,\mathbb{C})^{p,q}))\). He employs the fact that \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\) is stratified according to the degeneration type of the stable maps. The compatibility of Serre characteristics with stratification allows him to compute of each stratum and add up the results to obtain \(\text{Serre}(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2))\). In the final section, he also gives an additive basis for the Chow ring of \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\). Chow ring; moduli space; stable map; Betti numbers (Equivariant) Chow groups and rings; motives, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) An additive basis for the Chow ring of \(\overline {\mathcal M}_{0,2}(\mathbb P^r,2)\) | 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 \(X\) be a smooth projective curve over \(\mathbb{C}\), and let \({\mathcal L}\) be a line bundle on \(X\). A \(g^ r_ d\) on \(X\) is associated with a linear subspace \(V\subset H^ 0(X,{\mathcal L})\), and the \(g^ r_ d\) is said to have an \(N\)-fold point along a divisor \(D\) of degree \(N\geq 2\) on \(X\) if \(\dim(V\cap H^ 0(X,{\mathcal L}(-D))\geq r\). Suppose \(X\) without nontrivial automorphism. If one denotes by \(N(X)\) the closed scheme parametrizing all the \(g^ r_ d\)'s on \(X\) having an \(N\)-fold point, then the author's main result is as follows: if \(g\geq N\geq 2\), \(g\geq 3\), \(r\geq 2\) then \(\dim N(X)\leq\rho-r(N-1)+N\), where \(\rho\) is the Brill-Noether number. This results extends a previous one by \textit{M. Coppens} [J. Reine Angew. Math. 374, 61-71 (1987; Zbl 0597.14028)] who proved the case \(n=2\). divisor; Brill-Noether number Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Divisors, linear systems, invertible sheaves Linear series with an \(N\)-fold point on a general 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 non-singular absolutely irreducible projective curve with a single point \(Q\) at infinity and such that the pole orders of the coordinate functions on the affine curve \(X\setminus\{Q\}\) generate the Weierstrass semigroup at \(Q\). Let \(D\) be a divisor on \(X\) whose support does not include \(Q\). The authors give an algorithm for computing a basis of the Riemann-Roch space \(L(D)\), with the basis functions having distinct pole orders at \(Q\). The main motivation for the algorithm is its usefulness for ``one-point'' algebraic-geometry codes. The algorithm is similar to the one given by \textit{D. Eisenbud} and \textit{M. E. Stillman} in their tutorial on divisors that can be found in the documentation on the Macaulay2 homepage \url{http://www.math.uiuc.edu/Macaulay2/}, but the authors are able to simplify this algorithm substantially given the assumptions on \(X\) and \(D\). One simplification involves the computation of a Gröbner basis of a quotient of two zero-dimensional ideals by using linear algebra; this is similar to the method in the exercise 2.3.21 in the text by \textit{W. W. Adams} and \textit{P. Loustaunau} [``An introduction to Gröbner bases''. Graduate Studies in Mathematics. 3 (Providence 1994; Zbl 0803.13015)], where it is attributed to the dissertation of \textit{Y. N. Lakshman}.
A more complete version of the article under review, including proofs and examples, has appeared in J. Symb. Comput. 30, No. 3, 309-324 (2000). affine algebraic curve; algebraic geometry code; Gröbner basis; Weierstrass semigroup; divisor; Riemann-Roch space Computational aspects of algebraic curves, Geometric methods (including applications of algebraic geometry) applied to coding theory, Riemann surfaces; Weierstrass points; gap sequences, Divisors, linear systems, invertible sheaves Computing a basis of \({\mathfrak L} (D)\) on an affine algebraic curve with one rational place at infinity | 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 object of the paper under review is a torsor \(f\colon Y\to X\) under an algebraic torus \(T\). In the first part the authors are interested in the study of the cokernel of the natural map \(f^*:\text{Br}(X)\to \text{Br}(Y)\). The main result of this part, Theorem 1.7, describes this cokernel in the case where \(k\) is a field of characteristic zero such that \(H^3(k,\overline k^*)=0\) and \(X\) is a smooth and geometrically integral \(k\)-variety such that \(\overline k[X]^*=\overline k^*\) and \(\text{Pic}(\overline X)\) is a free abelian group of finite type. In the case of the universal torsor (i.e. when the natural map \(\widehat T\to \text{Pic}(\overline X)\) is an isomorphism) this description is particularly simple: the map \(\text{Br}(X)\to \text{Br}(Y)/\text{Br}(k)\) can be identified with the map \(\text{Br}(X)\to \text{Br}(\overline X)^{\Gamma}\), where \(\Gamma =\text{Gal} (\overline k/k)\).
The second part of the paper is devoted to the case where \(k\) is a number field. The authors continue the search for new classes of varieties in which the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. More precisely, they consider the case where a smooth projective variety \(X\) admits a dominant morphism \(\pi\colon X\to \mathbb P^1\) with geometrically integral generic fibre and ask whether assuming the Brauer-Manin obstruction is the only one for the smooth fibres of \(\pi\) one can deduce the same for \(X\). Several earlier works showed that this approach is successful when the fibration has a few ``bad'' fibres. In the paper under review several new instances of this principle have been found. Explicit examples of applications include conic bundles over the plane and some varieties of norm type. Brauer group; Hasse principle; universal torsor D. Harari and A. N. Skorobogatov, ''The Brauer group of torsors and its arithmetic applications,'' Ann. Inst. Fourier \((\)Grenoble\()\), vol. 53, iss. 7, pp. 1987-2019, 2003. Varieties over global fields, Rational points The Brauer group of torsors and its arithmetic 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 Let \(C\) be a smooth irreducible projective curve of genus \(g\) defined over the field \(\mathbb{C}\) of the complex numbers. Let \(J(C)\) be the Jacobian variety of \(C\). Fixing a base point \(P_0\) on \(C\) one defines naturals subschemes \(W^r_d\) di \(J(C)\). Those subschemes parametrize invertible sheafs \(L\) on \(C\) of degree \(d\) whose space of global sections have dimension at least \(r+1\). When \(C\) is a smooth curve embedded in some projective space \(\mathbb{P}^r\), using elementary constructions, such embeddings give rise to some naturally defined irreducible subset of some schemes \(W_e^1\).
In this article, the author finds sufficient conditions implying that such subset \(Z\) is a multiple irreducible component of \(W_e^1\). He shows that those sufficient conditions are satisfied for some natural types of embedded curves.
Those results are in sharp contrast to the known results in case \(C\) is a general curve. As a matter of fact, by the Brill-Noether Theory, in case \(C\) is a general curve of genus \(g\), many fundamental results concerning those schemes \(W_d^r\) are known. In particular, the scheme \(W_d^r\) is nonempty if and only if \(\rho_d^r(g)=g-(r+1)(g-d+r)\geq 0\). Moreover, in case \(\rho_d^r(g)\geq 0 \) then \(W_d^r\) is a reduced scheme of dimension \(\rho_d^r(g)\).
A lot of interesting curves do not satisfy the properties of general curves described by Brill-Noether Theory (e.g. smooth plane curves, complete intersection curves, double coverings of curves of low genus, and so on). Quite often by definition of the curve, they do not satisfy the nonexistence part. Nevertheless one could expect that the structure of those schemes satisfies nice properties such as reducedness for such ``natural'' curves.
It is important to observe that, in particular, the results of this article do indicate that one cannot expect that naturally occurring curves have always nice structures for their schemes \(W_d^r\).
An indication that such expectation could be wrong was already obtained by the author in a previous paper [J. Algebr. Geom. 4, No. 1, 1--15 (1995; Zbl 0842.14020)]. There, the author studied linear systems \(g_e^1\) on a smooth plane curve \(C\) of degree \(d\). As a corollary of the results one finds many schemes \(W_e^1\) that are not reduced. More concretely the following fact is proved. Let \(C\) be a smooth plane curve of degree \(d\). Let \(n, i, e\) be the integers satisfying \(1 \leq n \leq (d-2)/2\), \(0 \leq i \leq (n-1)(n-2)/2\), \(e=nd-n^2+i\). Then \(W_e^1\) has an irreducible component \(Z\) of dimension \(3n+i-2\) such that the Zariski tangent space to \(W^1_e\) at a general point of \(Z\) has dimension \(3n+2i-2\).
It should be remarked that, in general, for some curve and some integer \(e\), describing all linear systems \(g_e^1\) on that curve is a very difficult problem.
In order to prove the results of this article, the author cannot use a classical tangent dimension argument because there are multiple components. Instead the paper concerns with the use of some special arguments based on lower bounds on the dimension of the intersection of \(W_{e-1}^1+W_1^0\) and components \(Z\) of \(W_e^1\) not contained in \(W_{e-1}^1+W_1^0\). Using those bounds, the description of the Zariski tangent space to \(W^1_e\) of points on such intersection give rise to restriction on dim\((Z)\).
The main theorem is stated as follows. Theorem: Let \(C\subset \mathbb{P}^3\) be a smooth linearly normal and \(2-\)normal irreducible projective curve of degree \(d\) and genus \(g>2d-g\). Let \(x\in W^3_d\) be the point corresponding to \(C\subset \mathbb{P}^3\). Assume \(C\) is not contained in a quadric, and assume \(C\) has a \(3-\)secant line divisor \(E\) satisfying the generic condition. Then \(Z=x-W^0_2\) is a \(2-\)dimensional component of \(W^1_{d-2}\) satisfying tdim\((Z)=3\). In particular, \(x-W^0_2\) is a multiple component of \(W^1_{d-2}\).
Let \(g_d^r\) be a linear system on a curve \(C\), and let \(E\) be an effective divisor of degree \(e\) on \(C\). We say that \(E\) is an \(e-\)secant \(f-\)space divisor for \(g_d^r\) if \(\dim(g_d^r(-E))\geq r-f-1\). In case \(f=1\), then we talk about an \(e-\)secant line divisor for \(g_d^r\). \(E\) satisfies the generic condition if \(E\) is not contained in some \((e+1)-\)secant \(f-\)space divisor and \(2E\) imposes \(2e\) independent conditions on quadrics in \(\mathbb{P}^r\).
Since each smooth curve can be realized as a smooth space curve, not all smooth space curves have multiple components for some scheme \(W^1_e\) as it is the case for smooth plane curves of degree \(d \geq 8\). On the other hand, a smooth plane curve of degree \(d\) is always linearly normal, and in case \(d \geq 3\) then it is \(2-\)normal, and it is not contained in a conic. It is remarkable that in the case of space curves, linear sections already give rise to multiple components. Applying liaison to an arbitrary smooth space curve, the author obtains many examples of space curves satisfying the assumption of the main theorem, hence space curves of some degree \(d\) having a multiple component for \(W^1_{d-2}\).
Finally, the author gives a generalization of the main theorem to higher dimensional linear systems. In this generalization, we have to include some other types of conditions implying restrictions on its applicability. Therefore, this generalization is not considered as being the main result by the author. As a matter of fact, the author does expect the existence of much better generalizations. complete intersection curves; Jacobian; space curves; smooth curves; special divisor Special algebraic curves and curves of low genus, Plane and space curves, Special divisors on curves (gonality, Brill-Noether theory) Curves having schemes of special linear systems with multiple components | 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, proper, geometrically connected curve of genus \(g\geq 2\) over a field \(K\), and let \(U_{(r)}\) be the \(r\)-th configuration scheme of \(X\), i.e., the fiber product of \(r\) copies of \(X\) over \(K\) minus all the weak diagonals. Let \(r\geq 3\) and consider \(U_{(r)}\) to be the moduli space of (ordered) \(r\) marked points on \(X\). Pick any three indices \(i,j,k\) from \([1,r]\). In this paper, it is shown that the étale fundamental group of \(U_{(r)}\) is generated by three local fundamental groups arising from the tubular neighborhoods of divisors each of which respectively generically represents stable degeneration of \(X\) with the \(i,j\)-th (resp. \(j,k\)-th, resp. \(i,j,k\)-th) marked points are isolated on a rational component from the other marked points. Based on this result, the author constructs a ``partial cuspidalization of \(U_{(r)}\)'' from a certain system of data consisting of the projection homomorphisms of fundamental groups of lower dimensional configuration schemes of \(X\) as well as of divisors at infinity: Roughly speaking, one can construct group theoretically \(r\) surjections from the free profinite product of the above three local fundamental groups to \(\pi_1(U_{(r-1)})\) so that they factor through the natural \(r\) projections \(\pi_1(U_{(r)})\to\pi_1(U_{(r-1)})\) respectively. The formulation of results and their proofs in the present paper are given in the language of log geometry. cuspidalization; algebraic curve; anabelian geometry Y. Hoshi, On the fundamental groups of log configuration schemes , Math. J. Okayama Univ. 51 (2009), 1-26. Coverings of curves, fundamental group, Families, moduli of curves (algebraic) On the fundamental groups of log configuration 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 the author's preface: ``These notes originated in several classes that I taught in the mid 60's to introduce graduate students to algebraic geometry. I had intended to write a book, entitled ``Introduction to Algebraic Geometry'', based on these courses and, as a first step, began writing class notes. The Harvard mathematics department typed them up and distributed them. They were called ``Introduction to Algebraic Geometry: Preliminary version of the first three chapters'' and were bound in red. The intent was to write a much more inclusive book. But as the years progressed, my ideas of what to include in this book changed. The book became two volumes, and eventually, with almost no overlap with these notes, the first volume appeared in 1976, entitled ``Algebraic Geometry. I: Complex projective varieties'' (1976; Zbl 0356.14002; second edition 1980). The present plan is to publish shortly the second volume, entitled ``Algebraic Geometry. II: Schemes and cohomology'', in collaboration with \textit{David Eisenbud} and \textit{Joe Harris}.
