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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 We study the étale homotopy theory of Brauer-Severi varieties over fields of characteristic 0. We prove that the induced Galois representations on geometric homotopy invariants (e.g., \(\ell\)-adic cohomology or higher homotopy groups) are all isomorphic for Brauer-Severi varieties of the same dimension. If the base field has cohomological dimension smaller or equal 2 then we can show more in the case of Brauer-Severi curves: There is even an isomorphism between the Hochschild-Serre spectral sequences computing cohomology with local coefficients. Further, we study homotopy rational and homotopy fixed points on Brauer-Severi varieties and their connections to genuine rational points. In particular, we show that under a suitable assumption on the first profinite Chern class map an analogue of the weak section conjecture for Brauer-Severi varieties turns out to be true. We can give a counter example to this analogue without the extra assumption over \(p\)-adic local fields. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Brauer groups of schemes, \(p\)-adic cohomology, crystalline cohomology Anabelian aspects in the étale homotopy theory of Brauer-Severi 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 (Review 1)
The classical construction of the Artin representation and Swan representation associated with henselian discretely valued fields of rank 1 with perfect residue field is generalized to a larger class of valued fields. This construction and a Lefschetz fixed point formula are applied to prove an Euler-Poincaré formula for the étale cohomology of rigid analytic curves.
(Review 2)
In previous papers the author has introduced the concept of adic spaces, see in particular his monograph `Étale cohomology of Rigid Analytic Varieties and Adic Spaces' (Vieweg 1996; Zbl 0868.14010). To any rigid analytic variety \(X\) over a complete nonarchimedean field one can construct an adic space \(X^{ad}\); in addition to the classical points of \(X\), \(X^{ad}\) contains points whose residue field is equipped with a valuation of rank \(>1\). \newline To a finite Galois extension \(L/K\) of complete discretely valued fields one can associate the Artin and the Swan representation of \(G=\text{Gal}L/K\). In the first part of the present paper the author generalizes this construction to a certain class of fields with a valuation of higher rank which includes all valuations arising in adic spaces. As in the classical situation one constructs class functions \(a_G\) and \(sw_G\) which in the new situation turn out to be characters of virtual representations only.
A central result of the paper is a Lefschetz fixed point formula for a finite morphism \(f:X \to X\) of a quasicompact smooth adic curve \(X\). Such a curve has a unique adic compactification, and to the formula contribute not only the fixed points of \(f\), but also the points in the completion (which all have residue fields with valuations of rank \(>1\)). \newline If \(F:Y \to X\) is a finite Galois covering of adic curves with Galois group \(G\), \(M \subset Y\) a locally closed constructible subset and \(L:=f^{-1}(M)\), then the Lefschetz fixed point formula can be applied to express the representations of \(G\) on the étale cohomology of \(L\) in terms of the Artin and Swan representations of \(G\). \newline In the last part of the paper \(F\) is a locally constant sheaf on the étale site of an adic curve \(X\) as above (or a slightly more general sheaf). The author proves an Euler-Poincaré formula for the restriction of \(F\) to a locally closed constructible subset \(L\) of \(X\). The main ingredient here is the Swan conductor of the restriction of \(F\) to the étale site of a point. The definition of this conductor is also generalized from discrete valuations to valuations of higher rank. Finally these results are compared to Ramero's results on meromorphically ramified sheaves [\textit{L. Ramero}, J. Alg. Geom. 7, No. 3, 405-504 (1998; Zbl 0964.14019)]. Euler-Poincaré formula; rigid analytic curves; adic space; valuation of higher rank; Artin representation; Swan representation; Lefschetz fixed point formula Huber, R.: Swan representations associated with rigid analytic curves. Journal für die Reine und Angewandte Mathematik \textbf{537}, 165-234 (2001) Rigid analytic geometry, Arithmetic ground fields for curves, Étale and other Grothendieck topologies and (co)homologies, Ramification and extension theory Swan representations associated with rigid analytic 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 theory of Néron models has been extensively developed over the last fifty years. The motivation behind their original introduction in [\textit{A. Neron}, Publ. Math., Inst. Hautes Étud. Sci. 21, 128 (1964; Zbl 0132.41403)] was to control the behaviour of heights on abelian varieties, and this later proved decisive in, for example, Falting's proof of the Mordell Conjecture, but the focus here is not on diophantine problems at all. The Néron model \({\mathcal A}\) of an abelian variety \(A\) over the quotient field \(K\) of a DVR \(R\) is essentially the minimal smooth \(R\)-model of \(A\) as a group scheme. As such it is naturally associated to \(A\) and has good functorial properties, but its association with the base is more problematic. It does not commute with ramified base change, and a central problem is to understand and control this failure.
Considerable work in this direction has been done, by the authors of this book among others, in the case of tame ramification. An important invariant is the component group \(\Phi(A)\), which is the group of components of the reduction, i.e.\ of the group scheme \({\mathcal A}\times_R k\), where \(k\) is the residue field of \(R\). The order of this group is a generalisation of the notion of Tamagawa number: in the case where \(A\) is the localisation of an abelian variety over a number field at a prime \(p\), the order of the component group is precisely the Tamagawa number at \(p\) that appears in the Birch--Swinnerton-Dyer conjecture.
In earlier work [Math. Ann. 348, No. 3, 749--778 (2010; Zbl 1245.11072)] and [Adv. Math. 227, No. 1, 610--653 (2011; Zbl 1230.11076)] the authors introduced two formal power series that are generating functions for the Néron models of tamely ramified extensions of \(K\). Fix a separable closure \(K^s\) of \(K\) and for \(d\) coprime to \(\text{Char}\, k\) denote by \(K(d)\) the unique degree \(d\) extension of \(K\) inside \(K^s\) and by \(A(d)=A\times_K K(d)\) the corresponding abelian variety, with Néron model \({\mathcal A}(d)\). The motivic zeta function
\[
Z_A(T)=\sum_d \left[{\mathcal A}(d)_k\right]{\mathbb L}^{\text{ord}_A(d)} T^d \in K_0({\mathbf{Var}}_k)[[T]]
\]
has coefficients in the Grothendieck ring of \(k\)-varieties, and is closely related to the in principle cruder component series
\[
S^\Phi_A(T)=\sum_d\left|\Phi(A(d))\right|T^d \in {\mathbb Z}[[T]].
\]
In particular the earlier work of the authors shows that both these series are rational functions if \(A\) is tamely ramified. Moreover, \(Z_A({\mathbb L}^{-s})\) has a unique pole where \(s\) is equal to Chai's base change conductor \(c(A)\).
The main aim of the present book is to extend these results and ideas to other group schemes and to some case of wildly ramification. A fully general theory still appears out of reach at the moment, but by focussing on semi-abelian varieties and on wildly ramified Jacobians the authors are able to identify the main features and provide some technical tools, as well as obtaining good results for those cases.
The general principle used here as well as in the authors' earlier work is that although Néron models do change under tame base change they do not do so more than they can help. If \(K'/K\) is a tame extension of degree~\(d\) then the size of the component group gets multiplied by \(d^{t(A)}\) (where \(t(A)\) is the rank of the torus part of \({\mathcal A}_k^0\)) in the good case where \({\mathcal A}_k^0\) has no unipotent part (\(A\) is said to have semi-abelian reduction in this case). This will not always be the case for arbitrary tame \(K'/K\) and arbitrary \(A\), but it does hold if, essentially, what we have to adjoin to \(K\) to get a field \(L\) over which \(A\) has semi-abelian reduction has nothing to do with \(K'\) (the authors say that \(K'\) is ``sufficiently orthogonal'' to \(L\)). Under these conditions one also expects the torus rank, unipotent rank and abelian part of the central fibres to be unchanged by the base change to \(K'\), and that amounts to saying that they give the same element of the Grothendieck ring \(K_0({\text{Var}}_k)\).
In the tame case it is easy to say what ``sufficiently orthogonal'' should mean: the degrees of \(L/K\) and \(K'/K\) being coprime is enough. If \(A\) is wildly ramified the situation is much less clear. However, for Jacobians \(A={\text{Jac}}\, C\), one can pass to the curves and extract a new invariant, introduced here, called the stabilisation index \(e(C)\): then one requires \(d\) to be coprime to \(e(C)\), and the desirable consequences above for the Néron model follow.
The definition of \(e(C)\) and the details of the argument above, leading to a proof that the Néron component series \(S^\Phi_A(T)\) is a rational function in the Jacobian case, occupy Chapter~4 of the book: Chapters~1--3 are in various ways introductory. The invariant \(e(C)\) is in the tame case simply the degree of the extension necessary to obtain a normal crossings curve, but the general definition is more complicated and depends on the multiplicities of components in a suitable model of~\(C\).
Chapter~5 deals with the semi-abelian case. The difficulty here is that one cannot remain in the algebraic setting. Instead the authors use the rigid analytic geometry approach of [\textit{S. Bosch} and \textit{X. Xarles}, Math. Ann. 306, No. 3, 459--486 (1996; Zbl 0869.14020)] (making a correction to that paper as they do so) to control the torsion part of the component group -- the component group itself is in this context no longer finite.
The next part of the book examines some other invariants of Néron models, starting with an analogue of Chai's base-change conductor called the tame base-change conductor, defined for any semi-abelian \(K\)-variety \(G\) and agreeing with Chai's conductor if (and only if) \(G\) is tamely ramified. It is the sum of the jumps in Edixhoven's filtration of the central fibre [\textit{B. Edixhoven}, Compos. Math. 81, No. 3, 291--306 (1992; Zbl 0759.14033)]: Chapter~6 establishes its basic properties. In Chapter~7 the authors attempt to relate it to other arithmetic invariants such as the Artin conductor, obtaining results for Jacobians of curves of genus~1 and genus~2 but in a rather ad hoc way, making for instance essential use of the hyperellipticity.
Finally we come to motivic zeta functions. Chapter~8 makes the promised use of the rationality results for the component series to deduce results about rationality and poles for \(Z_A\) from those for \(S^\Phi_A\), in the cases where the latter have now been proved (tamely ramified semi-abelian varieties, or Jacobians, with a mention also of Pryms). Chapter~9 adds a cohomological interpretation of the motivic zeta function via a trace formula, and deduces some consequences. Chapter~10 lists some open problems and future directions, some of which have been implicitly mentioned earlier in the book. Néron model; semi-abelian variety; conductor; base change; motivic zeta function Halle, L.H., Nicaise, J.: Néron Models and Base Change. Vol.~2156 of Lecture Notes in Mathematics. Springer, New York (2016) Arithmetic ground fields for abelian varieties, Arcs and motivic integration, Rigid analytic geometry, Jacobians, Prym varieties Néron models and base change | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(U\) be a smooth connected affine curve over an algebraically closed field \(k\) of characteristic \(p>0\). An explicit description of the set of finite quotients of the étale fundamental group \(\pi_1(U)\) was conjectured in 1957 by Abhyankar, and proved by \textit{M. Raynaud} [Invent. Math. 116, 425--462 (1994; Zbl 0798.14013)] (in the affine line case) and by \textit{D. Harbater} [Invent. Math. 117, 1--25 (1994; Zbl 0805.14014)] (in general case), giving a necessary and sufficient condition for a finite group to be a Galois group of an étale cover of \(U\). This paper investigates questions of which covers of \(U\) are dominated by other covers having specified Galois groups, and of which inertia groups can arise over points ``at infinity''. The main results give constructions of modified covers of a given cover (enlarging a given Galois group by a quasi \(p\)-group, or enlarging the \(p\)-parts of inertia subgroups of a given Galois group) with special controls of branching data. As an application, a tame analogue of the geometric Shafarevich conjecture is proved. It implies, for example, the following result: Let \(\pi_1^t(U,\Sigma)\) be the Galois group of the maximal extension of the function field of \(U\) that is at most tamely ramified over the places in \(\Sigma\subset U\), and is étale over all places corresponding to other points of \(U\). If \(k\) is finite and \(\Sigma\) is a dense open subset of \(U\), then \(\pi_1^t(U,\Sigma)\) is isomorphic to the semidirect product of \(\hat\mathbb Z\) and the free profinite group of countably infinite rank. Galois covers David Harbater, ``Abhyankar's conjecture and embedding problems'', J. Reine Angew. Math.559 (2003), p. 1-24 Coverings of curves, fundamental group, Inverse Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Abhyankar's conjecture and embedding problems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 generalize the Galois correspondence and existence of Galois closure of a field to certain algebraic stacks over a field. More precisely, they introduce the notions of pseudo-properness and inflexibility of stacks. Then they consider a pseudo-proper and inflexible algebraic stack \(\mathcal X\) of finite type over a field \(k\) and an essentially finite cover \(f: {\mathcal Y}\to {\mathcal X}\). They also need some additional assumptions if \({\mathrm char} \, k>0\): either \(f\) is étale or \(\dim H^1 ({\mathcal X} , E)<\infty\) for all vector bundles \(E\). Then they show that there exists a unique (up to equivalence) finite map to the Nori fundamental gerbe \(\Pi ^{\mathrm N}_ {{\mathcal X}/k}\) of \({\mathcal X}/k\), whose base change along \({\mathcal X} \to \Pi ^{\mathrm N}_{{\mathcal X}/k}\) gives \(f\). They also prove some additional criteria on when \({{\mathcal Y}/k}\) is inflexible in case \(f\) is étale or a torsor. As a corollary they get a Galois correspondence between pointed essentially finite covers \(({\mathcal Y}, y)\to ({\mathcal X}, x)\) with inflexible \(\mathcal Y\) and subgroups of finite index in the Nori fundamental group of \(\mathcal X\).
The authors prove also existence of a Galois closure for (pointed) towers of torsors under finite group schemes over a pseudo-proper and inflexible algebraic stack of finite type over a field. They also show that previous attempts to construct such closures fail and their assumptions in positive characteristic are necessary. In particular, the construction provided in [\textit{M. A. Garuti}, Proc. Am. Math. Soc. 137, No. 11, 3575--3583 (2009; Zbl 1181.14053)] is incorrect.
Part of the paper is devoted to extension of the above results from the Nori set-up to the so called S-fundamental gerbes that are defined using numerically flat bundles. Nori fundamental gerbe; essentially finite bundle; essentially finite cover; S-fundamental gerbe; algebraic stack Homotopy theory and fundamental groups in algebraic geometry, Stacks and moduli problems, Group schemes Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75-109 (1969; Zbl 0181.48803)] defined stable curves and proved that the moduli space \(\overline{{\mathcal M}_g}\) of stable curves of genus \(g\) is a ``compactification'' of the moduli space \({\mathcal M}_g\) of smooth curves. For any given integer \(m\geq 3\) (invertible on some base scheme) Mumford has constructed a fine moduli scheme \({\mathcal M}_{g,m}\) of curves of genus \(g\) with level-\(m\)-structure; moreover \({\mathcal M}_{g,m} \to {\mathcal M}_g\) is a Galois covering. It is useful to have a compactification of \({\mathcal M}_{g,m}\),
\[
\begin{matrix} {\mathcal M}_{g,m} & \hookrightarrow & ?\\ \downarrow & & \downarrow\\ {\mathcal M}_g & \hookrightarrow & \overline{{\mathcal M}_g}. \end{matrix}
\]
We find definitions and properties of such a compactification in the paper cited above, p. 106, in the book by \textit{H. Popp}, ``Moduli theory and classification theory of algebraic varieties'', Lect. Notes Math. 620 (1977; Zbl 0359.14005); lecture 10, and in \S 2 of a paper by \textit{J.-L. Brylinski} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 295-333 (1979; Zbl 0432.14004)]. With the convenient definitions, the results are not so difficult to find, and in this note we put these properties together. The main results are:
Theorem 2.1. A compactification \(\overline{{\mathcal M}_{g,m}}\), with a tautological family \({\mathcal D}\to \overline {{\mathcal M}_{g,m}}\) exists.
This space is not constructed as a coarse or a fine moduli scheme associated with a moduli functor.
Theorem 3.1. The compactification \(\overline{{\mathcal M}_{g,m}}\) is a normal space, singular for \(g\geq 3\).
The results of this note were written up in preprint 301 (August 1983) of the Mathematics Department of the University of Utrecht. Our results were partly contained in a paper by \textit{S. M. Mostafa} [J. Reine Angew. Math. 343, 81-98 (1983; Zbl 0526.14012)], and we never published this preprint. compactification of moduli space B. van Geemen and F. Oort, A compactification of a fine moduli space of curves, University of Utrecht, Preprint 301, 1983. To be published in the Proceedings of Resolutions of Singularities, September 1997, Obergurgl, Austria. Families, moduli of curves (algebraic), Fine and coarse moduli spaces A compactification of a fine moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0635.00006.]
The authors consider the Hilbert scheme \(Hilb^ 3(P^ 2)\) which parametrizes the triplets in the projective plane, and its ring of rational equivalence CH. Their aim is to describe the multiplicative structure of this ring and to give some enumerative applications. The details and other applications are announced in two other papers.
The authors introduce the incidence variety \(Hilb^ 3(P^ 3)\) of pairs (t,d) where d is a doublet and t a triplet containing d. They prove that this variety is smooth. No such result is known for a general \(Hilb^ d(P^ 2)\), although \(Hilb^ d(P^ 2)\) is known to be smooth (Hartshorne, Fogarty). Next the authors recall the computation of the additive structure of CH by Ellingsrud and Strømme, and then they give explicit generators of the ring by cycles having a geometric meaning and prove that the \(CH^ i\) are free groups generated by monominals on these cycles.
The paper ends with three enumerative applications: number of curves on a net \({\mathcal N}\) osculating a given curve or having a singular point on it, number of singular points appearing in two given nets. Hilbert scheme; rational equivalence; number of curves; net; number of singular points G. Elencwajg and P. le Barz, Explicit computations in Hilb3P2, à paraître. Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Enumerative problems (combinatorial problems) in algebraic geometry Explicit computations in \(Hilb^ 3\,{\mathbb{P}}^ 2\) | 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 \((A,\mathscr{L},\Theta_n)\) be a dimension \(g\) abelian variety together with a level \(n\) theta structure over a field \(k\) of odd characteristic. We thus denote by \((\theta^{\Theta_{\mathscr{L}}}_i)_{(\mathbb{Z}/n \mathbb{Z})^g}\in\Gamma (A,\mathscr{L})\) the associated standard basis. For a positive integer \(\ell\) relatively prime to \(n\) and the characteristic of \(k\), we study change of level algorithms which allow one to compute level \(\ell n\) theta functions \((\theta_i^{\Theta_{\mathscr{L}^\ell}}(x))_{i\in (\mathbb{Z}/\ell n\mathbb{Z})^g}\) from the knowledge of level \(n\) theta functions \((\theta^{\Theta_{\mathscr{L}}}_i(x))_{(\mathbb{Z}/n \mathbb{Z})^g}\) or vice versa. The classical duplication formulas are an example of change of level algorithm to go from level \(n\) to level \(2n\). The main result of this paper states that there exists an algorithm to go from level \(n\) to level \(\ell n\) in \(O(n^g \ell^{2g})\) operations in \(k\). We derive an algorithm to compute an isogeny \(f:A\rightarrow B\) from the knowledge of \((A,\mathscr{L},\Theta_n)\) and \(K\subset A[\ell]\) isotropic for the Weil pairing which computes \(f(x)\) for \(x\in A(k)\) in \(O((n\ell)^g)\) operations in \(k\). We remark that this isogeny computation algorithm is of quasi-linear complexity in the size of \(K\). isogenies; abelian varieties; computational algebraic geometry Isogeny, Computational aspects of higher-dimensional varieties, Theta functions and abelian varieties, Abelian varieties of dimension \(> 1\) Fast change of level and applications to isogenies | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present doctoral dissertation deals with two topics in the theory of irreducible affine curves. More precisely, the aim of the work is to study differential operators with a view towards fixed point formulae and isomorphism theorems for affine curves.
In Chapter I the author considers non-singular affine plane curves over a finite field of characteristic \(p\) and the so-called Dwork operators on them. The traces of these operators play an important role in computing the \(\mathbb{F}_{q^ s}\)-rational points in that curve, where \(q=p^ f\) is the cardinality of the finite ground-field and \(s\) any positive integer [cf. \textit{B. Dwork}, Am. J. Math. 82, 631-648 (1960; Zbl 0173.485)]. The main result in this chapter consists in relating the Dwork operator to the Cartier operator on differential 1-forms and, furthermore, to the Hasse-Witt invariant of the affine curve. The link between the Dwork operator and the Cartier operator is proven in the more general case of an affine hypersurface. From this interrelation, and a trace formula for a modified Dwork operator the author derives both an elementary proof of the fixed point formula \(\pmod p\) for the Cartier operator on curves and a new fixed point theorem for the Hasse-Witt invariant.