D. Gieseker and several others have, however, convinced me to let reprint the original notes, on the grounds that they serve a quite distinct purpose. These old notes had been intended only to explain in a quick and informal way what varieties and schemes are, and give a few key examples illustrating their simplest properties. The hope was to make the basic objects of algebraic geometry as familiar to the reader as the basic objects of differential geometry and topology.
This volume is a reprint of the old notes without change, except that the title has been changed to clarify their aim.
It may be of some interest to recall how hard it was for algebraic geometers to find a satisfactory language. At the time these notes were written, the ``foundations'' of the subject had been described in at least half a dozen different mathematical ``languages''.
Then Grothendieck came along and turned a confused world of researchers upside down, overwhelming them with the new terminology of schemes as well as with a huge production of new and very exciting results. These notes attempted to show something that was still very controversial at that time: that schemes really were the most natural language for algebraic geometry.''
The language of schemes is now fully accepted as the correct one for the problems of algebraic geometry. Therefore the book is welcome as an introduction to this topic and serves very well as such in spite of the long time since it was first published.
The contents covers: Varieties (chapter I); preschemes (chapter II); local properties of schemes (chapter III). Varieties; preschemes Mumford, D. (1988). \(The Red Book of Varieties and Schemes\), Springer, Berlin. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Varieties and morphisms, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Schemes and morphisms The red book of varieties and 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 book is the third part of the textbook on algebraic geometry by Kenji Ueno. The Japanese original of this volume was published in 1998, shortly after the first two parts ``Algebraic geometry. 1: From algebraic varieties to schemes'' (1999; Zbl 0937.14001) and ``Algebraic geometry. 2: Sheaves and cohomology'' (2001; Zbl 0965.14008), and has been again translated into English by Goro Kato.
In this third volume, which contains the remaining chapters 7 to 9 of the entire text, the author presents selected topics from the theory of schemes and their sheaf cohomology beyond the introductory level, mainly with the goal of studying those refined properties of schemes and coherent sheaves that are necessary for a deeper understanding of modern algebraic geometry as a whole, on the one hand, and with the idea of providing concrete applications to classical complex algebraic geometry, on the other hand.
As to the precise contents, chapter 7 is entitled ``Fundamental properties of scheme theory'' and covers nearly two thirds of the present volume. After a brief reminder of some known elementary facts on varieties and schemes, the author discusses the (Krull) dimension theory for rings and schemes, normal and regular schemes, normalization morphisms, Weil and Cartier divisors, flat and proper morphisms, flat families of schemes, Chow's lemma and the cohomology of proper morphisms, regular schemes and smooth morphisms, sheaves of relative differential forms, non-singular algebraic varieties, completions, formal schemes, and Zariski's main theorem. This very thorough and detailed discussion also includes, always at the right place, special concepts and results such as Hilbert polynomials, Bertini's theorem, base change and cohomology, Hilbert schemes, Stein factorization, and other important topics in modern algebraic geometry.
Chapter 8 turns then to the study of complete non-singular algebraic curves. As this chapter is meant as an application of the much more general previous chapter, the methods developed here differ remarkably from the classical ``orthodox'' approaches to the theory of algebraic curves. That means, the author treats (parts of) the theory of algebraic curves in the context of general modern algebraic geometry, as it has been developed before, and so the whole subject appears in a much more coherent, elegant and sophisticated manner, which is extremely efficient at this level. The topics discussed here include the Riemann-Roch theorem for complete non-singular curves, projective embeddings, curves and algebraic function fields, the Riemann-Hurwitz formula for coverings of curves, curves in characteristic \(p>0\), Frobenius morphisms, étale morphisms, Lüroth's theorem, the geometry of elliptic curves, generalities on group schemes, the Picard functor, and the construction of the Jacobian variety of a curve.
The final chapter 9 (of this volume and of the entire textbook) explains the connection between complex algebraic geometry and complex analytic geometry. This chapter is comparatively short, sketchy and more survey-like, but nevertheless a very beautiful, enlightening and perfect epilogue. The author gives an outline of Serre's fundamental GAGA theorems, together with Grothendieck's generalization to proper schemes over the field of complex numbers, sketches the so-called Lefschetz principle and concludes, as an application of the Lefschetz principle, this final chapter with a full proof of the famous Kodaira vanishing theorem.
The entire text, as a whole, ends with an appendix entitled ``Overview and references''. This is basically an annotated list of books and research papers for further reading, enhanced by historical remarks and hints at more recent developments in algebraic geometry and related fields. This additional service to the reader is very useful, inspiring and stimulating.
As in the previous two volumes, the author has larded the text with numerous concrete examples, exercises, and problems. Solutions to the exercises and problems are again provided at the end of the book, and that anew in great detail. Also, just as in the previous volumes, each chapter comes with its own short summary, helping the reader to grasp the essentials and highlights of each single chapter.
Altogether, this final third part of the whole textbook on algebraic geometry is just as masterly composed as the first two parts of it. The author leads the reader here into the deeper grounds and to the powerful applications of modern algebraic geometry, and that in an amazingly gentle and effective way. Together with the two previous, more introductory parts, this text is perfectly appropriate for a graduate course on modern algebraic geometry or for profound self-study. It must be seen as a distinguished feature of the whole three-volume text that it is sufficiently comprehensive, adequately rigorous and detailed, nearly self-contained, didactically functional, always appealing and motivating, technically well-balanced, extremely reader-friendly and appetizing. Also, the author has paid much attention to intradisciplinary connections, disciplinary up-to-dateness, and research-oriented hints. As for further applications of algebro-geometric methods in contemporary mathematics and physics, the interested reader should consult the volume ``Advances in moduli theory'' by \textit{Y. Shimizu} and \textit{K. Ueno} [Transl. Math. Monogr. 206 (2002; Zbl 0987.14001)], which might be regarded as sort of a fourth part of the present textbook. At any rate, K. Ueno's three-volume text on algebraic geometry has already become very popular among Japanese students and teachers, and now it is very likely to become one of the great international standard texts, too. algebraic schemes; algebraic varieties; Jacobians; group schemes; Riemann-Roch theorem; dimension theory; divisors; morphisms; Chow's lemma; differential forms; completions; formal schemes; Zariski's main theorem; non-singular algebraic curves; GAGA; vanishing theorem Schemes and morphisms, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Curves in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Algebraic geometry. 3: Further study of schemes. Transl. from the Japanese by Goro Kato | 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 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. The famous volumes ``Éléments de géométrie algébrique'' as a series in Publ. Math., Inst. Hautes Étud. Sci. (1960-1967) by \textit{A. Grothendieck} and \textit{J. Dieudonné} were entirely written in the new language of schemes, without being linked up with the classical roots, and the so far existing textbooks just dealed with classical methods. It was \textit{David Mumford}, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ``Introduction to algebraic geometry: Preliminary version of the first three chapters'' (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title ``The red book of varieties and schemes'' [cf. Lect. Notes Math. 1358 (1988; Zbl 0658.14001)]. In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). -- In 1976, the author published the first volume under the title ``Algebraic geometry. I: Complex projective varieties'' (1976; Zbl 0356.14002; corrected second edition 1980; Zbl 0456.14001), where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies.
The book under review is an unchanged reprint of this corrected second edition from 1980. Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's ``Volume I'' is, together with its predecessor ``The red book of varieties and schemes'', now as before, one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!
As to the intended volume II of the book under review, the author planned to publish it in collaboration with \textit{D. Eisenbud} and \textit{J. Harris}. This would have been based on existing but unpublished notes of the author (partially revised by \textit{S. Lang}), but then the author and his co-authors came to the conclusion that such a second volume was not really what was needed anymore, because \textit{R. Hartshorne}'s famous book ``Algebraic geometry'' (1977; Zbl 0367.14001) already covered a good part of the material they had planned to include. Instead, D. Eisenbud and J. Harris published what they felt is needed more: a brief introduction to schemes [cf. \textit{D. Eisenbud} and \textit{J. Harris}, ``Schemes: The language of modern algebraic geometry'' (1992; Zbl 0745.14002)]. Their booklet may be regarded as a bridge between D. Mumford's thorough classic (under review) and the now existing several textbooks on ``scheme- theoretic'' algebraic geometry, including Hartshorne's book as well as Mumford's other classic, the ``Red book of varieties and schemes''. affine varieties; projective varieties; correspondences; linear systems; algebraic curves; algebraic surfaces; birational algebraic geometry David Mumford, \textit{Algebraic geometry. I}, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Complex projective varieties, Reprint of the 1976 edition. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Varieties and morphisms, Curves in algebraic geometry, Relevant commutative algebra, Rational and birational maps, Surfaces and higher-dimensional varieties Algebraic Geometry. I: Complex projective 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 \(X\) be a smooth projective curve of genus \(g\geq 2\) over \(\mathbb C\), \(\text{M}\) the moduli space of semistable rank \(r\) vector bundles on \(X\) with trivial determinant and \(\mathcal D\) the determinant bundle on \(\text{M}\). Improving substantially results of \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] and \textit{J. Le Potier} [in: Moduli vector bundles, Lect. Notes Pure Appl. Math. 179, 83--101 (1996; Zbl 0890.14017)], \textit{M. Popa} [Duke Math. J. 107, 469--495 (2001; Zbl 1064.14032)] showed that \({\mathcal D}^{\otimes h}\) is generated by global sections for \(h\geq [r^2/4]\). His proof is based on an effective upper bound for the dimension of Grothendieck schemes of quotients with fixed Hilbert polynomial of a vector bundle on \(X\) (this bound was improved by \textit{M. Popa} and \textit{M. Roth} [Invent. Math. 152, 625--663 (2003; Zbl 1024.14015)]).
In this paper, the author proves an analogous result for the parabolic determinant bundle \({\mathcal L}^{\text{par}}\) on the moduli space \(\text{M}^{\text{par}}\) of semistable parabolic bundles of rank \(r\), with trivial determinant, and a fixed parabolic structure at a finite subset \(I\) of points of \(X\). She firstly defines analogues of Grothendieck schemes for parabolic quotients of a parabolic vector bundle, and finds an effective upper bound for the dimension of these schemes. The proof of this bound is by induction, the initial step of the induction being the above mentioned estimate of Popa and Roth. Then she produces global sections of \({\mathcal L}^{\text{par}\otimes h}\) of theta type, extending a result of \textit{C. Pauly} [Math. Z. 228, No.1, 31--50 (1998; Zbl 0905.14016)] who considered the case \(r = 2, h = 1\). Finally, using M. Popa's strategy, she proves the result on the global generation of \({\mathcal L}^{\text{par}\otimes h}\).
As a byproduct, the author obtains the following characterization of parabolic semistability: Let \({\mathcal M}_{\underline n}\) denote the moduli stack of quasi-parabolic bundles on \(X\) at \(I\), with multiplicities \(\underline n = ((n_i(p))_{p\in I}\). To each system \(\underline \alpha = ((\alpha _i(p))_{p\in I}\) of rational parabolic weights, one can associate a ``determinant'' line bundle \({\mathcal L}_{\underline \alpha}\) on \({\mathcal M}_{\underline n}\). The author shows that, for all \(l\geq [r^2/4]\), the closed substack of \({\mathcal M}_{\underline n}\) consisting of \(\underline \alpha \)-unstable quasiparabolic bundles is the locus where all the theta type global sections of \({\mathcal L}_{\underline \alpha}^{\otimes l}\) vanish. projective curves; generalized theta functions; Grothendieck's quot scheme Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes) Theta functions on the moduli space of parabolic 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 For a positive integer r, set \(N=r(r+3)/2\). Write G for \(PGL(3)=Aut({\mathbb{P}}^ 2)\). Take N plane curves \(D_ 1,...,D_ N\), and denote by T the set of solutions of the tangency problem given by the \(D_ i\). In other words, T is the set of reduced plane curves of degree \(r,\) tangent to all the \(D_ i\). By a result proved by \textit{W. Fulton, S. Kleiman} and \textit{R. MacPherson} [Algebraic geometry - open problems, Proc. Conf., Ravello/Italy 1982, Lect. Notes Math. 997, 156-196 (1983; Zbl 0529.14030)]. T is finite if the \(D_ i\) are replaced, if necessary, by general translates. Moreover, card(T) is independent of the translations if these are, if need be, still more general; write s for this number. - More precisely, denote by U the dense open subset in \(G^ N\) which provides the appropriate translations. Varying the translations, the solutions T trace a variety Z, finite over U, and if the \(D_ i\) are smooth, it is known that Z is irreducible. Write K (respectively L) for the function field of U (respectively the normalization, over K, of the function field of Z). If the base field is \({\mathbb{C}}\), if \(r\geq 2\), and if each \(D_ i\) is smooth of degree \(\geq 2\), the author shows that the Galois group of L/K is the full symmetric group on s letters.