Chapter II is devoted to the study of the ring of global differential operators \(\hbox{Diff}_ \mathbb{C}(X)\) of irreducible affine curves over the field of complex numbers. The starting point of the author's investigations is the question of whether the isomorphism class of \(X\) is uniquely determined by the isomorphism class of its ring of differential operators \(\hbox{Diff}_{\mathbb{C}}(X)\). The central result is here an affirmative answer in the case of those curves whose normalization is either
(i) different from the affine line \(\mathbb{A}^ 1_ \mathbb{C}\) or
(ii) equal to \(\mathbb{A}^ 1_ \mathbb{C}\), but the normalization map is not injective.
These curves, as the author shows, turn out to be precisely those affine curves which are not topologically simply connected. The present result improves an earlier theorem of \textit{L. Makar-Limanov} [cf. Bull. Lond. Math. Soc. 21, No. 6, 538-540 (1989; Zbl 0693.16003)] which says that two curves of positive genus are isomorphic if and only if their rings of global differential operators are so. The author's proof is based upon a detailed study of the relation between the coordinate ring \({\mathcal O}(X)\) of \(X\) and the subalgebra \(\hbox{adnil}(X)\subseteq\hbox{Diff}_ \mathbb{C}(X)\) of ad-nilpotent differential operators. The concept of ad- nilpotent differential operators is due to \textit{J. Dixmier} [cf. Bull. Soc. Math. Fr. 96, 209-242 (1968; Zbl 0165.04901)], and the relation \(\hbox{adnil}(X)={\mathcal O}(X)\) turns out to be equivalent to the above statement (i) and (ii). --- The remaining part of chapter II provides some deeper insight into the structure of maximal commutative subalgebras of \(\hbox{Diff}_ \mathbb{C}(X)\) contained in \(\hbox{adnil}(X)\) but different from \({\mathcal O}(X)\). characteristic \(p\); Dwork operators; fixed point formula; Cartier operator; Hasse-Witt invariant; differential operator Arithmetic ground fields for curves, Finite ground fields in algebraic geometry, Commutative rings of differential operators and their modules, Differential algebra, Special algebraic curves and curves of low genus, Rings of differential operators (associative algebraic aspects) Contributions to the investigation of irreducible 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 curve over a local field \(K\) of residue characteristic \(p\). This paper studies the divisibility properties of the rational points on the Jacobian of \(C\), under the simplifying assumption that \(C\) has totally degenerate semi-stable reduction.
Firstly, the paper gives a necessary and sufficient condition for a line bundle \(L\) on \(C\) to be \(r\)-divisible, when \(r\) is prime to \(p\). Clearly it is necessary that \(L\) be representable by a divisor \(D\) whose closure on the semi-stable reduction of \(C\) has degree divisible by \(r\) on each component. The main contribution of the paper is to provide a finer, sufficient condition.
Secondly, the paper applies the condition above to certain hyperelliptic curves, in order to determine whether or not they have rational theta characteristic. Furthermore, for the hyperelliptic curves above, the author computes the full prime-to-\(p\) \(K\)-rational torsion group of the Jacobian.
All the computations rely on the fact that the special fiber of the Néron model of the Jacobian of a curve \(C/K\) with totally degenerate semi-stable reduction is an extension of a finite group scheme by a torus. The author is able to prove his criterion by explicitly and carefully relating the character theory of the torus to the homology of the dual graph of the special fiber of a semi-stable model of \(C\). semi-stable reduction; tori over finite fields; Néron models; Picard group torsion; theta characteristic Local ground fields in algebraic geometry, Curves over finite and local fields, Picard groups, Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves A descent map for curves with totally degenerate semi-stable reduction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 classic book ``Algèbre Locale -- Multiplicités'' [Lect. Notes Math. 11 (1957/58; Zbl 0661.13008)], \textit{J.-P. Serre} introduced a definition of intersection multiplicity in a general algebraic setting. In Serre's words this multiplicity ``is equal to some Euler-Poincaré characteristics constructed by means of the Tor functor of Cartan-Eilenberg''. This notion agrees with the classical intersection multiplicity considered in algebraic geometry (in the case of proper intersections). Because of this general setting some properties which should hold for the intersection multiplicity are not obviously satisfied. The author presents a recent proof of Gabber of the non-negativity conjecture for this intersection multiplicity:
If \(R\) is a regular local ring and \(M,N\) finitely generated \(R\)-modules such that \(M\otimes_R N\) has finite length, then the Serre intersection multiplicity \( \chi(M,N)\) is non-negative.
He also gives a new proof of the vanishing conjecture. non-negativity conjecture for intersection multiplicity; local rings; Hilbert-Samuel polynomial ----, Recent developments on Serre's multiplicity conjectures : Gabber's proof of the nonnegativity conjecture , L'Enseignment Mathématique 44 (1998), 305-324. Homological conjectures (intersection theorems) in commutative ring theory, Multiplicity theory and related topics, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 develop geometry of algebraic subvarieties of \(K^n\) over arbitrary Henselian valued fields \(K\) of equicharacteristic zero. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem to the effect that the projections \(K^n\times\mathbb{P}^m(K)\to K^n\) are definably closed maps. It enables, in particular, application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses, among others, the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to \textit{R. Cluckers} and \textit{I. Halupczok} [Confluentes Math. 3, No. 4, 587--615 (2011; Zbl 1246.03059)]. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Hölder continuity of functions definable on closed bounded subsets of \(K^n\), the existence of definable retractions onto closed definable subsets of \(K^n\) and a definable, non-Archimedean version of the Tietze-Urysohn extension theorem. In a recent paper, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with several applications. Non-Archimedean valued fields, Quantifier elimination, model completeness, and related topics, Henselian rings, Schemes and morphisms, Singularities in algebraic geometry, Other nonalgebraically closed ground fields in algebraic geometry, Semialgebraic sets and related spaces A closedness theorem and applications in geometry of rational points over Henselian valued 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 Serre's conjecture \((3.2.4_ ?)\) claims that a continuous, irreducible and odd representation \(\rho: G_ \mathbb{Q}:= \text{Gal} (\overline {\mathbb{Q}}/\mathbb{Q})\to \text{GL}_ 2 (\overline {\mathbb{F}}_ p)\) with invariants (as defined by Serre) \((N, k, \varepsilon)\) comes from a Hecke cusp form \(\pmod p\) of type \((N, k, \varepsilon)\). Here, inspired by a numerical experiment, the notions of minimal representation \(\rho(m)\) for \(\rho\) and of companion representations \(\rho (n_ 1)\) and \(\rho (n_ 2)\) for \(\rho\) are introduced. It is shown that (i) if \(\rho(m)\) satisfies Serre's conjecture \((3.2.4_ ?)\) then so does \(\rho\), and (ii) if \(\rho\) admits companion representations \(\rho(n_ 1)\) and \(\rho(n_ 2)\) then \(\rho(n_ 1)\) verifies Serre's conjecture if and only if \(\rho(n_ 2)\) does.
This result is applied to give a (new) proof of Serre's conjecture \((3.2.4_ ?)\) for the irreducible representation \(\rho: G_ \mathbb{Q}\to \Aut (E_ p) \simeq \text{GL}_ 2 (\overline {\mathbb{F}}_ p)\) provided that \(E\) is a vertical Weil curve at the prime \(p>7\). This last result was also obtained in joint work with \textit{P. Bayer} [Compos. Math. 84, 71-84 (1992; Zbl 0770.11029)]. Galois representation; Serre's conjecture; Hecke cusp form; minimal representation; companion representations; Weil curve Holomorphic modular forms of integral weight, Elliptic curves over global fields, Elliptic curves, Representation-theoretic methods; automorphic representations over local and global fields On Serre's conjecture (3.2.4\(_ ?\)) and vertical Weil 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 book is at the same time an introduction, addressed to sophisticated mathematicians, to a purely algebro-geometric theory of \(D\)-modules, and a summary of the main results, most of which due to the author, in that field. The necessary and sufficient prerequisite requested of the reader, aside from basic experience with algebraic and analytic sheaves, is a good knowledge of homological algebra, and familiarity with the formalism and basic properties of derived categories. Relying on this background, the proofs are complete, with two exceptions: Kashiwara's constructibility theorem and the theorem of faithful flatness of the ring of differential operators of infinite order over the ring of differential operators (due to Sato-Kawai-Kashiwara). The original micro-differential proofs of those results, did not in fact fit into the spirit of this book: proofs based purely on the algebro-geometric formalism of \(D\)-modules were later provided by the author, but only the first could be included, as a final note (with \textit{L. Narváez-Macarro}) in the present book.
According to Grothendieck's philosophy, \(D\)-modules are regarded in this book as general coefficients for the cohomology of smooth analytic or algebraic varieties over the complex numbers. (A Monsky-Washnitzer type theory of \(D\)-modules is, however, also in the author's mind, as well as a generalization to singular varieties.) On the model of the theory of quasi-coherent coefficients over a scheme, developed in \textit{R. Hartshorne's} book ``Residues and duality'' [Berlin etc.: Springer-Verlag (1966; Zbl 0212.26101) and of the theory of discrete coefficients contained in \(SGA 4 and 5\) (SGA \(=\) Sémin. Géom. Algébr.), the author establishes in the first chapter of this book a complete formalism for the present type of coefficients including ``les six opérations de Grothendieck'' for a smooth morphism \(f: X\to Y,\) and Grothendieck's local and global duality. In particular, the global duality results here obtained include as special cases the most general known formulations of Serre and Poincaré dualities. While the author makes a point of treating the algebraic and analytic situations in parallel, some asymmetries still remain: only the hypothesis of existence of a good global filtration allows one to prove the coherence of the direct image via a proper morphism of a coherent \(D\)-module in the analytic case (see section \(I\quad 5.4).\) In the algebraic case that hypothesis is automatically verified.
The main result of the second chapter is the interpretation of discrete constructible coefficients in terms of regular holonomic \(D\)-module coefficients. This is ``Mebkhout's equivalence'' between the derived category of complexes of \(D\)-modules with bounded regular-holonomic cohomology and the derived category of complexes of sheaves of vector spaces with bounded constructible cohomology. This equivalence, obtained via the (contravariant) solution functor, and the dual equivalence via the (covariant) de Rham functor, represent the widest known generalization of the so-called Riemann-Hilbert correspondence; they are compatible with Grothendieck's operations and are interchanged by duality (in any of the two categories). The previous statements hold in both the analytic and the algebraic case; their proofs depend strongly on Hironaka's resolution of singularities, since they rely on Deligne's work [cf. \textit{P. Deligne}, ``Équations différentielles à points singuliers réguliers'', Lect. Notes Math. 163. Berlin etc.: Springer Verlag (1970; Zbl 0244.14004)] on connections with regular singularities. All of the known comparison theorems between algebraic and complex-analytic cohomology follow here from more general statements of equivalence between several seemingly independent definitions of regularity (the author \textit{proves} that the structural sheaf is a regular \(D\)-module!), together with their behaviour under direct images. Here, relevant tools are the local algebraic cohomology of analytic \(D\)-modules, and the functor of their solutions in formal functions along a closed analytic subspace. The author has recently been able to free all of the previous results of dependence upon Hironaka's resolution of singularities (see the author's article in [Publ. Math., Inst. Hautes Étud. Sci. 69, 47--89 (1989; Zbl 0709.14015)], opening the way to an extension of the theory to schemes in positive characteristic: it is clear that, if a new systematic treatment of the theory of D-modules were to be written, this new approach by the author should be followed.
These main results are complemented by a variety of topics: the problem of Cauchy-Kowalewski in the \(D\)-module setting, the study of \(D\)-modules over a 1-dimensional complex disk, the characterization of perverse sheaves and finally the theory of vanishing cycles (in the sense \(of SGA 7)\). Considerable attention is devoted to this last topic in chapter three, where the author and \textit{C. Sabbah}, show how to recover the properties of the \(V\)-filtration of Fuchs-Malgrange-Kashiwara, from the general results on \(D\)-modules obtained in the first two chapters.
We found this book extremely interesting, rich and pleasant, and recommend its reading to any mathematician possessing the necessary background. local duality; D-modules; derived categories; global duality; Mebkhout's equivalence; Riemann-Hilbert correspondence; local algebraic cohomology of analytic D-modules; problem of Cauchy-Kowalewski; perverse sheaves; vanishing cycles Mebkhout, Z., Le formalisme des six opérations de Grothendieck pour les DX-modules cohérents, Travaux en Cours, vol. 35, (1989), Hermann: Hermann Paris, With supplementary material by the author and L. Narváez Macarro Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry, Sheaves of differential operators and their modules, \(D\)-modules, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Duality theorems for analytic spaces Differential systems. The formalism of the six Grothendieck operations for coherent \(D_X\)-modules) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 field \(k\), \(X\) an integral \(k\)-variety and \(p : X \to C\) a dominant \(k\)- morphism. The paper gives a formula for the relative Chow group \(\text{CH}_0 (X/C) = \ker (p_* : \text{CH}_0 (X) \to \text{CH}_0 (C))\) under the hypotheses that for each closed point \(P\) of \(C\), every zero-cycle of degree zero on the fibre \(X_P\) is rationally equivalent to 0 and the group \(p_* \text{CH}_0 (X_P)\) is generated by the image of a zero-cycle on \(X_P\) supported on the regular locus of \(X\).
For any variety \(Y\) over a field \(F\), let \(N_Y (F)\) be defined as the subgroup of \(F^*\) generated by all norms \(N_{L/F} (L^*)\) for all finite extensions \(L\) of \(F\) such that \(Y(L) \neq \emptyset\). Define \(k(C)^*_{dn}\) to be the group of nonzero rational functions on \(C\) which at any point of \(P \in C\) can be written as a product of a unit at \(P\) and an element of \(N_{X_\eta} (k(C))\). Then
\[
\text{CH}_0 (X/C) \simeq k (C)^*_{dn}/k [C]^* \cdot N_{X_\eta} \bigl( k(C) \bigr).
\]
This theorem is then applied to the case of admissible quadric bundles. These are morphisms \(p:X\to C\) which are proper, surjective, with generic fibre \(X_\eta\) a smooth quadric hypersurface, whose localisation at any closed point \(P\) of \(C\) is isomorphic to a projective scheme over the local ring \(A_p\) at \(P\), given by a homogeneous equation \(\sum^n_{i = 1} a_i X^2_i = 0\) where the valuations \(v_p (a_i) \in \{0,1\}\) and \(v_p (a_i) = 0\) for \(i \leq (n + 1)/2\). If \(n \geq 3\) and \(k\) is a field of cohomological dimension one, then \(\text{CH}_0 (X/C) = 0\). If the relative dimension is two, the study of \(\text{CH}_0 (X/C)\) is related to the study of \(\text{CH}_0 (Y/ \widetilde C)\) for an associated conic bundle \(Y \to \widetilde C\) over a curve \(\widetilde C\) which is a double cover of \(C\) associated with the discriminant of the quadric bundle. When \(\widetilde C\) is geometrically integral, \(\text{CH}_0 (X/C)\) injects into \(\text{CH}_0 (Y/ \widetilde C)\). In the case where \(k\) is a number field or a local field, this implies that CH\(_0 (X/C)\) is finite. dominant morphism; integral curve; relative Chow group; zero-cycle; admissible quadric bundles J.-L. Colliot-Thélène, A. Skorobogatov, Groupe de Chow des zéro-cycles sur les fibrés en quadriques. K-Theory 7(5), 477-500 (1993) Parametrization (Chow and Hilbert schemes), Algebraic cycles, Arithmetic ground fields for surfaces or higher-dimensional varieties, Quadratic forms over global rings and fields Chow groups of zero cycles on pencils of quadrics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 developed in this paper arose from two main sources. They are the theory of varieties of group representations developed recently by \textit{M. Culler} and the second author [ibid. 117, 109-146 (1983; Zbl 0529.57005)] and Thurston's construction of a compactification of Teichmüller space. As an application of their ideas, the authors give a new construction of this compactification. As they state, their ''methods are drawn from the mathematical mainstream, and therefore help to explain Thurston's results by putting them in a wider framework.''
The central topic of the paper is a construction of compactifications of real and complex algebraic varieties. While there is an obvious way to compactify curves, which was used by Culler-Shalen, the problem of compactification of higher dimensional varieties is anything but routine. The authors' approach to this problem is motivated by the construction of Thurston's compactification of Teichmüller space. They consider an affine algebraic set V and an indexed family \((f_ j)_{j\in J}={\mathcal F}\) with countable index set J of functions which belong to the coordinate ring of V and generate it as an algebra. A compactification of V is canonically defined by \({\mathcal F}\) as follows. Let \({\mathcal P}\) be the quotient of \([0,\infty)^ J\setminus \{0\}\), where \([0,\infty)^ J\) is the Cartesian power, by the diagonal action of positive reals: \(\alpha (t_ j)_{j\in J}=(\alpha t_ j)_{j\in J}.\) Define a map \(\theta\) : \(V\to {\mathcal P}\) by \(\theta (x)=[\log (| f_ j(x)| +2)]_{j\in J}.\) Then the closure of \(\theta\) (V) in \({\mathcal P}\) is compact. This closure is the compactification in question. This compactification is studied on three different levels of generality.
The first one is that of a general variety. This is the theme of Chapter I. The points added to V are interpreted as valuations of the coordinate ring of V over a countable field of definition of V. These valuations are neither discrete nor of rank 1 in general. An important result says that there is a dense subset of added points consisting of discrete, rank 1 valuations.
At the second level V specializes to be the variety of characters X(\(\Gamma)\) of representations of a discrete group \(\Gamma\) in \(SL_ 2({\mathbb{C}})\). In this case there is a natural choice of \({\mathcal F}\). The corresponding \(f_ j\) are the values of characters on conjugacy classes in \(\Gamma\). Now the added points can be interpreted as actions of \(\Gamma\) on some generalized trees. On the vertices of an ordinary tree there is an integer-valued distance function. On generalized trees a similar distance function takes values in an ordered abelian group. The most important case is that of a subgroup of \({\mathbb{R}}\). The theory of such trees is developed from scratch up to a generalization of the well known Bass-Serre theory of trees associated to \(SL_ 2(F)\), where F is a field. While in the Bass-Serre theory a tree is associated with a discrete valuation of F, here the valuation can be nondiscrete. These trees are used for compactification. All this is the theme of Chapter II.
Finally, in Chapter III this theory is applied to the case \(\Gamma =\pi_ 1(S)\), where S is a surface. The Teichmüller space of S turns out to be a component of X(\(\Gamma)\) and the compactification of X(\(\Gamma)\) constructed in Chapter II leads to the Thurston's compactification of Teichmüller space.
In the second part of this paper, existing now in preprint form, these ideas are applied to the study of 3-manifolds. In particular, another important result of Thurston is re-proved from an entirely new point of view. representations of a dicrete group in \(SL_ 2({\mathbb{C}})\); actions on generalized trees; hyperbolic structures on surfaces; varieties of group representations; compactification of Teichmüller space; compactifications of real and complex algebraic varieties; affine algebraic set; valuations of the coordinate ring J. Morgan, P. Shalen. Valuations, trees, and degenerations of hyperbolic structures. I, \textit{Ann. of Math. } 120 (1984), 401--476. General low-dimensional topology, Abelian varieties and schemes, Valuations and their generalizations for commutative rings, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Compactification of analytic spaces, Group rings of finite groups and their modules (group-theoretic aspects), Classification theory of Riemann surfaces, Topology of Euclidean 2-space, 2-manifolds, Topology of general 3-manifolds Valuations, trees, and degenerations of hyperbolic structures. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be an algebraic group acting linearly on a projective space \(\mathbb{P}(V)\) and \(X\subseteq \mathbb{P}(V)\) a \(G\)--invariant subvariety. A groupoid--theoretic approach to compute invariants is proposed by slicing the groupoid \(G\times X\rightrightarrows X\) by a subvariety \(W\subset X\) which is suitably transverse to the generic orbit to produce a groupoid \(R|_W\rightrightarrows W\) where it is easier to compute invariants. This idea is illustrated giving generalizations of the classical Gelfand-MacPherson correspondence and Gale transform. The slicing technique is used to give Zariski--local descriptions of the moduli space of \(n\) order points in \(\mathbb{P}^1\). Finally a global description of the Kontsevich moduli space of stable maps \(M_0(\mathbb{P}^1, 2)\) (the coarse moduli space parametrizing non--constant, degree 2 morphisms \(\mathbb{P}^1\to \mathbb{P}^1\) modulo automorphisms of the source) is given and its GIT compactification. Invariant theory; groupoids; Artin stacks J. Alper, Computing invariants via slicing groupoids: Gelfand--MacPherson, Gale and positive characteristic stable maps, Math. Nachr. (in press) eprint arXiv:1011.3448~[math.AG]. Actions of groups on commutative rings; invariant theory, Stacks and moduli problems Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 studies the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension.