The author bases his proofs on arguments of \textit{A. Hefez} and \textit{G. Sacchiero} [Math. Scand. 56, 171-190 (1985; Zbl 0566.14024)]. (In his own words: ''we steal them.'') The point of the present article is that one obtains the full conclusion modulo projective rather than just algebraic equivalence, in the cases at hand. monodromy; Galois group of function field; tangency problem Curves in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry On the monodromy group for the contact problem for 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 The book under review grew out of a course of lectures delivered by Professor U. Storch at the Indian Institute of Science, Bangalore in 1998. As the authors point out in the preface, the objective of these lectures was to introduce some fundamental principles of both algebraic geometry and commutative algebra simultaneously, with strong emphasis on their close interplay.
Representing a carefully elaborated version of the course notes in book form, the current text comprises seven chapters reflecting the subject matter of the original lectures quite faithfully. Geared toward graduate students, the book only assumes some familiarity with the basic concepts of modern abstract algebra as necessary prerequisites, whereas it is utmost detailed and largely self-contained otherwise.
As for the general approach, the authors have adopted A. Grothendieck's fundamental viewpoint and his language of modern algebraic geometry, which is heavily based on the conceptual and methodological framework of commutative algebra, thereby displaying the corresponding path of historical development in a particularly in a particulary accentuated manner. The precise contents of the seven chapters can be briefly summarized as follows:
Chapter 1 starts with some basic topics from commutative algebra, including algebras over a ring, factorization in rings, Noetherian rings and modules, graded rings and modules, integral ring extensions, the Noether Normalization Lemma, and the Hilbert Nullstellensatz.
The geometric interpretation of these algebraic concepts and results is provided in Chapter 2, where affine algebraic varieties over a ground field and their Zariski topology are explained. Chapter 3 turns to the generalization of the latter, mainly by analyzing the prime spectrum of a commutative ring, on the one hand, and by developing the dimension theory of Noetherian rings and their spectra on the other.
The main topics of Chapter 4 are sheaves of rings, general schemes and their morphisms, finiteness conditions on schemes, and products of schemes, while Chapter 5 is devoted to projective schemes, the Main Theorem of Elimination, and Chevalley's Mapping Theorem on constructible sets in Noetherian schemes. Chapter 6 gives a detailed account of regular local rings, normal domains, the normalization of a scheme, and smoothness properties. In addition, this chapter contains a detailed treatment of the theory of Kähler differentials, which is then used to exhibit the sheaf of Kähler differentials as a fundamental example of a (quasi-) coherent module sheaf on a scheme.
The concluding Chapter 7 presents the highlight of this versatile introductory textbook: the classical Riemann-Roch Theorem for projective algebraic curves over an arbitrary ground field. In fact, the authors offer a rather general approch to this fundamental theorem in algebraic geometry inasmuch as they allow arbitrary curve singularities and state it for arbitrary coherent sheaves. The authors' proof of the Riemann-Koch Theorem does not involve any sheaf cohomology. Instead it is based on a fine analysis of coherent and quasi-coherent module sheaves on projective schemes by means of the theory of graded rings and modules in commutative algebra.
In the course of the entire exposition, a wealth of fundamental concepts, methods, techniques, and results from both algebraic geometry and commutative algebra is lucidly provided, with full proofs and profound explanations all through. A large number of exercises after each section, many of which come with precise hints for solution, invite the reader to get acquainted with related topies, further importent results, or additional instructive examples in the context of the core material of the book. All together, this masterly written text must be seen as an excellent introduction to modern algebraic geometry and advanced commutative algebra in their inseparable relationship. algebraic geometry (textbook); commutative algebra (textbook); finitely generated algebras; schemes; projective schemes; local rings; Kähler differentials; sheaves; algebraic curve Patil, D.P., Storch, U.: Introduction to algebraic geometry and commutative algebra, IISc Lecture Notes Series, vol. 1. World Scientific Publishing Co., Pte. Ltd., Bangalore. IISc Press, Hackensack, NJ (2010) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Schemes and morphisms, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Regular local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Introduction to algebraic geometry and commutative algebra | 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 central problem in curve theory is to describe algebraic curves in projective space with fixed degree and genus. One wants to know the extrinsic geometry of the curve. Koszul cohomology groups in some sense carry `everything one wants to know' about the extrinsic geometry of curves: the number of equations of each degree defining the curve, the relations between the equations, etc. The paper under review is an excellent survey on the Koszul cohomology of curves and its applications to moduli. The outline of the paper is as follows.
In section 2, the authors review the definition of Koszul cohomology groups and the main technical tools to study them.
Section 3 is devoted to the applications of Koszul cohomology groups to moduli. After a brief review of the classical Brill-Noether divisors on the moduli space \(M_g\) of smooth curves of genus \(g\) and their closure in the Deligne-Mumford compactification \(\overline{M}_g\), the authors introduce Koszul cycles on \(M_g\): fix the Brill-Noether number \(\rho=g-(r+1)(g-d+r)=0\), and consider
\[
Z_{g,p}=\{C\in M_g\;|\;\exists\;g^r_d\;L \text{ such that } K_{p,2}(C,L)\neq0\}
\]
When \(g=s(2s+sp+p+1)\), \(r=2s+sp+p\), and \(d=rs+r\) for some \(s\geq1\), \(Z_{g,p}\) is a virtual divisor on \(M_g\). The authors explain the computation of the slope of these virtual divisors in a partial compactification of \(M_g\). These virtual divisors turn out to have small slope compared to the Brill-Noether divisors and give conter-examples to the Harris-Morrison Slope conjecture provided one can verify they are honest divisors on \(M_g\).
Section 4 consists of a nice survey of some recent progress on some of the main conjectures. Starting with Green's conjecture and the Green-Lazarsfeld gonality conjecture and continuing with the Prym-Green conjecture.
Finally in section 5, the authors formulate the strong Maximal Rank conjecture and the Minimal Syzygy conjecture. These conjectures are significant not only because they imply the virtual Koszul divisors \(Z_{g,p}\) described in sections 2 are honest divisors but also give a complete prediction of the resolution of an embedded curve with general moduli at least for some special range of \(g\), \(r\), \(d\). Koszul cohomology; Koszul cycles; Green's conjecture; Gonanity conjecture; Strong maximal rank conjecture Aprodu, M.; Farkas, G., Koszul cohomology and applications to moduli, Clay math. proc., 14, 25-50, (2011) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Vanishing theorems in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Koszul cohomology and applications to moduli | 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 connected Dedekind scheme with field of rational functions \(K\) and let \(X_ K\) be a smooth and separated \(K\)-scheme of finite type. A Néron model of \(X_ K\) is a smooth separated \(S\)-scheme \(X\) of finite type with generic fibre \(X_ K\) satisfying the following universal property: for each smooth \(S\)-scheme \(Y\) and each \(K\)-morphism \(u_ K: Y_ K\to X_ K\) there exists a unique \(S\)-morphism \(u: Y\to X\) extending \(u_ k\).
This book is devoted to the construction of the Néron models and to the study of their properties. In particular, in the case of a relative curve \(X\to S\), the Néron model of the Jacobian \(J_ K\) of the generic fibre \(X_ K\) is studied and its relationship with the relative Picard functor is explained. The book contains also an ample exposition of the main tools and methods used so that, for example, chapter 8 and chapter 9 are a useful reference for questions related to the Picard functor and to its representability. Néron model of the Jacobian; relative Picard functor Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron models, (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, (1990), Springer-Verlag: Springer-Verlag Berlin), x+325 pp Schemes and morphisms, Picard schemes, higher Jacobians, Group schemes, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Picard groups Néron 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 A part of Grothendieck's program for studying the Galois group \(G_Q\) of the field of all algebraic numbers \(\overline{Q}\) emerged from his insight that one should lift its action upon \(\overline{Q}\) to the action of \(G_Q\) upon the (appropriately defined) profinite completion of \(\pi_1(\mathrm{P}^1\{0,1,\infty\})\). The latter admits a good combinatorial encoding via finite graphs, ``dessins d'enfant''.
This part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara.
This chapter concerns another part of Grothendieck's program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labeled points. Our main goal is to show that dual graphs of such curves may play the role of ``modular dessins'' in an appropriate operadic context. Polycategories/dioperads, properads, PROPs, cyclic operads, modular operads, Operads (general), Quantum equilibrium statistical mechanics (general), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Dessins d'enfants theory Dessins for modular operads 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 notion of dicritical divisors of a pencil of plane curves \((F,G)\) was developed by many authors: e.g. as horizontal divisors see \textit{A. Campillo} et al. [J. Algebra 293, No. 2, 513--542 (2005; Zbl 1086.14006)] or in terms of Rees valuations \textit{I. Swanson} [in: Commutative algebra. Noetherian and non-Noetherian perspectives. New York, NY: Springer. 421--440 (2011; Zbl 1237.13013)] and, more generally, from an algebraic point of view by Abhyankar, in the local and in the polynomial case.
In the present paper, following Abhyankar, the authors give geometrical interpretations of dicritical divisors and new proofs of their existence and unicity.
In particular they prove the equivalence between dicritical divisors and Rees valuations. Furthermore they show that, in the case where \(G_{\mathrm{red}}\) is regular at the base points of the pencil \((F,G)\), \(F/G\) is residually a polynomial along any dicritical divisor; this result clarifies geometrically the theorem of \textit{S. S. Abhyankar} and \textit{I. Luengo} [Am. J. Math. 133, No. 6, 1713--1732 (2011; Zbl 1232.13012)] and generalizes the connectedness theorem [\textit{Lê Dũng Tráng} and \textit{C. Weber}, Kodai Math. J. 17, No. 3, 374--381 (1994; Zbl 1128.14301)]. dicritical divisors; Rees valuations; horizontal divisors; pencil of curves Singularities of curves, local rings, Singularities in algebraic geometry, Valuation rings Existence of dicritical divisors, after S.S. Abhyankar | 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 constructs a compactification for the relative degree-\(d\) Picard variety associated to a family of (proper) stable curves. The problem arises because the Picard functor neither is proper nor separated when fibers of \(X/S\) are not smooth. For instance one can find families of invertible sheaves specializing to sheaves not locally free when the central fiber has nodes. The author uses a new functor to solve this problem. One crucial idea is that in this new functor such families specialize to invertible sheaves on the curve resulting from replacing the node (in the central fiber) by \(\mathbb{P}^1\). [The same idea was used by \textit{D. Gieseker} for rank-2 vector bundles on nodal curves; cf. J. Differ. Geom. 19, 173-206 (1984; Zbl 0557.14008).]The proof relies on GIT (=``geometric invariant theory'') theory. moduli problem; semistable curves; geometric invariant theory; compactification; Picard variety Lucia Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves , J. Amer. Math. Soc. 7 (1994), no. 3, 589-660. JSTOR: Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Picard groups A compactification of the universal Picard variety over the moduli space of stable 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 his paper [\textit{J.-P. Serre}, Invent. Math. 15, 259--331 (1972; Zbl 0235.14012)], Serre shows that for a fixed elliptic curve \(E\) over a number field that the representation \(\rho_{E,\ell}\) on the \(\ell\)-torsion points of \(E\) without complex multiplication is surjective for sufficiently large primes \(\ell\). He asks in the same paper whether there is a uniform bound to ensure surjectivity of \(\rho_{E,\ell}\) for all elliptic curves \(E\) over \(\mathbb{Q}\) without complex multiplication.
This question is still open and is equivalent to the determination of rational points on the modular curves classifying non-surjective \(\rho_{E,\ell}\) for sufficiently large \(\ell\). The case of a Borel subgroup was proven by [\textit{B. Mazur}, Invent. Math. 44, 129--162 (1978; Zbl 0386.14009)] and provided a framework to determine and restrict rational points on modular curves with a rational cusp. The last remaining case, which is what prevents a resolution of Serre's question, is the normalizer of a non-split Cartan subgroup.
The case of the normalizer of a split Cartan subgroup was recently settled by [\textit{Y. Bilu} and \textit{P. Parent}, Ann. Math. (2) 173, No. 1, 569--584 (2011; Zbl 1278.11065)] and [\textit{Y. Bilu} et al., Ann. Inst. Fourier 63, No. 3, 957--984 (2013; Zbl 1307.11075)], except for the case \(\ell = 13\). Interestingly, it was shown in [\textit{B. Baran}, J. Number Theory 145, 273--300 (2014; Zbl 1300.11055)] that the normalizer split Cartan and normalizer non-split Cartan modular curve of level \(13\) are isomorphic over \(\mathbb Q\), which provides an explanation for the unsuccessful application of the methods of [\textit{Y. Bilu} et al., Ann. Inst. Fourier 63, No. 3, 957--984 (2013; Zbl 1307.11075)] for \(\ell = 13\), but also the possibility that the methods in the paper under review may lead to a resolution of Serre's question.