As an application of this study, the effectiveness of the height jump divisors for families of pointed abelian varieties is proved. The effectiveness of the height jump divisor was conjectured in the more general case of variations of polarized Hodge structures of weight \(-1\) by \textit{R. Hain} [in: Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press. 527--578 (2015; Zbl 1322.14049)].
After studying the period map associated to a family of pointed polarized abelian varieties, a local expansion is given for the metric on the pullback of the Poincaré bundle under this period map. Then normlike functions (which are the functions that appear as the logarithm of the norm of a section of the pullback of the Poincaré bundle) are studied. Several estimates on their growth and that of their derivatives i are given. Then the main results on local integrability and positivity of the height jump are established. singularity; invariant metric; Poincare bundle; abelian variety; Hodge structure; period map; height jump divisor Families, moduli of curves (algebraic), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Variation of Hodge structures (algebro-geometric aspects) Singularities of the biextension metric for 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 [For the entire collection see Zbl 0528.00004.]
This is mainly an expository article of previous results by Ph. Griffiths and the author. Poincaré had introduced the notion of normal function associated to a ''sufficiently general'' curve C of an algebraic surface. An important fact was that viceversa given a normal function the homology class of C can be reconstructed. Griffiths has generalized Poincaré methods to an algebraic manifold X of any dimension, getting for every k a group \(N^ k(X)\) of ''normal functions'' and an associated group of integral cohomology classes of degree 2k, clearly related to the group of algebraic cycles of X of codimension k. Precisely one of the main results of this theory [cf. \textit{P. A. Griffiths}, Am. J. Math. 101, 94-131 (1979; Zbl 0453.14001), and the author, Invent. Math. 33, 185-222 (1976; Zbl 0329.14008) and Ann. Math., II. Ser. 109, 415-476 (1979; Zbl 0446.14002)] is the following: consider the normal function associated to a smooth projective morphism \(f: X\to S\) where S is a non singular algebraic curve and the ''analogous'' notion for the completions: \(\bar f:\) \(\bar X\to \bar S\). There is a subgroup of \(N^ k(\bar X)\) (defined by a differential equation), the group of horizontal normal functions, whose cohomology classes are exactly those of type (k,k) in \(H^{2k}(\bar X,{\mathbb{Z}})\). The precise statement requires several definitions. Relations with the Hodge conjecture are obvious. group of algebraic cycles of codimension k; intermediate Jacobians; normal function; horizontal normal functions; Hodge conjecture S. Zucker, Intermediate Jacobians and normal functions , Topics in Transcendental Algebraic Geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton University Press, Princeton, NJ, 1984, pp. 259-267. Transcendental methods, Hodge theory (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Transcendental methods of algebraic geometry (complex-analytic aspects), Picard schemes, higher Jacobians Intermediate Jacobians and normal 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 \(E\) be a reflexive sheaf of rank \(r\) on a normal projective surface \(X\). The first Chern class of \(E\) is \(c_1(E) = (\wedge ^rE)^{**} \in\) Weil \(X\). The author defines the second Chern class of \(E\) by using a generalization of Wahl's relative local Chern classes of vector bundles on a desingularisation of \(X\). The author proves a Riemann-Roch type formula for \(E\) and shows that the terms in this formula which come from singularities of the sheaf are bounded. Using this, the inequality in Wahl's conjecture on the relative asymptotic Riemann-Roch formula for rank \(2\) vector bundles is proved. It is shown that local orbifold Chern classes and Wahl's local Chern classes coincide at quotient singularities. In particular, Wahl's conjecture holds at quotient singularities. Finally, the Wahl conjecture is proved for quotients of cones over elliptic curves. reflexive sheaves; normal surfaces; Riemann-Roch formula; Chern class Langer, A., \textit{Chern classes of reflexive sheaves on normal surfaces}, Math. Z., 235, 591-614, (2000) Riemann-Roch theorems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Chern classes of reflexive sheaves on normal 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 authors generalize \textit{A. A. Rojtman}'s theorem [Ann. Math., II. Ser. 111, 553-570 (1980; Zbl 0504.14006)] which states that the Abel-Jacobi map induces an isomorphism of the torsion subgroups of the Chow group \(A_0(X)\) and the Albanese variety \(\text{Alb}(X)\), in the case of smooth complex projective surfaces. This result was already generalized by \textit{A. Collino} [J. Pure Appl. Algebra 34, 147-153 (1984; Zbl 0589.14007)] and \textit{M. Levine} [Am. J. Math. 107, 737-757 (1985; Zbl 0579.14007)] to normal surfaces. In this work, the result is proved for any singular complex projective surface but some modifications are made. First, the authors define a modified version of \(A_0(X)\), which can be interpreted by means of algebraic \(K\)-theory, and replace \(\text{Alb} (X)\) by Griffith's intermediate Jacobian \(J^2(X)\) which is defined with the help of Hodge theory. A generalization of Abel-Jacobi map gives an isomorphism between their torsion subgroups. As a consequence, the authors deduce the isomorphism of \(A_0(X)\) and \(J^2(X)\) in some cases.
In the proof the authors construct Mayer-Vietoris sequences for \(K\)-theory and Deligne cohomology in order to compute \(J^2(X)\). They apply the statement of the Collino and Levine generalization (loc. cit.) to the normalized of \(X\) to get the result. \(K\)-theory; Abel-Jacobi map; Chow group; Albanese variety; intermediate Jacobian; Hodge theory Barbieri-Viale, L.; Pedrini, C.; Weibel, C., \textit{roitman's theorem for singular complex projective surfaces}, Duke Math. J., 84, 155-190, (1996) Picard schemes, higher Jacobians, Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry Roitman's theorem for singular complex projective 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 book under review grew out of a one-semester course that the author recently taught, in 2005 and in 2006, at the Australian National University. Geared toward fourth-year undergraduate students, this course was to provide a broad, panoramic glimpse of some of the central topics in modern algebraic geometry, without striving for a thorough introduction to the field as a whole. As the guiding subject matter for such a special, topical and wide-ranging survey course in algebraic geometry, thereby taking into account the presumed basic mathematical background knowledge of undergraduate students in their final year, the author choose the close relations between algebraic geometry and complex-analytic geometry as reflected by \textit{J.-P.Serre's} famous GAGA principle. Serre's pioneering paper ``Géométrie algébrique et géométrie analytique'' from more than fifty years ago [Ann Inst. Fourier 6, 1--42 (1955/56; Zbl 0075.30401)], now usually referred to as ``GAGA'', was not only a decisive milestone in the refoundation'of algebraic geometry by himself and A. Grothendieck, but also established crucial comparison theorems relating algebro-geometric objects (varieties, regular morphisms and algebraic sheaves) with complex-analytic objects (analytic spaces, holomorphic mappings and analytic sheaves) within the modern sheaf-theoretic framework.
In fact, a detailed explanation of Serre's cohomological GAGA theorem for coherent algebraic sheaves on projective spaces constitutes both the punchline and the ultimate goal of the present, fully elaborated and significantly extended course notes. Along the way, the author provides an introduction to those concepts, methods and techniques of modern algebraic geometry and complex analysis that are necessary to reach this goal. This approach to giving a sufficiently substantial, streamlined, accessible and inspiring introduction to some important aspects of modern algebraic geometry for beginners is not only highly original, and even fairly unique in the relevant textbook literature, but also didactically interesting and exemplary, all the more as a thorough discussion of Serre's GAGA principle is still sadly neglected in most primers of algebraic geometry, despite its fundamental importance and wide range of applications.
In view of these particular facts, the current book must be seen as out-standing, utmost valuable enhancement of the modern textbook literature in advanced undergraduate algebraic and analytic geometry, and the author is due to highest appreciation for having provided such a complementary, quite alternative introduction to the subject, very much to the benefit of interested students and instructors likewise.
Starting from scratch and throughout trying to keep the involved abstract machinery of modern algebraic geometry at a level digestible for beginners, the author has organized the material in ten chapters, each of which is subdivided into up to ten sections.
Chapter 1 introduces some of the basic objects of study and illustrates a few motivating examples. The reader gets here acquainted with algebraic and analytic subspaces of \(\mathbb{C}^n\), with the statement of Chow's theorem, and with the analytic concept of elliptic curves.
Chapter 2 is to prepare the ground for the introduction of algebraic schemes from the more familiar analytic viewpoint. This is done by discussing differentiable manifolds, first in the traditional way and then via sheaves (of functions) and ringed spaces.
Chapter 3 is devoted to affine algebraic schemes, together with their allied commutative algebra and topology, and, in the sequel, to general schemes and their morphisms. Ringed spaces over \(\mathbb{C}\) and schemes of finite type over \(\mathbb{C}\) are treated at the end of this chapter.
The comparison between the Zariski topology of a scheme locally of finite type over \(\mathbb{C}\) on the one hand, and the complex topology of its underlying analytic space (of closed points), on the other hand, is the subject of Chapter 4.
This is followed by developing the analytic theory of schemes over \(\mathbb{C}\) in the subsequent Chapter 5, where rings and sheaves of analytic functions are appropriately explained and predominantly utilized. The constructions given here are of local character, but are then globalized in the more sketchy Chapter 6 for the keen, particularly interested reader, mainly by outlining the basic facts on Fréchet spaces and a coordinate-free approach to polydiscs.