Mazur's method in [\textit{B. Mazur}, Invent. Math. 44, 129--162 (1978; Zbl 0386.14009)] analyzes the rational points on a modular curve by embedding it into its Jacobian and then using precise information about the arithmetic of its Jacobian. A precondition is the need for the Jacobian to have a non-zero rank 0 quotient or for the Jacobian to have a quotient with rank less than its dimension [\textit{M. H. Baker}, Proc. Am. Math. Soc. 127, No. 10, 2851--2856 (1999; Zbl 0931.11017)]. This fails in the case of the normalizer of a non-split Cartan subgroup, and in particular for \(X_{\text{s}}(13) \simeq X_{\text{ns}}(13)\).
In [\textit{M. Kim}, Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006)] and [\textit{M. Kim}, Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)], Kim initiated a program to generalize the Chabauty method beyond the natural barrier when the rank is less than the dimension. In this theory, the Jacobian variety is replaced by torsors for the maximal \(n\)-unipotent quotient of the \(\mathbb{Q}_p\)-étale fundamental group of the curve. It has been successfully applied to showing finiteness of the S-unit equation, both theoretically and computationally [\textit{I. Dan-Cohen} and \textit{S. Wewers}, Int. Math. Res. Not. 2016, No. 17, 5291--5354 (2016; Zbl 1404.11093)].
A special case of this approach which is more amenable to explicit computation, called quadratic Chabauty, was initiated in [\textit{J. S. Balakrishnan} et al., Math. Comput. 86, No. 305, 1403--1434 (2017; Zbl 1376.11053)] and [\textit{J. S. Balakrishnan} and \textit{N. Dogra}, Duke Math. J. 167, No. 11, 1981--2038 (2018; Zbl 1401.14123)]. This approach is related to the case \(n = 2\) of Kim's program. It has been shown in [\textit{S. Siksek}, ``Quadratic Chabauty for modular curves'', Prperint, \url{arXiv:1704.00473}] that for moduar curves of genus \(\ge 3\), the quadratic Chabauty method satisfies a necessary precondition needed to work beyond the classical Chabauty barrier, namely that the Néron-Severi rank of the Jacobian of the modular curve is \(\ge 2\).
The authors first show how to construct the quadratic Chabauty pairs needed for the quadratic Chabauty method using Nekovář's theory of \(p\)-adic height functions on Selmer varieties [\textit{J. Nekovář}, Prog. Math. 108, 127--202 (1993; Zbl 0859.11038)]. An explicit description of this \(p\)-adic height function is then derived using by solving explicit \(p\)-adic differential equations. Finally, to carry out the quadratic Chabauty argument, one uses a number of known rational points to solve for the explicit equations which give a finite \(p\)-adic set containing the rational points of the curve.
The authors successfully apply these methods to determine the rational points on \(X_{\text{s}}(13)\), the modular curve associated to the normalizer of a split Cartan subgroup, thus completing a resolution of Serre's question in the case of the normalizer of a split Cartan subgroup. Diophantine equations; modular curves; \(p\)-adic heights; non-abelian Chabauty Rational points, Computer solution of Diophantine equations, Heights, Arithmetic aspects of modular and Shimura varieties Explicit Chabauty-Kim for the split Cartan modular curve of level 13 | 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 conjecture of Grothendieck and Serre on torsors dating back to 1958 (see [Séminaire Claude Chevalley (2e année) Tome 3. (French) École Normale Supérieure. Paris: Secrétariat Mathématique (1958; Zbl 0098.13101)] predicts that for a regular local ring \(R\) with fraction field \(K\) and a reductive group \(R\)-scheme \(G\), the specialization map
\[
H^1_{\text{ét}}(R, G) \to H^1_{\text{ét}}(K, G)
\]
is always injective.When \(R\) contains an infinite field \(k\), this has been proved in [\textit{R. Fedorov} and the author, Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)]. That proof hinges upon the Main Theorem 1.0.3 in this article stated as follows. Let \(\mathcal{O}\) be a semi-local ring with finitely many closed points on a smooth affine variety \(X\) over \(k\). Assume that for all semisimple simply connected \(\mathcal{O}\)-group scheme \(H\), the map \(H^1_{\text{ét}}(\mathcal{O}, H) \to H^1_{\text{ét}}(k(X), H)\) is injective, then the same holds for any reductive \(\mathcal{O}\)-group scheme \(G\), with \(K = \text{Frac}(\mathcal{O})\) replacing \(k(X)\).
This is in turn built on two purity Theorems 1.0.1 and 1.0.2, which are of independent interest. The first one asserts that in the circumstance above, consider a smooth \(\mathcal{O}\)-morphism of reductive group schemes \(\mu: G \to C\) such that \(\text{Ker}(\mu)\) is a reductive \(\mathcal{O}\)-group scheme, then there is an exact sequence
\[
0 \to C(\mathcal{O})/\mu(G(\mathcal{O})) \to C(K)/\mu(G(K)) \to \bigoplus_{ \text{ht}(\mathfrak{p})=1 } C(K) \big/ C(\mathcal{O}_p)\mu(G(K)) \to 0.
\]
For the second purity result, let \(i: Z \hookrightarrow G\) be a central closed subgroup scheme of a semisimple \(\mathcal{O}\)-group scheme \(G\), with \(G' = G/i(Z)\). Then there is an exact sequence
\[
0 \to \dfrac{ H^1_{\text{fppt}}(\mathcal{O}, Z) }{\text{im}(\delta_{\mathcal{O}}) } \to \dfrac{ H^1_{\text{fppt}}(K, Z) }{\text{im}(\delta_K) } \to \bigoplus_{ \text{ht}(\mathfrak{p})=1 } \dfrac{ H^1_{\text{fppt}}(K, Z) }{ H^1_{\text{fppt}}(\mathcal{O}_{\mathfrak{p}}, Z) + \text{im}(\delta_K) } \to 0
\]
where \(\delta_\star\) stand for the coboundary maps \(G'(\star) \to H^1_{\text{fppt}}(\star, Z)\), and fppt indicates the topology of finitely presented flat morphisms. reductive group schemes; principal bundles; Grothendieck-Serre conjecture [78] Panin I., On Grothendieck--Serre's conjecture concerning principal \(G\)-bundles over reductive group schemes. II, 2009, 28 pp., arXiv: Group schemes, Linear algebraic groups over adèles and other rings and schemes, Exceptional groups, Linear algebraic groups and related topics On Grothendieck-Serre's conjecture concerning principal \(G\)-bundles over reductive group schemes. II | 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 0635.00006.]
The author studies the moduli problem for basic surfaces of Picard \(number\quad n,\) i.e. for surfaces obtained as ordered blowings-up of \({\mathbb{P}}^ 2\) at n-1 points. He constructs:
(1) a versal family \(P=\{\pi_ n: X_{n+1}\to X_ n\}\), such that any family locally for the étale topology ``comes'' from P by fibre- products.
(2) a scheme \(I_ n\) representing the functor \(Isom_ n\) defined as follows: for any scheme U and any pair of morphisms \(f,g: U\to X_ n,\) \(Isom_ n(U)\) is the set of U-isomorphisms \(U\times_ fX_{n+1}\cong U\times_ gX_{n+1},\)
(3) natural morphisms s,t: \(I_ n\to X_ n.\)
Since in this case the moduli functor is not representable, the existence of such a family is the best result one can hope to find.
Then the author studies in detail the Picard functor and the scheme structure of \(I_ n\). moduli problem for basic surfaces of Picard \(number\quad n,\); Picard functor Harbourne, B.: Iterated blow-ups and moduli for rational surfaces. Lecture notes in math. 1311, 101-117 (1988) Families, moduli, classification: algebraic theory, Rational and birational maps, Picard groups, Rational and unirational varieties Iterated blow-ups and moduli for rational 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 In the course of its history, algebraic geometry has undergone several fundamental changes with respect to its conceptual foundations, methods, and techniques. A brilliant analysis of the historical developments in algebraic geometry can be found in the first volume of \textit{J. Dieudonné}'s ``History of algebraic geometry''
[``Cours de Géometrie Algébrique'', Presses Universitaires de France, Paris 1974, Engl. translation by Judith D. Sally, Wadsworth (1985; Zbl 0629.14001)], according to which the history of algebraic geometry is characterized by about seven main epoches. The so far most recent epoch of algebraic geometry began in the 1950s, when A. Grothendieck launched his revolutionary program of refounding the entire theory by means of a completely new, much more general conceptual framework, including arbitrary algebraic schemes, sheaves and their cohomology, methods of categorical and homological algebra, relative geometry, classifying spaces, and other powerful tools. Grothendieck first sketched his new theories, which turned over a new leaf in the development of algebraic geometry, in a series of talks at the Seminaire Bourbaki between 1957 and 1962. The notes of these talks were then published in the famous volume [``Fondements de la Géométrie Algébrique'' (Paris: Secrétariat mathématique) (1962; Zbl 0239.14004)], commonly abbreviated FGA. These mimeographed notes became of central importance in the sequel, as they contained the only available outlines of Grothendieck's fundamental new constructions such as descent theory, Hilbert schemes and Quot schemes, formal algebraic geometry, Picard schemes, and other pioneering approaches toward a better understanding of classical problems in algebraic geometry.
Grothendieck's various theories sketched in FGA were of crucial significance for the tremendous progress in algebraic geometry thereafter, and most of them even became indispensable ingredients in allied fields of contemporary mathematics such as arithmetic, complex-analytic geometry, mathematical physics, and others. Certainly, much of Grothendieck's FGA is now common knowledge, or at least utilized folklore, but a good deal of it is still less well-known, and perhaps only a few experts are familiar with its full scope.
In view of the undiminished significance of Grothendieck's ideas outlined in FGA for the current and future research in algebraic geometry, the book under review aims at explaining its rich contents in full detail, thereby taking into account recent developments and improvements. Written by six experts in the field, this book contains the elaborated lectures delivered by the authors at the ``Advanced School in Basic Algebraic Gecmetry'', which was held at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, from July 7 to July 18, 2003, and which addressed advanced graduate students as well as beginning researchers in algebraic geometry.
As the authors point out, this book is not intended to replace Grothendieck's celebrated FGA, after more than forty years. Rather, it is to fill in Grothendieck's ingenious sketches with detailed proofs, on the one hand, and to present both newer ideas and more recent developments wherever appropriate, on the other hand. In accordance with Grothendieck's original FGA, the present book consists of five parts written by different authors, in which the following topics are treated: (1) theory of descent; (2) Hilbert schemes and Quot schemes; (3) local properties of Hilbert schemes of points; (4) Grothendieck's existence theorem in formal geometry; and (5) Picard schemes.
Part 1 was written by A. Vistoli (Bologna). This part comes with the headline ``Grothendieck topologies, fibered categories, and descent theory'' and is subdivided into four chapters. Differing from Grothendieck's original approach in FGA, descent theory is here presented in the language of Grothendieck topologies, which Grothendieck actually introduced somewhat later. Chapter 1 reviews some basic notions of scheme-theoretic algebraic geometry and of category theory, whereas chapter 2 continues the warm-up with a brief introduction to representable functors, Grothendieck topologies and their sheaves, and to group objects in categories. Chapter 3 discusses fibered categories, which provide the appropriate abstract set-up for explaining descent theory in its full generality. The main example, in this context is the category of quasi-coherent sheaves over the category of schemes. Chapter 4 develops descent theory for algebraic stacks, that is for those fibered categories in which it properly works. The author discusses, in full detail, the various ways of defining descent data, including those for quasi-coherent sheaves, morphisms of schemes, and quasi-coherent sheaves along torsors. Overall, the author has tried to develop descent theory in its greatest generality, beyond the scope of FGA, which is certainly necessary for deeper understanding and more advanced applications.
Part 2, comprising Chapter 5, explains Grothendieck's construction of Hilbert schemes and Quot schemes. Written by N. Nitsure, this chapter provides a modern treatment of these basic constructions using more recent basic tools such as faithtully flat descent, flattening stratifications, semi-continuity techniques, and Castelnuovo-Mumford regularity of sheaves. This chapter concludes with some concrete applications due to D. Mumford, A. Altman and S. Kleiman, A. Grothendieck himself, and others.
Part 3 is formed by chapters 6 and 7. Containing the lectures by B. Pantechi and L. Göttsche (Trieste), this chapter discusses local properties and Hilbert schemes of points. Chapter 6 introduces elementary deformation theory in algebraic geometry, involving the infinitesimal study of schemes, the infinitesimal deformation functor, a tangent-obstruction theory for such a functor, and local moduli spaces. The theory is thoroughly worked out in some special cases, and sketched in a few others. This is used in chapter 7, where the Hilbert scheme of points on a smooth quasi-projective variety is studied. The reader meets here the Hilbert-Chow morphism, stratifications of Hilbert schemes of points, Betti numbers of Hilbert schemes, and further more recent cohomological results in this direction.