Chapter 7 turns to the crucial concept of coherent sheaves in both the algebraic and the analytic context. According to its significance for the main theme of the entire book, Serre's cohomological GAGA theorem, this chapter is worked out in greater detail, culminating in the description of the analytification of coherent algebraic sheaves and finally, in the statement of Serre's GAGA theorem, in its weak versions.
Chapter 8 and Chapter 9 provide the essential concepts and results regarding the algebraic and analytic geometry of projective spaces \(\mathbb{P}^n_{\mathbb{C}}\) and their quotients with respect to the action of affine group schemes. The sheaf-theoretic main results are explained and illustrated in Chapter 8, while their proofs, including the involved methods from invariant theory, are thoroughly carried out in Chapter 9.
The final Chapter 10, the highlight of the book, is devoted to the (almost) complete proof of Serre's GAGA theorem for projective spaces \(\mathbb{P}^n_{\mathbb{C}}\) in its following version: The analytification functor, which takes a coherent algebraic sheaf on \(\mathbb{P}^n_{\mathbb{C}}\) to its associated coherent analytic sheaf, is an equivalence of categories.
In addition to the original course notes, the present book also contains the proof of the cohomological variant of Serre's GAGA theorem, which states that a coherent algebraic sheaf on \(\mathbb{P}^n_{\mathbb{C}}\) and its analytification have canonically isomorphic \(i\)-th cohomology groups for any integer \(i\in \mathbb{Z}\).
Also, in an appendix, the predominantly technical proofs concerning some analytification results, as used in Chapters 5--7, are given as a supplement for the inquisitive reader.
All in all, the book under review is a masterpiece of expository writing in modern algebraic geometry. It is exactly what the author promised: no comprehensive text to train future algebraic geometers, but rather an attempt to convince students of the fascinating beauty, the tremendous power, and the high value of the methods of algebraic and analytic geometry.
The author has reached his declared goal in an admirable, truly brilliant manner. His utmost lucid exposition of these modern, fairly advanced topics in the field for beginners breathes his passion for the subject and for grippingly teaching it, and his style of writing bespeakes a good sympathetic understanding towards students and their needs. Numerous synoptic sections, asides for the expert reader, and additional remarks help the student grasp the vast material thoroughly, and both the careful glossary and the detailed index facilitate the active working with this highly inspiring book. Also, the author makes concrete, instructive proposals for how to use his text for different variants of self-study or teaching, which is particularly useful as its current extended contents now offer by far more than can be covered in a one-semester course.
It remains to be wished that this very special introduction to the realm of (complex) algebraic geometry find the deserved wide-spread interest within the mathematical community. algebraic schemes; complex manifolds; sheaves; transcendental methods of algebraic geometry; projective geometry; invariant theory; GAGA-type theorems; sheaf cohomology Neeman, A.: Algebraic and Analytic Geometry. Cambridge University Press, Cambridge (2007) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Schemes and morphisms, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Analytic sheaves and cohomology groups, Actions of groups on commutative rings; invariant theory Algebraic and 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 For two monic polynomials depending on a single variable, the following identity holds: The product of the values of the first polynomial at the roots of the second polynomial is equal to the product of the values of the second polynomial at the roots of the first one apart from the sign. -- André Weil found a far-reaching generalization of this identity. It can be used for any pair of non-zero meromorphic functions on a compact complex curve.
We present definitions necessary for the formulation of Weil's theorem. Let
\[
f=c_1u^{b_1}+\cdots, \quad g=c_2u^{b_2} +\cdots,\tag{*}
\]
where \(c_1\neq 0\) and \(c_2\neq 0\) are the highest-order terms of the Laurent series of meromorphic functions \(f\) and \(g\) in a neighbourhood of a point \(a\), and \(u\) is a local parameter such that \(u(a)=0\).
For a vector-function \((f,g)\), we call a non-cancellable integer-valued vector \(\vec n=(n_1,n_2)\) proportional to its exponent vector \(\vec b=(b_1,b_2)\) with a positive integer coefficient \(k\), \(\vec b=k\vec n\), the type of the germ at the point \(a\). The coefficient \(k\) is called the multiplicity of the germ of the vector-function. We call the number \(c^{n_1}_2c_1^{-n_2}\) the reduced Weil number of the germ (*), where \((n_1,n_2)\) are the components of the type \(\vec n\) of this germ. -- Starting from a compact complex curve \(\Gamma\) and a meromorphic function \((f, g)\) on it, we define a function \(\text{Mul}_{\Gamma fg}\) mapping the product \(\mathbb{Z}^2_{\text{ir}}\times\mathbb{C}^*\), where \(\mathbb{Z}^2_{\text{ir}}\) is a set of non-cancellable integer-valued vectors on the plane, to the set \(\mathbb{C}^*\) of non-zero complex numbers. The function \(\text{Mul= Mul}_{\Gamma fg}\) has non-negative integer values and is equal to zero everywhere except at a finite number of points. By definition, its value on a pair \((\vec n,c)\) is equal to the total multiplicity of points on the complex curve \(\Gamma\) at which the germ \((f,g)\) has the type \(\vec n\) and the reduced Weil number \(c\). -- In these terms, Weil's theorem is formulated as follows:
\[
\prod(-c)^{\text{Mul}(\vec n,c)}=1.\tag{1}
\]
The degrees of the divisors \(f\) and \(g\) vanish on a compact curve \(\Gamma\). In terms of the function \(\text{Mul=Mul}_{\Gamma fg}\), these relations are of the form
\[
\sum \text{Mul} (\vec n,c)\vec n=0. \tag{2}
\]
The present results are based on the following simple observation: The Weil numbers of the germ \((f,g)\) and of the germ \((F,G)\), where \(F=f^{a_{11}}g^{a_{12}}\), \(G=f^{a_{21}}g^{a_{22}}\) and \(A=\{a_{ij}\}\) is an integer-valued matrix with determinant 1, are equal to one another. This observation suggests that Weil's theorem must be related to the two-dimensional torus geometry (the matrix \(A\) specifies an automorphism of the two-dimensional torus \(\mathbb{C}^{*2})\) and to the Newton polygon theory. We show that this is indeed so. First, the Weil numbers simplify and refine the classical theorem on Newton polygons (see \S 2). On the other hand, by using Newton polygons, one can offer a very simple proof of Weil's theorem: It is reduced to the Vieta formula for the product of roots of a polynomial (see \S\S 1-2 and \S 9).
Is Weil's theorem invertible? That is, is it true that for any function Mul satisfying conditions (1)--(2), there exists a triple \(\Gamma,f,g\) such that \(\text{Mul=Mul}_{\Gamma fg}\)? We offer a complete description of triples \(\Gamma,f,g\) such that \(\text{Mul=Mul}_{\Gamma fg}\).
With a triple \(\Gamma,f,g\) we associate a polygon called the Newton polygon. This polygon can be recovered from the function \(\text{Mul= Mul}_{\Gamma fg}\). We show that it plays the same role as the usual Newton polygon (see \S 11).
Another theme of our paper is the following one. Let \(D\) be a divisor lying on the union \(M_\infty\) of one-dimensional orbits of a torus surface \(M\). Is there a curve \(\Gamma\subset M\) that does not cross null-dimensional orbits of the surface \(M\) for which the divisor \(D\) is a divisor of the intersection of the curve \(\Gamma\) with the union of one-dimensional orbits \(M_\infty\)? We denote the space of such curves \(\Gamma\) by \({\mathcal R}(D)\), and call the divisor \(D\) admissible if the space \({\mathcal R}(D)\) is not empty. We encode divisors \(D\) with some function \(\text{Mul=Mul}_D\) on \(\mathbb{Z}^2_{\text{ir}}\times \mathbb{C}^*\). The admissibility condition for the divisor \(D\) proves to be identical to the condition for the existence of a triple \(\Gamma,f,g\) such that \(\text{Mul=Mul}_{\Gamma fg}\). In \S\S 6-7 a general curve in the space \({\mathcal R}(D)\) is described. The admissible divisor \(D\) is related to a polygon \(\Delta_D\). curves on torus surfaces; admissible divisor; multiplicity of the germ of the vector-function; reduced Weil number; total multiplicity of points; torus geometry; roots of a polynomial; Newton polygon Khovanskii, AG, Newton polygons, curves on torus surfaces, and the converse Weil theorem, Russ. Math. Surv., 52, 1251, (1997) Toric varieties, Newton polyhedra, Okounkov bodies, Polynomials in real and complex fields: location of zeros (algebraic theorems) Newton polygons, curves on torus surfaces, and the converse of Weil's theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem of compactifying the (generalized) Jacobian of a singular curve has been studied since \textit{J. Igusa} [Am. J. Math. 78, 171--199, 745--760 (1956; Zbl 0074.15803)]. He constructed a compactification of the Jacobian of a nodal and irreducible curve \(X\) as limit of the Jacobians of smooth curves approaching \(X.\) Igusa also showed that his compactification does not depend on the considered family of smooth curves. When the curve \(X\) is reducible and nodal, \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1--90 (1979; Zbl 0418.14019)] produced a family of compactified Jacobians \(\text{Jac}_{\phi}\) parameterized by an element \(\phi\) of a real vector space. \textit{C. S. Seshadri} [``Fibrés vectoriels sur les courbes algébriques'', Astérisque 96 (1982; Zbl 0517.14008)] dealt with the general case of a reduced curve considering sheaves of higher rank as well. \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] showed how to compactify the relative Jacobian over the moduli of stable curves and described the boundary points of the compactified Jacobian of a stable curve \(X\) as invertible sheaves on certain Deligne-Mumford semistable curves that have \(X\) as a stable model. Most of the above papers are devoted to the construction of the compactified Jacobian of a curve, not to describe it.
In the paper under review, the author gives an explicit description of the structure of these Simpson schemes, \(\text{Jac}^{d}(X)_{s},\) and \(\overline{\text{Jac}^{d}(X)}\) of any degree \(d,\) for \(X\) a polarized curve of the following types: tree-like curves and all reduced and reducible curves that can appear as singular fibers of an elliptic fibration. moduli of stable curves; Deligne-Mumford semistable curves; Simpson scheme López~Martín, A.C., Simpson Jacobians of reducible curves, J. reine angew. math., 582, 1-39, (2005) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Vector bundles on curves and their moduli, Picard groups, Schemes and morphisms, Families, moduli of curves (analytic) Simpson Jacobians of reducible 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 Isogeny classes of abelian varieties of the finite field with \(q\) elements are parametrized by the characteristic polynomial of Frobenius, which is a polynomial over the integers, called a \(q\)-Weil polynomial. Conversely, given such a polynomial, it is a natural problem to calculate the number of principally polarized abelian varieties contained in the corresponding isogeny class.