Part 4, written by L. Illusie (Paris), is devoted to Grothendieck's FGA exposé, where he established a fundamental comparison theorem of GAGA-type between algebraic geometry and his newly created formal geometry, on the one hand, and outlined some possible applications to the theory of the algebraic fundamental group and to infinitesimal deformation theory, on the other hand. Shortly afterwards, \textit{A. Grothendieck} gave a detailed account of this particular topic in EGA III [Publ. Math., Inst. Hautes Étud. Sci. 17, 137--223 (1963; Zbl 0122.16102)] and in SGA1 [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Here, in part 4 of the book under review, the author revisits this topic in great detail, and in a more introductory manner for non-experts, ending with a discussion of J.-P. Serre's celebrated examples of varieties in positive characteristic that do not lift to characteristic zero.
Being one of the leading experts in this realm, the author has taken the opportunity to give various improvements and updatings of Grothendieck's original approach, mainly by using the toolkit of derived categories and, especially, perfect complexes, and he has enhanced the entire discussion by several more recent applications to lifting problems in algebraic or formal geometry, respectively. Part 4 comprises chapter 8 of the book, whereas the final part 5 is identical with chapter 9. This concluding part, written by S. Kleiman (Boston), is perhaps the show-piece of the entire collection. In a masterly manner, beginning with a highly enlightening introduction of 15 pages, the author develops in great detail most of Grothendieck's theory of the Picard scheme, together with its further developments later on.
After the extensive historical introduction, which is also of independent cultural interest, the author describes the four common relative Picard functors in their comparison, and proves then Grothendieck's existence theorem for the Picard scheme. This is followed by the study of both the connected component of the identity and the torsion component of the identity of the Picard scheme, including the related deep finiteness theorems, and the entire treatise closes with two appendices. Appendix A provides detailed answers to all the exercises scattered in this chapter, and appendix B contains an elementary treatment of basic intersection theory, as far as it is used freely in some proofs.
All together, this book must be seen as a highly valuable addition to Grothendieck's fundamental classic FGA, and as a superb contribution to the propagation of his pioneering work just as well. It is fair to say that, for the first time, the wealth of Grothendieck's FGA has been made accessible to the entire community of algebraic geometers, including non-specialists, young researchers, and seasoned graduate students. The authors have endeavoured to elaborate Grothendieck's ingenious, epoch-making outlines in the greatest possible clarity and detailedness, with complete proofs given throughout, and with various improvements, simplifications, updatings, and topical hints wherever appropriate. Due to this rewarding undertaking, FGA has come down to earth that is much closer to the community of all algebraic geometers, and therefore the book under review ought to be in the library of anyone using modern algebraic geometry in his research. textbook; algebraic geometry; Grothendieck topologies; fibred categories; descent theory; Hilbert schemes; Picard schemes; formal geometry B. Fantechi, L. Göttsche, L. Illusie, S.L. Kleiman, N. Nitsure and A. Vistoli, Fundamental Algebraic Geometry. Grothendieck's FGA Explained, Math. Surveys Monogr., \textbf{123}, Amer. Math. Soc., Providence, RI, 2005. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Divisors, linear systems, invertible sheaves, Formal methods and deformations in algebraic geometry, Picard schemes, higher Jacobians, Fibered categories Fundamental algebraic geometry: Grothendieck's FGA explained | 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 There is a strong analogy between number fields and function fields, hence between the theories of the two kinds of global fields. For instance, one may consider analogies between the \(p\)-adic numbers and Laurent series over \(\mathbb{F}_p\), or, much deeper, one has the notion of the \textit{genus} of a number field, which is analogous to the \textit{genus} of an algebraic curve on which a given function field is the field of rational functions (see [\textit{A.~Weil}, Rev. Sci. Paris 77, 104--106 (1939; JFM 65.1140.01)]). These analogies have brought important number-theoretic applications. For example the genus notion makes the statement of the Riemann-Roch theorem for algebraic curves extend to arithmetic geometry. It is thus expected that the geometry of arithmetic schemes, i.e. schemes of finite type over \(\mathbb{Z}\), should behave similar to the algebraic geometry of schemes of finite type over a regular projective curve.
However, this is just an expectation and it seems that the tools that have been developed up to the present point make it difficult to approach both settings in an uniform way. For example, the core problem of arithmetic schemes is that they are not compact. The main idea of the Arakelov intersection theory of arithmetic schemes is to ``compactify'' the scheme by considering the associated complex analytic variety at infinity, endowed it with analytic data such as hermitian metrics, Green currents, etc. This approach has led to rich number theoretical applications such as the proof of Mordell's conjecture by \textit{G. Faltings} [Invent. Math. 73, No. 3, 349--366; erratum ibid. 75, 381 (1983; Zbl 0588.14026)] and to the Bogomolov conjecture (see e.g. [\textit{S. W. Zhang}, Ann. of Math. (2) 147, No. 1, 159--165 (1998; Zbl 0991.11034)]). Even though this approach has led to such important results, the proofs mostly involve subtle tools in complex analysis and the transition between algebraic and analytic techniques is sometimes hard to track. This makes that statements in the number field case do not necessarily lead to analogous statements in the function field case and viceversa.
The present manuscript provides a uniform fundament for Arakelov geometry for both settings. This is done by introducing the notion of an \textit{adelic curve}. This is a field equipped with a family of absolute values parametrized by a measure space such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space with \(0\) as its integral. The latter is called the \textit{product formula}. With this notion, the authors develop an Arakelov theory over adelic curves. As the authors point out, the notion of an adelic curve has already been present in the literature, although in somewhat less general settings (see e.g. [\textit{W. Gubler}, Symp. Math. 37, 190--227 (1997; Zbl 0991.11034)]).
In the present book, the authors first formalize the notion of adelic curves and discuss algebraic covers thereof, which are fundamental for the notion of \textit{heights} in this setting. Then they set up the theory of adelic vector bundles over adelic curves and discuss adelic analogues of stability and slope theory. They also discuss arithmetic invariants of adelic vector bundles such as the Arakelov degree. Then they study metrized line bundles on arithmetic varieties over adelic curves. These differ from the classical setting of adelic metrics in [\textit{S.~W. Zhang}, J. Algebr. Geom. 4, No. 2, 281--300 (1995; Zbl 0861.14019)] and [\textit{A. Morowaki}, Mem. Am. Math. Soc. 242, No. 1144, 122 p. (2016; Zbl 1388.14076)] which are based on the notion of \textit{model metric}, and this notion is no longer adequate in this general setting. Positivity properties of these metrized line bundles are also discussed. In the last chapter the authors give a generalization of the Nakai-Moishezon's criterion settled in the setting of Arakelov geometry over an adelic curve.
Assuming basic knowledge of algebraic geometry and algebraic number theory, this manuscript is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics. Arakelov geometry; adelic curves; heights; adelic vector bundles; slope theory; arithmetic verieties; adelic metrics Research exposition (monographs, survey articles) pertaining to algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights, Global ground fields in algebraic geometry, Heights Arakelov geometry over adelic 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 [For the entire collection see Zbl 0755.00010.]
Let \(S\) be a stable curve, i.e. a Riemann surface with at most nodes as singularities, with arithmetic genus \(g\). In Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 43-55 (1974; Zbl 0294.32016) \textit{L. Bers} defines a deformation space \(D=D(S)\) parametrizing all stable curves which can be deformed into \(S\). If \(S\) is nonsingular, \(D(S)\) equals the Teichmüller space \(T_ g\). For \(t\in D\) denote by \(S_ t\) the curve associated to \(t\). To any \(S_ t\) one associates the generalized Jacobian \(J(S_ t)\) and the generalized theta function \(\theta_ \delta(\cdot,t):\mathbb{C}^ g\to\mathbb{C}\). In the nonsingular case \(\theta_ \delta\) is a translate of the classical theta function. \(\theta_ \delta\) defines a divisor \(\Theta\) on \(J(S_ t)\). As a generalization of Lefschetz's Theorem it is shown that \(3\Theta\) is very ample. Also in the context of differential equations the stable curves \(S_ t\) and the associated theta functions \(\theta_ \delta\) behave similar as in the nonsingular case: it is shown that they lead to solutions of the \(K\)-\(P\)-equation. period matrix; Riemann surface; theta function [Go] González-diéz, G.: Theta functions on the boundary of moduli space. In: Donagi, R. (ed.) Curves, Jacobians, and Abelian Varieties, Contemp. Math.136, Providence RI: Amer. Math. Soc., 1992, pp. 185--208 Period matrices, variation of Hodge structure; degenerations, Theta functions and abelian varieties, Jacobians, Prym varieties, KdV equations (Korteweg-de Vries equations) Theta functions on the boundary of moduli space | 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 \(\varphi:\widetilde{C}\to C\) be an unramified morphism of plane affine curves defined over a number field \(K\). In this paper we obtain a quantitative version of the classical Chevalley-Weil theorem for curves giving an effective upper bound for the norm of the relative discriminant of the field \(K(Q)\) over \(K\) for any integral point \(P\in C(K)\) and \(Q\in \varphi^{-1}(P)\). Draziotis, K; Poulakis, D, Explicit Chevalley-Weil theorem for affine plane curves, Rocky Mt. J. Math., 39, 49-70, (2009) Plane and space curves, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings of curves, fundamental group, Classification of affine varieties Explicit Chevalley-Weil theorem for affine 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 Belyi's theorem for curves states that a smooth projective curve \(X\) over \(\mathbb{C}\) is definable over \(\overline{\mathbb{Q}}\) if and only if there exists a meromorphic function \(f:X \to \mathbb{P}^{1}_{\mathbb{C}}\) ramified over at most three points. This was the starting point of Grothendieck's theory of dessins d'enfants, in which he proposed the study of the absolute Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}})\) through its action on Belyi functions. The result of Belyi was later extended to complex surfaces by \textit{G. González-Diez} [Am. J. Math. 130, No. 1, 59--74 (2008; Zbl 1158.14015)], where the role of Belyi functions is played by Lefschetz functions, that is, compositions of Lefschetz pencils \(X \dashrightarrow \mathbb{P}^{1}_{\mathbb{C}}\) with rational functions \(\mathbb{P}^{1}_{\mathbb{C}}\to \mathbb{P}^{1}_{\mathbb{C}}\).
In this paper, the author follows a similar strategy to give a Belyi-type characterisation of smooth complete intersections \(X\) of general type over \(\mathbb{C}\) which can be defined over \(\overline{\mathbb{Q}}\). More precisely, it is proved that such a variety \(X\) can be defined over \(\overline{\mathbb{Q}}\) if and only if there exists a Lefschetz function \(X \dashrightarrow \mathbb{P}^{1}_{\mathbb{C}}\) with at most three critical points. Contrary to the 2-dimensional case, the general proof poses several technical difficulties which require new results on finiteness of maps to varieties of general type and rigidity of Lefschetz pencils of complete intersections. Belyi's theorem; minimal model program; rigidity of families of varieties; Shafarevich boundedness conjecture; Lefschetz pencils Structure of families (Picard-Lefschetz, monodromy, etc.), Families, moduli, classification: algebraic theory, Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic), Dessins d'enfants theory Belyi's theorem for complete intersections of general type | 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 \(X\rightarrow S\) be a smooth family of projective curves over an algebraically closed field \(k\) of characteristic zero. Assume that both \(X\) and \(S\) are smooth projective varieties and let \(E\) be a vector bundle of rank \(r\) over \(X\) and \(\mathbb{P}(E)\) be its projectivization. Fix \(d\ge 1\). Let \(\mathcal{Q}(E,d)\) be the relative Quot scheme of torsion quotients of \(E\) of degree \(d\). Then we show that if \(r\ge 3\), then the identity component of the group of automorphisms of \(\mathcal{Q}(E,d)\) over \(S\) is isomorphic to the identity component of the group of automorphisms of \(\mathbb{P}(E)\) over \(S\). We also show that under additional hypotheses, the same statement is true in the case when \(r=2\). As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of \(\mathcal{O}^r_C,\) where \(r\ge 3\) and \(C\) a smooth projective curve of genus \(\ge 2\) is computed. Quot schemes; flag schemes; automorphisms Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties, Homogeneous spaces and generalizations Automorphisms of relative Quot 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 Algebraic geometry has undergone a tremendous development over the last century. In its first stage, during the 19th century, algebraic geometry mainly consisted in studying concrete varieties in projective space by purely geometric constructions. This ``classical'' approach reached its culmination in the work of the Italian school around the turn of the century, when it ultimately became apparent that the purely geometric language and techniques of the subject were exceedingly exploited and a deeper foundation was needed. This was done in the twenties of our century, basically by \textit{O. Zariski} and, independently, by \textit{A. Weil} who (under the influence of the German school of abstract algebraists) put the whole subject of algebraic geometry on a firm algebraic foundation. Finally, using the new framework of sheaf theory and cohomology, \textit{J.-P. Serre} and (above all) \textit{A. Grothendieck} invented an even more powerful, radical new conceptual and methodical footing for algebraic geometry. Grothendieck's theory of schemes, developed in the 1960's, is since then (and for now) an extremely rich and striking tool in algebraic geometry, arithmetical geometry, algebraic number theory, complex-analytic geometry, hermitean differential geometry and nowadays also mathematical physics. Modern algebraic geometry, in its scheme-theoretic foundation, has flourished enormously over the past thirty years, and much of the brilliant, advanced work in classical (Italian) algebraic geometry could be firmly re-established and carried further. Thus everybody who wants to study algebraic geometry today will find himself in a position of having to acquire a vast spectrum of concepts, methods and powerful tools. This certainly raises a didactical problem: what is the best way to study (or to teach) topics in algebraic geometry? In the meantime, many outstanding textbooks on algebraic geometry, written from different viewpoints, on different levels and for different purposes, are available. However, most of them introduce the modern approach from the beginning on and develop the whole subject to be treated in these terms. This is perfect for someone whose ultimate goal is to work in that field, or to apply its methods precisely.
If, on the other hand, someone just wants to know what algebraic geometry is about, what kind of objects (and their properties) are studied, what sort of results one can obtain, and whether it is worth to learn the sophisticated, vast framework of modern algebraic geometry systematically, then it might be better to start with the classical geometric part of the theory, in its modern setting, and to avoid the (perhaps discouraging) technical side for the beginning. -- This latter aspect is the guiding issue for the present textbook. It represents the author's very welcome attempt at giving an introduction to algebraic geometry from the more geometric and elementary algebraic point of view, stressing the still glorious classical aspects with their fascinating wealth of concrete examples, deep problems and beautiful intrinsic structure. The book grew out of his various courses on the subject given at Harvard and Brown University, during the past decade, and the circulating manuscript of it was popular and used long before this book appeared.
This textbook is unique, both in its (just explained) aim and its content, and it certainly fills a gap in the current literature in algebraic geometry. It is really a ``very first'' introduction to algebraic geometry, omitting any fancy algebraic, sheaf-theoretic, scheme-theoretic, or cohomological framework, but nevertheless it is not elementary altogether. The text leads the reader from the discussion and illustration of various kinds of projective varieties and their correspondences to the frontiers of current research in the field, provided by topics such as classifying spaces for certain types of algebraic varieties (moduli spaces), families of varieties and their parameter spaces (Chow varieties, Hilbert varieties), geometric invariant theory and algebraic groups. All this is done with an absolute minimum of technical machinery, but (instead) with an extremely skillful emphasis on the geometric ideas behind everything, on typical examples and encroaching links. Many examples are dealt with several times, in the light of each new conceptual development in the text, and the reader is invited to study them more thoroughly by working on the numerous exercises (and the hints to them). As for the prerequisites from commutative algebra, it will suffice to use one of the small textbooks simultaneously, for example the concise introduction by \textit{M. F. Atiyah} and \textit{I. G. MacDonald} [``Introduction to commutative algebra'' (1969; Zbl 0175.036)] or, even better, the forthcoming book by \textit{D. Eisenbud} ``Commutative algebra: With a view towards algebraic geometry'' (in press), which has just been written as the algebraic counterpart to the author's present textbook on algebraic geometry. On the other hand, as for further reading, \textit{D. Eisenbud} and the author have already published the book ``Schemes: The language of modern algebraic geometry'' [(1992; Zbl 0745.14002)], which provides a brief introduction to Grothendieck's theory of algebraic schemes. However, any of the recent advanced books on algebraic geometry will be perfectly suited for continued studies, and the reader of the author's present introductory text will certainly be well-equipped with a profound geometric background, concrete examples, motivation, and appreciation for the subject to be studied.
The content of the present book is of great ampleness, much too stratified to be discussed in detail here. Basically, the text is divided into two main parts. Part one, entitled ``Examples of varieties and maps'', is devoted to introducing basic varieties (such as affine and projective varieties, Grassmannians, cones, determinantal varieties, secant varieties, quadrics, flag varieties, etc.) and maps (regular maps, projections, incidence correspondences, rational and birational maps, etc.), whereas part two, entitled ``Attributes of varieties'', is concerned with the basic notions associated with varieties (such as degree, dimension, smoothness, tangent spaces, Hilbert functions, Gauss maps, parameter spaces, moduli, etc.) and their significance.
The author even explains assertions and theorems whose proofs could not be given, because they are much beyound the scope of the text, but whose statements are already enlightening and inspiring. The book contains a lot of material that cannot be found in other places in the textbook literature (so far), for example such topics like Fano varieties (of determinantal varieties), their tangent varieties, varieties of secant lines, varieties of incident planes, etc. Such varieties are objects of intense research, and it is very inspiring, even for the actively working algebraic geometer, to have them discussed in a modern, user-friendly textbook.
Altogether, the present work is a highly welcome enrichment of the textbook literature in algebraic geometry. It helps to make the existing, more advanced textbooks and the current research literature easier accessible, and it perfectly serves its purpose of providing a very first, albeit far-going introduction to the fascinating, beautifully intricate field of algebraic geometry. It is really a joy, both mathematically and aesthetically, to study this book, in particular as one can easily do it by jumping around in the text, without losing track of the essentials. correspondences; quadrics; moduli spaces; Chow varieties; Hilbert varieties; geometric invariant theory; algebraic groups; Grassmannians; cones; determinantal varieties; maps; degree; dimension; smoothness J.~Harris, \textit{Algebraic geometry: a first course}, Graduate Texts in Mathematics \textbf{133}, Springer-Verlag, New York, 1992. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Rational and birational maps, Varieties and morphisms, Projective and enumerative algebraic geometry Algebraic geometry. A first course | 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 Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When \(n \geq 2\), the space \(\overline{Q}_{g}({\mathbb {P}}^{n-1},d) = \overline{Q}_{g}(G(1,n),d)\) compactifies the space of degree \(d\) maps of smooth genus \(g\) curves to \({\mathbb {P}}^{n-1}\), while \(\overline{Q}_{g}(G(1,1),d) \simeq \overline{M}_{1, d \cdot \epsilon }/S_d\) is a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne-Mumford stacks \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\), for all \(n \geq 1\). We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, the cones of ample divisors, and in the case \(n=1\) the cones of effective divisors. We conclude that \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\) is Fano if and only if \(n(d-1)(d+2) < 20\). Moreover, we show that \({\overline{Q}}_{1}({\mathbb {P}}^{n-1},d)\) is a Mori Fiber space for all \(n\), \(d\), hence always minimal in the sense of the minimal model program. In the case \(n=1\), we write in addition a closed formula for the Poincaré polynomial. Y Cooper, The geometry of stable quotients in genus \(1\), in preparation Families, moduli of curves (algebraic), Geometric invariant theory, Group actions on varieties or schemes (quotients), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The geometry of stable quotients in genus one | 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 our previous paper [the author, Commun. Algebra 45, No. 8, 3422--3448 (2017; Zbl 1408.14145)], we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus \(g>1\), Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley-Weil formula gives a description of it. fundamental group schemes; vector bundles; Tannaka duality Group schemes, Coverings of curves, fundamental group, Vector bundles on curves and their moduli Semifinite bundles and the Chevalley-Weil formula | 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 prove the following analog of the result of A. Grothendieck on the cohomology of a coherent sheaf.
Theorem. Let \(U\) be a local regular scheme of geometric type over a field \(k\) and \(T\to U\) be a smooth proper morphism. Let \(F\) be a locally constant constructible torsion étale sheaf on \(T\) with torsion prime to characteristic of \(k\). Then there exists a finite complex \(L\) of locally constant constructible sheaves on \(U\) and a functor isomorphism between étale hypercohomology \(H_{\text{ét}}^q(T\times_U U',F) \cong H_{\text{ét}}^q(U',L\times_U U')\), where \(q\geq 0\) and \(U'\) denotes an \(U\)-scheme. constructible étale sheaf; smooth proper base change Panin, I.; Smirnov, A.: On a theorem of Grothendieck. Zap. nauchn. Sem. S.-peterburg. Otdel. mat. Inst. Steklov (POMI) 272, 286-293 (2000) Étale and other Grothendieck topologies and (co)homologies On a theorem of Grothendieck | 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 investigates the cohomological correspondence between analytic cycles on a complex manifold and Cartier divisors such that the isomorphism class of the associated line bundle is given by integration of the fundamental class. This correspondence satisfies some interesting functorial properties, admits a relative version and coincides with the theory developed by \textit{D. Barlet} and \textit{J. Magnusson} [Ann. Sci. Éc. Norm. Supér. 31, No. 6, 811--842 (1998; Zbl 0963.32019)]. Deligne cohomology; line bundle; Cartier divisor; fundamental class Local cohomology of analytic spaces, Integration on analytic sets and spaces, currents, Complex-analytic moduli problems, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Divisors, linear systems, invertible sheaves Line bundles and Cartier divisors on cycle spaces in analytic 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 \(K\) be an algebraic function field in one variable with a finite constant field of characteristic \(p\geq 5\), and let \(C\) be an algebraic curve over \(K\) defined by the equation \(y^2=x^p+a\) (\(a\not \in K^p\)). It is a singular curve of absolute genus \(0\) computed over a separable closure \(K^{\text{sep}}\), and relative genus \(g=(p-1)/2\) computed over \(K\). By the result of \textit{J. F. Voloch} [Bull. Soc. Math. Fr. 119, 121-126 (1991; Zbl 0735.14018)] \(C\) has a finite number of points defined over \(K\). Let now \(\text{Pic}^0(C)\) be the Picard group of divisors of degree zero on \(C\), and let \(\text{Pic}^0_K(C)\) be a subgroup in \(\text{Pic}^0(C)\) consisting of those divisor classes which are invariant under the action of the absolute Galois group \(\text{Gal}(K^{\text{sep}}/K)\). The author constructs a group variety \(X\) over \(K\) of dimension \(g\), such that \(X\) has a finite number of \(K\)-rational points, and then embeds \(\text{Pic}^0_K(C)\) into \(X(K)\). Applying the Albanese map, this result also gives a proof of the finiteness of \(K\)-rational points on the curve \(C\). algebraic curve; Picard group; Galois group; rational point Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points Rational points on Picard groups of some genus-changing curves of genus at least 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 \(K=k(C)\) be the function field of an algebraic curve \(C\) over an algebraically closed ground field \(k\). Let \(\Gamma/K\) be a smooth projective curve of genus \(g>0\) with a \(K\)-rational point \(O\in\Gamma(K)\). There is a smooth projective algebraic surface \(S\) with genus \(g\) fibration \(f:S\to C\) which has \(\Gamma\) as its generic fibre and which is relatively minimal. Let \(J/K\) denote the Jacobian variety of \(\Gamma/K\). Further let \((\tau,B)\) be the \(K/k\)-trace of \(J\). The main purpose of the present paper is to give the Mordell-Weil group \(J(K)/\tau B(k)\) (modulo torsion) the structure of Euclidean lattice via intersection theory on the algebraic surface \(S\). rational point; function field; Jacobian variety; Mordell-Weil group; intersection theory Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry (Warwick 1996), pp. 359-373, London Mathematical Soceity, Lecture Note on Series , vol. \textbf{264}. Cambridge University Press, Cambridge (1999) Rational points, Algebraic functions and function fields in algebraic geometry, Jacobians, Prym varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Special surfaces Mordell-Weil lattices for higher genus fibration 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 The authors study the generalization of the classical Torelli morphism from the moduli of stable curves to the moduli of principally polarized stable semi-abelic pairs. The main results are concerned with the study of the fibres and of its injectivity locus.
Let us be more precise. The classical result of \textit{R. Torelli} [``Sulle varietà di Jacobi'', Rom. Acc. L. Rend. (5) 22, No. 2, 98--103, 437--441 (1913; JFM 44.0655.03)] claims the injectivity of the map from the moduli scheme of smooth projective curves of genus \(g\), \(M_g\), to the moduli scheme of principally polarized abelian varieties of dimension \(g\), \(A_g\), that maps a curve to its jacobian variety together with the theta divisor. The most common compactification of \(M_g\) is the moduli space of Deligne-Mumford stable curves, \(\bar M_g\). Thus, one is naturally concerned with the study of those compactifications of \(A_g\) together with a map from \(\bar M_g\) to it whose restriction to \(M_g\) coincides with the Torelli map.
An instance of such a pair is given by the second Voronoi toroidal compactification of \(A_g\) together with a suitable map. However, this map fails to be injective and, indeed, may have positive-dimensional fibers. We refer the reader to [\textit{Y. Namikawa}, Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics. 812. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0466.14011)] and [\textit{V. Vologodsky}, The extended Torelli and Prym maps. Univ. of Georgia PhD thesis (2003)].
In this paper, the authors deal with a second instance. Namely, they consider the coarse moduli space of principally polarized semi-abelic stable pairs introduced by Alexeev as well as an extension of the Torelli map for this case [\textit{V. Alexeev}, ``Complete moduli in the presence of semiabelian group action'', Ann. Math. (2) 155, No. 3, 611--708 (2002; Zbl 1052.14017); ``Compactified Jacobians and Torelli map'', Publ. Res. Inst. Math. Sci. 40, No. 4, 1241--1265 (2004; Zbl 1079.14019)].