The present paper computes this number for a special class of \(q\)-Weil polynomials in the case of abelian surfaces. The idea is to, define for every prime and for \(\infty\), a local factor which measures its relative frequency and show that the number in question is given by the (infinite) product of these factors. This was inspired by a result of \textit{E.-U. Gekeler} who proved in [Int. Math. Res. Not. 2003, No. 37, 1999--2018 (2003; Zbl 1104.11033)] such a formula in the case of elliptic curves. abelian surfaces; over finite fields; random matrices Isogeny Local heuristics and an exact formula for abelian surfaces over 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 The fundamental theorems of classical Brill-Noether theory of smooth projective curves have not been extended yet to stable curves, due to the technical difficulties of combinatorial nature. The goal of this paper is to extend some of them to binary curves. A binary curve is a stable nodal curve having two irreducible rational components, intersecting at \(g+1\) points. Their moduli space \(B_g\subset \overline{M_g}\) is irreducible of dimension \(2g-4\). The analogue of theorems of Riemann, Clifford and Martens are proved to hold for any binary curve and for line bundles parametrized by the compactified Jacobian scheme. An analogue of Brill-Noether theorem is proved for general binary curves and for \(r\leq 2\). More precisely, let
\[
B_d(g)=\{(d_1,d_2) \mid d_1+d_2=d, \frac{d-g-1}{2}\leq d_i\leq \frac{d-g+1}{2}, i=1,2\}
\]
be the set of balanced multidegrees.
If \(\underline{d}\in B_d(g)\), put \(W^r_{\underline{d}}(X)= \{L\in Pic^{\underline{d}}(X)\mid h^o(L)\geq r+1\}\), where \(X\) is a binary curve. Then it is proved that, for a general binary curve \(X\) and for \(r\leq 2\), \(\dim W^r_{\underline{d}}(X)\leq \rho_d^r(g)\), the usual Brill-Noether number, and equality holds for some \(\underline{d}\). Moreover \(\dim \overline{W^r_{\underline{d}}(X)}= \rho_d^r(g)\). stable curve; moduli space; Clifford theorem; Picard scheme; Brill-Noether Caporaso L.: Brill-Noether theory of binary curves. Math. Res. Lett. 17(2), 243--262 (2010) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) Brill-Noether theory of binary 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 We use tools of mathematical logic to analyse the notion of a path on a complex algebraic variety, and are led to formulate a ``rigidity'' property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to étale topology. These appear as criteria for \(\aleph_{1}\)-categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space.
Technically, for a variety \(\mathbf A\) defined over a finite field extension of the field \(\mathbb Q\) of rational numbers, we introduce a countable language \(L(A)\) describing the universal covering space of \(\mathbf A(\mathbb C)\), or, equivalently, homotopy classes of paths in \(\pi_1(\mathbf A(\mathbb C))\). Under some assumptions on \(\mathbf A\) we show that the universal covering space of \(\mathbf A(\mathbb C)\) is an analytic Zariski structure [\textit{B. Zilber}, Zariski geometries. Geometry from the logician's point of view. Cambridge: Cambridge University Press (2010; Zbl 1190.03034)], and present an \(L_{\omega _{1}\omega }(L(A))\)-sentence axiomatising the class containing the structure and that is stable and homogeneous over elementary submodels. The ``rigidity'' condition on fundamental groups says that projection of the fundamental group of a variety is the fundamental group of the projection, up to finite index and under some irreducibility assumptions, and is used to prove that the projection of an irreducible closed set is closed in the analytic Zariski structure.
In particular, we define an analytic Zariski structure on the universal covering space of an abelian variety defined over a finite extension of the field \(\mathbb Q\) of rational numbers. categoricity; fundamental group; analytic spaces; complex algebraic variety; bona fide topology; stability; homotopy classes of paths; universal covering space; analytic Zariski structure M. Gavrilovich, Covers of algebraic varieties, preprint Applications of model theory, Categoricity and completeness of theories, Classification theory, stability, and related concepts in model theory, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Topological aspects of complex manifolds Covers of abelian varieties as analytic Zariski structures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The moduli space \({\mathcal M}_ g\) for curves of genus \(g\geq2\) can be constructed by using geometric invariant theory (GIT) to construct a quotient of either the Chow variety or the Hilbert scheme which parameterizes \(n\)-canonically embedded curves in \(\mathbb{P}^{(2n-1)(g-1)- 1}\) for \(n\geq3\). The quotients of the Chow variety and the Hilbert scheme are complete. Thus the closure of \({\mathcal M}_ g\) in the quotient gives a compactifications of \({\mathcal M}_ g\). \textit{D. Mumford} [Enseign. Math., II. Sér. 23, 39-100 (1977; Zbl 0363.14003){]} has shown using the Chow variety and \textit{D. Gieseker} [``Lectures on moduli of curves'', Tata Inst. Fund. Res. (1982; Zbl 0534.14012){]} has shown using the Hilbert scheme that when \(n\geq5\) the associated compactification of \({\mathcal M}_ g\) is a moduli space for stable curves. We recall that a curve is said stable if it is reduced, connected, complete and (i) has only ordinary double points as singularities, (ii) every subcurve of genus 0 meets the rest of the curve at at least 3 points.
In the paper under review it is shown that when \(g\geq3\), the associated compactification of \({\mathcal M}_ g\) is a moduli space for a class of curves which the author calls pseudo-stable (p-stable). Precisely a reduced, connected, complete curve is said p-stable if (i) it has only ordinary double points and ordinary cusps as singularities, (ii) every subcurve of genus 1 or 0 meets the rest of the curve at at least 2 or 3 points, respectively. The method the author uses follows closely that of Mumford and Gieseker, and can be summarized as follows:
Let \(g\geq3\), \(d=6(g-1)\) and \(N=5(g-1)-1\). Let \(H\) be the Hilbert scheme parametrizing connected subcurves of \(\mathbb{P}^ N\) with Hilbert polynomial \(P(n)=dn+1-g\). Let \(Ch\) be the Chow variety parametrizing 1-cycles of degree \(d\) in \(\mathbb{P}^ N\). Let \(H_ 3\subset H\) be the subscheme of 3- canonically embedded curves of genus \(g\) in \(\mathbb{P}^ N\). Let \(Ch_ s\) be the open subset of \(Ch\) consisting of GIT-stable points under the action of \(SL(N+1)\). By GIT there exists a morphism \(\phi:H\to Ch\). Let \(U_ c=\phi^{-1}(Ch_ S)\cap\bar H_ 3\) and let \(Z\to H\) be the universal curve over H, and for \(h\in H\), let \(X_ h\) denote the fiber over \(h\) of \(Z\to H\). Again by GIT one can consider the GIT-quotient \(Q\) of \(U_ c\) by the action of \(SL(N-1)\).
Then the author proves that \(Q\) is a complete moduli space for p-stable curves by showing that: (1) if \(h\in U_ c\) then \(X_ h\) is p-stable; (2) if \(C\) is p-stable then there is an \(h\in U_ c\) such that \(X_ h\cong C\) (i.e. \(Q\) is a moduli space for p-stable curves), (3) \(Q\) is complete. compactification of the moduli space of curves; geometric invariant theory; Chow variety; Hilbert scheme; stable curves D. Schubert, ''A new compactification of the moduli space of curves,'' Compositio Math., vol. 78, iss. 3, pp. 297-313, 1991. Families, moduli of curves (algebraic), Geometric invariant theory, Compactification of analytic spaces, Group actions on varieties or schemes (quotients) A new compactification of the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper studies the Weil-étale topology which is to the usual étale topology what is the cyclic group \(\mathbb{Z}\) to its profinite completion. Namely for a scheme over a finite field \(k\) a Weil-étale sheaf on \(X\) is an étale sheaf on the base-extension to the algebraic closure \(k\), together with an isomorphism to its Frobenius-pullback. (Note that the total Frobenius is the product of the geometric and the arithmetic Frobenius, and it induces a natural equivalence on sheaves.) The properties of the corresponding cohomology are studied, also for non-torsion coefficients like the integers \(\mathbb{Z}\). This relies heavily on previous work of \textit{J. Milne} [``Étale cohomology'', Princeton Univ. Press (1980; Zbl 0433.14012); Am. J. Math. 108, 297--360 (1986; Zbl 0611.14020)]. Finally, the author states, and proves in some cases, a conjecture relating Euler-characteristics and zeta-values at \(s= 0\). étale topology; Weil-group Lichtenbaum, S., The Weil-étale topology on schemes over finite fields, Compositio Math., 141, 3, 689-702, (2005) Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The Weil-étale topology on schemes over 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 The present paper is an enlightening work on motivic stable homotopy theory. The authors study the fundamental question how the motivic stable homotopy category behaves under base change. The main theorem is that the base change along an extension of algebraically closed fields whose exponential characteristic is prime to \(\ell\) induces a full and faithful functor between the mod-\(\ell\)-motivic stable homotopy categories over the fields. Such a behavior under base change is generally known as \textit{rigidity}. \textit{A. Suslin} has shown a similar property for algebraic \(K\)-theory in his celebrated paper [Invent. Math. 73, 241--245 (1983; Zbl 0514.18008)]. Since then several generalizations and applications in other theories have been achieved. The strategy of the proof is almost always based on Suslin's ingenious idea in [loc. cit.]. Suslin showed that certain maps in algebraic \(K\)-theory induced by different rational points on a smooth projective curve agree by proving that the maps factor through the group of divisors of degree zero of the curve and using the divisibility of this group. He called this result the Rigidity Theorem in loc. cit. which turned out to be the mother of all rigidity results. The authors of the present paper also use Suslin's idea. But on their way to prove the main result, they also show several other interesting properties of the motivic stable homotopy category and the category of motivic symmetric spectra which are needed to generalize Suslin's input, but which are of independent interest as well. The second main ingredient is the construction of suitable transfer maps. The authors construct them over general base schemes in two different ways for finite étale maps and for what they call linear maps between smooth schemes. The third main point is a very interesting discussion of motivic Moore spectra and an explicit fibrant replacement functor in the mod-\(n\)-localized category of motivic symmetric spectra. The authors conclude the paper with an Appendix on Homological Localization giving an alternative foundation for a localization theory of motivic symmetric spectra. motivic homotopy theory; rigidity; transfer maps; motivic symmetric spectra; localization Röndigs, Oliver; Østvær, Paul Arne, Rigidity in motivic homotopy theory, Math. Ann.. Mathematische Annalen, 341, 651-675, (2008) Motivic cohomology; motivic homotopy theory, Stable homotopy theory, spectra, Abstract and axiomatic homotopy theory in algebraic topology Rigidity in motivic homotopy theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Krichever correspondence between algebraic curves with additional structures and certain commutative algebras of differential operators, on which the Mulase-Shiota proof of the Novikov conjecture and solution of the problem of Riemann-Schottky are based, is refined and generalized to form an equivalence between a fibered category over the category of algebraic curves and some category of commutative algebras of matrix pseudo-differential operators, with the aim of a characterization of special (e.g. d-gonal) curves by differential equations for their theta functions. A generalization and reinterpretation of the generalized Jacobian and the singularization procedure of Rosenlicht-Serre is used to show that a deformation of the curve data is reflected by the matrix KP hierarchy on the operator side. Krichever correspondence; problem of Riemann-Schottky; matrix pseudo- differential operators; theta functions; generalized Jacobian; matrix KP hierarchy Theta functions and curves; Schottky problem, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Jacobians, Prym varieties, Pseudodifferential operators, General theory of partial differential operators, Partial differential equations of mathematical physics and other areas of application, Theta functions and abelian varieties Über den Kričever-Funktor in der Theorie der algebraischen Kurven. (On the Krichever functor in the theory of 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 The paper under review is a summary of Ohmoto's minicourse delivered at the School on Real and Complex Singularities in Sao Carlos 2012. It contains important new results on Thom polynomial theory and its applications.