The first main result is that the compactified Torelli map is injective at curves having \(3\)-edge-connected dual graph (e.g. irreducible curves, curves with two components intersecting in at least three points). The second one offers different charactizations of curves having the same image by the compactified Torelli map. Torelli map; Jacobian variety; theta divisor; stable curve; stable semi-abelic pair; compactified Picard scheme; semiabelian variety; moduli space; dual graph 9 L. Caporaso and F. Viviani, 'Torelli theorem for stable curves', \textit{J. Eur. Math. Soc.}13 (2011) 1289-1329. Jacobians, Prym varieties, Special divisors on curves (gonality, Brill-Noether theory), Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Torelli theorem for stable 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 \(X\) be a smooth projective variety of dimension \(d\) over the field of complex numbers. In regard of the various cohomology theories for \(X\), there are three famous classical conjectures: the Hodge conjecture, the Grothendieck standard conjecture, and the Tate conjecture. It is well known that, especially for abelian varieties, these three fundamental conjectures are closely related to each other, and some implications among them have been proved in special cases. In the paper under review, the author derives some further results concerning the interrelation of these conjectures and their geometric interpretations. First, an earlier approach by \textit{Y. André} [Publ. Math., Inst. Hautes Étud. Sci. 83, 5--49 (1996; Zbl 0874.14010)] is extended, culminating in a reduction of the Hodge conjecture for abelian schemes to the question of the existence of an algebraic isomorphism
\[
H^2(C, R^{2d-i}\pi_*\mathbb{Q}) \widetilde\rightarrow H^0(C, R^i\pi_*\mathbb{Q})
\]
for all \(i\geq 2\) and all families \(\pi: X\to C\) of principally polarized complex abelian schemes of relative dimension \(d\) over nonsingular projective curves. Moreover, it is shown that, if some canonically defined Hodge cycles in \(H^0(C, R^i\pi_*\mathbb{Q})\otimes H^0(C, R^i\pi_*\mathbb{Q})\) are algebraic for all integers \(i\geq 2\), then the Grothendieck standard conjecture holds for the scheme \(X\). Using this, the validity of the Grothendieck standard conjecture [\textit{A. Grothendieck}, Algebr. Geom., Bombay Colloq. 1968, 193--199 (1969; Zbl 0201.23301)] is finally established for an abelian scheme \(X@>\pi>> C\) under the assumption that the Hodge group of the generic geometric fibre is a simple algebraic group of a certain type. In particular, this assumption holds if \(\text{End}(X_s)=\mathbb{Z}\) for some geometric fibre \(X_s\) and \(d\) is a non-exeptional dimension (e.g.,if \(d\) is an odd number).
This very thorough and detailed paper, subdivided into eleven sections, provides a major contribution towards a better understanding of the great classical conjectures for abelian varieties. The important new results are derived in a very complete, rigorous and lucid manner, and the entire paper may even serve as a profound introduction to the fascinating area of the celebrated conjectures on algebraic cycles on (abelian) varieties. Hodge theory; Hodge conjecture; algebraic cycles; Tate conjecture; cohomology С. Г. Танкеев, ``О стандартной гипотезе для комплексных абелевых схем над гладкими проективными кривыми'', Изв. РАН. Сер. матем., 67:3 (2003), 183 -- 224 Transcendental methods, Hodge theory (algebro-geometric aspects), Abelian varieties of dimension \(> 1\), Algebraic cycles, Analytic theory of abelian varieties; abelian integrals and differentials, Classical real and complex (co)homology in algebraic geometry On the standard conjecture for complex abelian schemes over smooth projective 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 According to the publication standards of Cambridge Tracts in Mathematics, the present book provides a thorough yet reasonably concise treatment of a specific topic by taking up a single thread in a wide subject, following various ramifications of it, and illuminating thus its different aspects from a central point of view. The topic of the treatise under review is the local theory of singularities of algebraic varieties, analytic spaces, or morphisms, and the thread taken up is the investigation of singularities via the cohomology theory for sheaves of differential forms, i.e., by means of methods of Hodge theory and its diverse generalizations.
Thus, in a body, this book provides both an introduction to, and a survey of, some central aspects of singularity theory, such as they have been intensively studied over the past thirty years. The text consists of three chapters, each of which is subdivided into several sections. To chapter I, which is preceded by a very thorough, detailed, motivating and masterly written introduction to the contents of the book, has been given the title ``The Gauss-Manin connection''. Here the author explains, always in a very concise but comprehensive and lucid manner, many of the fundamental ideas, methods, techniques, and results centred around this crucial concept. This includes: Milnor fibration, Picard-Lefschetz monodromy transformations, locally constant sheaves and systems of linear differential equations, integrable connections, relative De Rham cohomology sheaves, meromorphic connections, Brieskorn lattices, quasi-homogeneous singularities, singular points of systems of linear differential equations, regularity of the Gauss-Manin connection, period matrices, the geometry of the Picard-Fuchs equation, the monodromy theorem, the Gauss-Manin connection in the case of non-isolated hypersurface singularities, and many other related topics.
Chapter II deals with the various kinds of Hodge structures and their variation behavior under deformations. This chapter is entitled ``Limit mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity''. Here some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and their associated period maps, are developed in the local situation of isolated singularities of holomorphic functions. The main topics of this chapter are, among others, the following ones: mixed Hodge structures, polarized Hodge structures, Deligne's theorem, limit Hodge structures in the sense of W. Schmid, limit mixed Hodge structures in the sense of J. Steenbrink, the Hodge theory of smooth hypersurfaces (after Griffiths-Deligne), the Gauss-Manin system of an isolated singularity, decompositions of meromorphic connections, the limit Hodge filtration due to Varchenko and Scherk-Steenbrink, and spectra of various types of singularities.
The concluding chapter III, entitled ``The period map of a Milnor-constant deformation of an isolated hypersurface singularity associated with Brieskorn lattices and mixed Hodge structures'', discusses the glueing of Milnor fibrations, meromorphic connections of Milnor-constant deformations of singularities, root components of geometric sections, the period map for embeddings of Brieskorn lattices and various types of singularities, the period map associated with the mixed Hodge structure on the vanishing cohomology, the infinitesimal Torelli theorem, the period map for quasi-homogeneous singularities, the Picard-Fuchs singularity, and the recent results of C. Hertling for hypersurface singularities.
Without any doubt, the author has covered a wealth of material on a highly advanced topic in complex geometry, and in this regard he has provided a great service to the mathematical community, first and foremost with a view to the systematic, comprehensive understanding of the vast realm of singularity theory. Although he had to relinquish almost all proofs of the numerous deep theorems, he has succeeded in providing a brilliant introduction to, and a comprehensive overview of, this contemporary central subject of complex geometry. This book is designed for researchers whose interests are closely bound up with singularity theory, algebraic geometry, and complex analysis, and in that it is an excellent source and guide for them. Also, the entire text represents an irresistible invitation to the subject, and may be seen as a dependable pathfinder with regard to the vast existing original literature in the field. local theory of singularities; cohomology theory; sheaves of differential forms; Gauss-Manin connection; Milnor fibration; Picard-Lefschetz monodromy; De Rham cohomology; Brieskorn lattices; period matrices; Picard-Fuchs equation; monodromy; mixed Hodge structure; hypersurface singularity Kulikov, V. S.: Mixed Hodge structures and singularities, (1998) Singularities in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local complex singularities, Complex surface and hypersurface singularities Mixed Hodge structures and singularities | 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 smooth projective geometrically integral curve over a perfect field \(k\).
Let \(q: X \to C\) be a Severi-Brauer \(C\)-scheme of relative dimension \(p-1\), where \(p\) is a prime number different from the characteristic of \(k\). We write \(CH_0(X/C)\) for the kernel of \(q_*: CH_0(X) \to CH_0(C)\). In the paper under review, the author constructs an isomorphism \(\ker[ \pi^*: CH^d(X) \to CH^d(\bar{X})] \cong CH_0(X/C)\) for any integer \(d\) such that \(2 \leq d \leq p\), where \(\pi: \bar{X} \to X\) is the base change to an algebraic closure of \(k\). As a corollary, the Chow ring \(CH^*(X)\) is shown to be finitely generated when \(k\) is a number field. In the proof, the isomorphism \(CH_0(X/C) \cong k(C)^*_{dn}/k^* \cdot Nrd(A^*)\) constructed by \textit{E. Frossard} [Compos. Math. 110, No. 2, 187--213 (1998; Zbl 0902.14007)] is essentially used.
Here \(k(C)^*_{dn}\) is a subgroup of \(k(C)^*\) called the group of divisorial norms of \(X/C\), and \(A\) is the central simple algebra over \(k(C)\) associated to the generic fiber of \(q\). algebraic cycles; Chow groups; Severi-Brauer schemes Algebraic cycles, Varieties over global fields, (Equivariant) Chow groups and rings; motives, Global ground fields in algebraic geometry Algebraic cycles on Severi-Brauer schemes of prime degree 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 [For the entire collection see Zbl 0562.00001.]
This article deals with a program to unify the Frobenius theorem of differential geometry and the Jacobson-Galois theory of purely inseparable field extensions. The Frobenius theorem characterizes integral submanifolds of a manifold in terms of Lie subalgebras of the Lie algebra of derivations of the algebra of smooth functions, or, dually, in terms of differential ideals in the De Rham complex. The Jacobson theory also deals with the Lie subalgebras, this time of the Lie algebra of all derivations of the field extensions. As the author notes, it has no dual formulation using the algebraic De Rham complex.
He defines a new complex in which a dual Frobenius type Galois theorem is possible. A commutative characteristic p base ring R is fixed, and A is a commutative R-algebra. S is the universal A-algebra with a designated R- linear derivation s, and T is the universal A-algebra with a designated R-linear derivation t satisfying \(t^ p=0\). S and T are graded commutative A-algebras. The complex Q (replacing De Rham) is made up of the graded pieces of T, beginning with \(T_ 0\), and the differentials alternatively t and \(t^{p-1}\), beginning with t. The authors' main theorem then asserts that if A is a purely inseparable exponent one field extension of R, there is a one-one correspondence between intermediate fields and submodules V of \(Q_ 1\) satisfying \(q_ 1(V)=\sum T_ it^{p-1-i}(V)\quad\) (sum from 1 to p-1). The homology of the complex Q is computed to be R in degree zero and zero in higher degree, when A has a p-basis over R. This is the algebraic analogue of the Poincaré lemma.
This article, which contains no proofs, is based on an introduction to a paper by \textit{M. Takeuchi} and the author which is in preparation. It is titled ''From differential geometry to differential algebra; analogues to the Frobenius theorem and Poincaré lemma.'' positive characteristic algebraic geometry; differential algebra; integral submanifolds of a manifold; differential ideals in the De Rham complex; characteristic p base ring; purely inseparable exponent one field extension; Poincaré lemma Finite ground fields in algebraic geometry, Galois theory and commutative ring extensions, Inseparable field extensions, Generalizations (algebraic spaces, stacks), Modules of differentials, Decidability and field theory Introduction to the algebraic theory of positive characteristic differential 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 For every prime \(p\) let denote by \(M_g^-(p)\) the moduli scheme of stable curves of genus \(g\) over an algebraically closed field of characteristic \(p\). In this paper it is proved the following theorem:
Fix an integer \(g\geq 5\) and a prime \(p> 84(g-1)\), then the Néron-Severi group \(\text{NS} (M_g^-(p))\) of \(M_g^-(p)\) is freely generated by the \([g/2]+2\) classes \(\lambda\) and \(\Delta_i\), \(0\leq i\leq [g/2]\).
The proof uses a reduction to the characteristic 0 case (i.e. to the Harer's topological theorem) and a result by \textit{M. Pikaart} and \textit{A. J. de Jong} [The moduli space of curves, Proc. Conf. Texel Island 1994, Prog. Math. 129, 483-509 (1995; Zbl 0860.14024)]. moduli scheme of stable curves; characteristic \(p\); Néron-Severi group Families, moduli of curves (algebraic), Picard groups, Finite ground fields in algebraic geometry On the Picard group of the moduli scheme of stable curves 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 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 III.
Number Theory
[1967c] The first part of the volume is basically a course taught at Princeton in 1961-1962. One sentence taken from the book: As will become apparent from the first pages of this book, I have rather tried to draw the conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number-theory.
[1968a] Dirichlet series [1967a] having been related to automorphic forms, Weil estimates that broad generalizations can be obtained; they promise to be of great importance for number-theory and group-theory.
[1968b] Hecke had established that Dirichlet series [1967a] satisfying certain functional equations are linked to modular forms. This observation was generalized. Weil proved a converse statement.
[1970] Weil develops ideas considered in [1968a] that will be presented in [1971a].
[1971a] Weil provides full developments of [1968a]. In his comments he speculates on further achievements reaching maybe the Langlands program.
[1974e] The article is a continuation of [1971a].
[1974b,c] Problems like the decomposition of an integer into a sum of three or four squares as well as the cyclotomy, i.e., the separation of a circle into equal arches, may lead to old and modern mathematical problems.
[1975b] The last sentence of Weil's text reads: Here, except for some shorter notes (interesting but of comparatively minor import), Kummer takes leave of number theory; and here we leave him. As the reader must realize, we have necessarily had to confine ourselves to the most superficial description of the contents of this volume. Even after a hundred years, an attentive stuy can richly repay his efforts.
[1977c] The Pell equation, named by Euler, concerns the solution by integers \(x, y\) of the relation \(Nx^2 + 1 = y^2\), \(N\) being a nonsquare integer \(N\).
[1974a] This text of two lectures provides delightful reading. They start from the times of Fermat. Weil ends with this: I hope to have convinced you that there is a complete-continuity in the main lines of development in number-theory at least ftom the days of Euler down to the present day. I could not hope to do more; if I have convinced you of this, I have more than accomplished my purpose.
Various subjects
[1967a] Explicit computations of solutions for functional equations are displayed.
[1972] Some parts of this article were rearranged in later editions.
[1974d] The article is a continuation of [1952d].
[1977a] We quote from Weil's first lines: Leibniz's discovery, early in his career, of the famous series for \(\pi\) was not only, in the eyes of his contemporaries, one of his most striking achievements; it has also paved the way for the non less sensational summation by Euler, some fifty years later, of the series we now denote by \(\zeta(2n)\). In retrospect, we see that these were the first examples for relations between periods of Abelian integrals of Dirichlet series or (more generally) of automorphic forms. Although a number of the best mathematicians of the last and of the present century have studied various aspects of this subject, it may fairly be said that only its surface has been stretched so far.
[1976a] One may ask the general question of the role played by the complex structure in the present theory of automorphic functions. Even on \(GL(2)\) that structure abandons us as soon as the fundamental field is not totally real.
[1976b] This is a technical paper situated at the confluence of several research themes.
Harmonic Analysis
[1964b] This long article as well as the next one are providing very useful information. Both articles are very carefully written. Their subjects belong to Functional analysis and Harmonic analysis. For specific groups unitary representations are computed. Examples are locally compact Abelian groups and metaplectic groups.
[1965] Foundations of [1965] are provided in [1964b]. Dixmier suggested to make use of the Schwartz--Bruhat distributions and to suppose them well tempered. The Fourier transform may be studied. As one of the simplest outcomes one may mention the famous Poisson formula.
[1966] The article is about a reinterpretation of the proof given by Iwasawa aand Tate for the functional equation satisfied by the zêta function.
[1977c] Weil expresses serious doubts about the Hodge conjecture.
General
[1967b] André Weil was probably the best mathematician who could write a review of Emil Artin's Collected Papers.
[1971b] André Weil has written more than a CV, an extensive description of the history of Jean Delsarte.
[1971c] André Weil covers the whole impressive professional life of Jean Delsarte. He concludes: \textit{Les travaux réunis suffiront amplement à confirmer le renom de Delsarte comme l'un des meilleurs analystes et l'un des esprits les plus originaux parmi les mathématiciens de notre temps}.
[1973] Two quotations from Weil's article and comments: First sentence. Nothing could be more welcome than a book on Fermat. Last sentence. Facit indignatio versum.
[1975a] Weil is not very enthusiastic about the book in spite if its richness.
[1976c] Eisenstein tells us that his love for mathematics came from studying first Euler and Lagrange, then Gauß; studying the great work of the past is still the best education.
[1978a] Weil critizises translations from Greek mathematics. To conclude: When a discipline, intermediary in some sense between two already existing ones (say A and B) becomes newly established, this often makes room for the proliferation of prasites, equally ignorant of both A and B, who seek to thrive by intimating to practitioners of A that they do not understand B, and vice versa. We see this happening now, alas, in the history of mathematics. Let us try to stop the disease before it proves fatal.
[1978b] The article covers Weil's plenary address at the Helsinki ICM congress in 1978. Weil: My original question `Why mathematical history?' finally reduces itself to the question `Why mathematics?', which fortunately I do not feel called upon to answer. 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. III (1964--1978) | 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 result of \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] says that if \(S\) is a characteristic \(p>0\) scheme, \(Y/S\) is a smooth \(S\)-scheme \(p>0\), of relative dimension less than \(p\) and liftable mod \(p^2\), then the de Rham complex of \(Y/S\) is quasi-isomorphic to the direct sum of its cohomology sheaves. In [Duke Math. J. 60, No. 1, 139--185 (1990; Zbl 0708.14014)], \textit{L. Illusie} generalized this to the case of de Rham complexes with coefficients in Gauss-Manin systems: if \(f:X\to Y\) is a semistable \(S\)-morphism, we get a module with logarithmic integrable connection \(H = \bigoplus_i R^i f_* \Omega^\bullet_{X/Y}\) (where \(Y\) and \(X\) are to be treated as logarithmic schemes with the compactifying log structures given by \(E\) and its preimage, respectively), and a version of the decomposition theorem holds for \(H\). In the paper under review this is generalized further to the case when the log structure on \(X\) has some horizontal components (Theorem 5.9). This allows the author to show a version of the Kollár vanishing theorem in positive characteristic (Theorem 6.3). The method of proof follows closely that of Illusie in op.cit. de Rham complex; semistable reduction; positive characteristic Vanishing theorems in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology Decomposition of de Rham complexes with smooth horizontal coefficients for semistable reductions | 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 note, we discuss a Galois theoretic topic where the two subjects of the title intersect. Three co-related sections will be arranged as follows. In part I, we review basic notion of tangential base points for étale fundamental groups of schemes of characteristic zero. Then, in part II, we introduce `Eisenstein power series' as a main factor of the Galois representation ``of Gassner-Magnus type'' arising from an affine elliptic curve with `Weierstrass tangential base point'. Part III is devoted to examining the Eisenstein power series in the case of the Tate elliptic curve over the formal power series ring \(\mathbb{Q}[[q]]\) [introduced by \textit{P. Roquette} in: ``Analytic theory of elliptic functions over local fields'', Göttingen (1970; Zbl 0194.52002), and \textit{P. Deligne} and \textit{M. Rapoport} in: Modular Functions one variable II, Proc. Int. Summer School, Univ. Antwerp 1972, Lecture Notes Math. 349, 143-316 (1973; Zbl 0281.14010)]. We deduce then a certain explicit relation (theorem 3.5) between such Eisenstein power series and Ihara's Jacobi-sum power series [\textit{Y. Ihara}, Ann. Math. (2) 123, 43-106 (1986; Zbl 0595.12003)]. Eisenstein power series; Galois category; tangential base points; Tate elliptic curve Nakamura, H., Tangential base points and Eisenstein power series, (Aspects of Galois Theory. Aspects of Galois Theory, Gainesville, FL, 1996. Aspects of Galois Theory. Aspects of Galois Theory, Gainesville, FL, 1996, London Math. Soc. Lecture Note Ser., vol. 256, (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 202-217 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Galois theory, Elliptic curves over local fields, Polylogarithms and relations with \(K\)-theory Tangential base points and Eisenstein power 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 This is a graduate text with an integrated treatment of several complex variables and complex algebraic geometry, with applications to the structure and representation theory of complex semisimple Lie groups.
The book begins with an exposition of the Cauchy-Riemann equations, Mittag-Leffler and Weierstrass theorems, partitions of unity, Cauchy's formula, power series expansions, Hartog's theorem and domain of holomorphy (Chs 1-2, Selected Problems in One Complex Variable, Holomorphic Functions of Several Variables).
The behavior of holomorphic functions in small neighborhoods of a point is discussed, leading to the notion of the germ of a function at a point. The algebraic properties of the local rings of regular and holomorphic functions, first on \(\mathbb C^n\) and then on varieties, are studied. The basic tools for this study are the Weierstrass preparation and division theorems. These allow to reduce problems involving germs of holomorphic functions in \(n \) variables to problems involving polynomials with coefficients which are germs of holomorphic functions in \(n-1\) variables. These results lead to the fact that the local ring of germs of holomorphic functions is Noetherian, and to the implicit and inverse mapping theorems, which do not have analogues for the local rings of rational function (Ch. 3, Local Rings and Varieties). The exact connection between germs of holomorphic varieties and ideals in the ring of germ of holomorphic functions is stated (Ch. 4, The Nullstellensatz). Three notions of dimension of the local ring -- topological, geometric, and tangential -- are discussed. The attention is focused on holomorphic varieties, turning to the study of dimension for algebraic varieties (Ch. 5, Dimension).
Abstract sheaf theory and sheaf cohomology provide the formal machinery for passing from local to global solutions for wide variety of problems, as well for classifying the obstructions to doing so when local solutions do not give rise to global solutions. Sheaves are defined and sheaf cohomology is developed as an application of homological algebra to the category of sheaves on a topological space. These results are used to give brief developments of two classical cohomology theories -- de Rham and Čech (Chs 6-7, Homological Algebra, Sheaves and Sheaf Cohomology). Abstract algebraic and holomorphic varieties are defined and classes of quasi-coherent and coherent algebraic and analytic sheaves on such varieties are studied. It is shown that the category of coherent analytic sheaves on holomorphic variety \(X\) is a full abelian subcategory of the category of sheaves of \(_X\mathcal H\)-modules (Chs 8-9, Coherent Algebraic Sheaves, Coherent Analytic Sheaves).
The category of Stein spaces is considered and the ground work for proving Cartan's theorems is laid. The key result here is a vanishing theorem which states that a coherent analytic sheaf defined in a neighborhood of a compact polydisc has vanishing higher cohomology on the polydisc (Ch. 10, Stein Spaces).
The approximation argument used to finally prove Cartan's theorems requires knowing that a coherent analytic sheaf has the structure of a Fréchet sheaf. It is shown that there always is such a structure and that it is unique subject to certain condition. It is also proved that morphisms between coherent sheaves are automatically continuous for this structure. Then Cartan's theorems, that on a Stein space, a coherent analytic sheaf has a rich supply of global sections and every coherent analytic sheaf is acyclic, are proved. The Cartan-Serre theorem, that the cohomology modules of a coherent algebraic sheaf on a compact holomorphic variety are finite dimensional, is also proved. It plays a key role, along with Cartan's theorems, in the proof of Serre's theorem (Ch. 11, Fréchet Sheaves -- Cartan's Theorems).
Several complex variables and complex algebraic geometry are not just similar; they are equivalent when done in the context of projective varieties. This is the content of the famous Serre's GAGA theorems. Complete proofs of these results are given after first studying the cohomology of coherent sheaves on projective spaces (Chs 12-13, Projective Varieties, Algebraic vs. Analytic -- Serre's Theorems).
The final three Chs 14-16, Lie Groups and Their Representations, Algebraic Groups, The Borel-Weil-Bott Theorem, are devoted to the study of complex semisimple Lie groups and their finite dimensional representations. The subject does provide significant insight into how the preceding results are used in practice. A survey of basic results from harmonic analysis -- specifically, topological groups, Lie groups, Lie algebras and group representations, with an emphasis on compact groups and the Peter-Weyl theorem, the structure of complex semisimple Lie groups and Lie algebras, and the finite dimensional representations of semisimple Lie algebras -- is included in order to provide the formulation and Miličić's proof of the Borel-Weil-Bott theorem which pinpoints the relationship between finite dimensional holomorphic representations of a complex semisimple Lie group \(G\) and the cohomologies of \(G\)-equivariant holomorphic line bundles on a projective variety constructed from \(G\). A brief introduction to the theory of complex algebraic groups is developed just enough to prove the key structure theorems for complex semisimple Lie groups using algebraic group methods. In particular, it provides nice applications of the algebraic geometry, the sheaf theory, the Cartan-Serre theorem, the material on projective varieties, Serre's theorems, and of course, the background material on algebraic groups and general Lie theory. For example, it is proved that a connected and semisimple complex Lie group is actually an algebraic group.
Each chapter ends with an exercise set. Many exercises involve filling in details of proofs in the text or proving results that are needed elsewhere in the text, while others supplement the text by exploring examples or additional material.
The book can serve as an excellent text for a graduate course on modern methods of complex analysis, as well as a useful reference for those working in analysis. holomorphic functions of several variables; algebraic geometry; complex semisimple Lie groups; abstract harmonic analysis; local rings and varieties; nullstellensatz; dimension; homological algebra; sheaf cohomology; coherent algebraic sheaves; coherent analytic sheaves; Stein spaces; Fréchet sheaves; Cartan's theorem; projective varieties; Serre's theorems; Dolbeault cohomology; chains of syzygies; Cartan's factorization; amalgamation of syzygies; algebraic groups; Borel-Weil-Bott theorem; equivariant line bundles; flag variety; Casimir operator J. L. Taylor, \textit{Several Complex Variables with Connections to Algebraic Geometry and Lie groups}, Graduate Studies in Mathematics, \textbf{46}, AMS, Providence, RI, 2002. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, General properties and structure of complex Lie groups, Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations, Analysis on other specific Lie groups Several complex variables with connections to algebraic geometry and Lie groups | 0 |
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