For a map \(f\) between complex algebraic or analytic varieties and a singularity type \(\eta\) one considers the \(\eta\)-singularity subset \(\eta(f)\) to be the collection of points of the domain where the map \(f\) has singularity \(\eta\). The classical starting point of global singularity theory is the result claiming that the cohomological fundamental class of \(\overline{\eta(f)}\) can be expressed as a multivariate polynomial depending only on the singularity, if one plugs in the Chern classes of the source and target manifolds. The polynomial is called the Thom polynomial \(tp(\eta)\) of the singularity \(\eta\).
The main addition of the present paper to global singularity theory is the extension of this result from the \textit{fundamental class} represented by \(\overline{\eta(f)}\) to the \textit{Chern-Schwartz-MacPherson (CSM) class} of \(\eta(f)\). The CSM class of a variety is an inhomogeneous deformation of its fundamental class, it is an additive invariant, it is consistent with push-forward maps, and equals the total Chern class of the tangent bundle if the variety is smooth. A version of the CSM class, called Segre-Schwartz-MacPherson (SSM) class is consistent with pullback. In Section 3 Ohmoto reviews the CSM and equivariant CSM theory (invented by himself), and in Section 4 he lays dows the foundations of the theory of SSM Thom polynomials (\(tp^{SM}(\eta)\)). A calculating method (``restriction method'') is used to calculate some terms, and an excursion to multisingularities is included.
The author's first application is the proof of several universal weighted Euler characteristics formulas for singularity loci, some of them reprove and generalize such formulas of the 19th century geometers. These results now follow not by case-by-case arguments but from the general framework of SSM-Thom polynomials.
The last two sections are devoted to another spectacular application: the conceptual treatment of the vanishing topology of finitely determined weighted homogeneous map germs. In particular, the multiplicities of stable map germs within degenerate ones are calculated, as well as the image and discriminant Milnor numbers. These are important singularity theory notions which had been studied before without Thom polynomials. Ohmoto's calculations follow from the universal formulas of SSM-Thom polynomials.
The paper is a very important contribution to global singularity theory. Thom polynomial; Chern-Schwartz-MacPherson class; Segre class; vanishing cohomology; Milnor number ; Ohmoto, School on real and complex singularities. Adv. Stud. Pure Math., 68, 191, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Singularities of maps and characteristic classes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper begins with an analysis of the divisor of a (higher order) differential on a Gorenstein curve. In particular the behaviour of such a divisor under desingularisation is investigated. Furthermore the notion of the ramification divisor for a possibly singular curve is introduced. Using this ramification divisor the authors find that the theory of Weierstrass points on a singular curve is a simple generalisation of the corresponding theory for non-singular curves. divisor of a differential; Gorenstein curve; desingularisation; ramification divisor; Weierstrass points on a singular curve De Carvalho, C. F.; Stöhr, K. -O.: Higher order differentials and Weierstrass points on Gorenstein curves. Manuscripta math. 85, 361-380 (1994) Riemann surfaces; Weierstrass points; gap sequences, Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Divisors, linear systems, invertible sheaves Higher order differentials and Weierstrass points on Gorenstein curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let K be a number field, S a finite set of places of K, \(R_ S\) the ring of S-integers, and \(A| K\) an abelian variety. A conjecture of \textit{S. Lang} [''Fundamentals of diophantine geometry'' (1983; Zbl 0528.14013); p. 219] asserts, for every affine subset U of A, the finiteness of the set of points of U with coordinates in \(R_ S\). In the paper under consideration, a first step towards this conjecture is proved.
(*) Assume \(A=J(C)\) is the Jacobian of a curve C of genus \(\geq 2\) over the number field k. There exists a positive, irreducible divisor \(D\in Div(A)\) such that for every finite extension K:k, every finite set S of places of K, every \(f\in K(A)\) with pole divisor \((f)_{\infty}\geq D\), the set \(\{x\in A(K)| f(x)\) is defined and lies in \(R_ S\}\) is finite.
Corresponding finiteness results can be given for arbitrary abelian varieties A, using a finite map of A into a Jacobian. The proof of (*) is straightforward, the ingredients being (a) a criterion of good reduction of curves which the author assigns to Szpiro and Ogus, and (b) (the main point), Faltings' proof of the Shafarevich conjecture. finite number of integral points; Lang conjecture; abelian variety; Jacobian [Sil] Silverman, J.H.: Integral points on abelian varieties. Invent. Math.81, 341-346 (1985); Corrigendum: Invent. Math.84, 223 (1986) Arithmetic ground fields for abelian varieties, Rational points, Jacobians, Prym varieties Integral points 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 Denote by \(X\) a separated Noetherian scheme of finite Krull dimension which has enough locally free sheaves of finite rank (i.e., every coherent sheaf on \(X\) is a quotient of a locally free sheaf of finite rank). Assume that \(U\subseteq X\) is an open subscheme.
\textit{D. O. Orlov} introduced [Proc. Steklov Inst. Math. 246, 227--248 (2004); translation from Tr. Mat. Inst. Steklova 246, 240--262 (2004; Zbl 1101.81093)] an invariant, called the singularity category of \(X\) defined to be the Verdier quotient triangulated category: \(\mathbf D_{sg}(X)=\mathbf D^b(\text{coh} X)/\text{perf} X\). Orlov proved (ibid) that if the singular locus of \(X\) is \(\subseteq U\), then the triangle functor \(\overline j^*:\mathbf D_{sg}(X)\longrightarrow \mathbf D_{sg}(U)\) is an equivalence. Another result of \textit{D. Orlov} [Adv. Math. 226, No.~1, 206--217 (2011; Zbl 1216.18012)] proves that \(\mathbf D_{sg}(X)=\text{thick}\langle q(\text{coh}_\mathbb Z X)\rangle\), under the assumption that the singular locus of \(X\) is \(\subseteq \mathbb Z\) and \(q:\mathbf D^b(coh X)\longrightarrow \mathbf D_{sg}(X)\) is the quotient functor. The author in a sense unifies (and generalizes) these results of Orlov (in a spirit resembling a result of \textit{H. Krause} [Compos. Math. 141, No. 5, 1128--1162 (2005; Zbl 1090.18006)]).
This unifying result is that the triangle functor \(\overline j^*:\mathbf D_{sg}(X)\longrightarrow \mathbf D_{sg}(U)\) induces a triangle equivalence \(\mathbf D_{sg}(X)/\text{thick}\langle q(\text{coh}_\mathbb Z X)\rangle\cong \mathbf D_{sg}(U)\). This is proved through some lemmas and it was shown that Orlov's results are then corollaries. The author also proves a non-commutative version of his main result, for a left Noetherian ring \(R\); here \(X\) is replaced by \(R\) and \(U\) by \(eRe\), for an idempotent \(e\) that is subject to some reasonable conditions. singularity category; quotient functor; Schur functor; Verdier quotient; triangle functor; triangle equivalence; left Noetherian ring; idempotents; Orlov; Krause Chen, Xiao-Wu, Unifying two results of Orlov on singularity categories, Abh. Math. Semin. Univ. Hambg., 0025-5858, 80, 2, 207-212, (2010) Chain complexes (category-theoretic aspects), dg categories, Singularities in algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Unifying two results of Orlov on singularity categories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the topology of the quotient variety of a complex algebraic projective variety \(X\) with an action of a complex algebraic torus \((\mathbb{C}^*)^ n\). As a main result, he obtains an inductive formula for the intersection Betti numbers. The formula includes singular quotients. One can always find a rationally nonsingular quotient and a canonical map that is a small resolution in the sense of Goresky- MacPherson. The most important quotients under consideration are those that can be understood as symplectic reduced spaces.
The author starts with a nice introduction to this subject. The main part of the paper contains an adequate stratification, the proof that there exist small resolutions and the decomposition theorem for the intersection homology in the symplectic case. In a further step this is done analogously in the so-called semigeometric case which generalizes the symplectic one. But the results are smaller: Vanishing property of intersection homology in odd degree and isomorphism to rational intersection groups are transferable from the fixed-point set to the quotient. Finally, an application to flag varieties is given. See also \textit{F. C. Kirwan}, ``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020).
[See also erratum to this paper in the following review.]. topology of the quotient variety; action of a complex algebraic torus; intersection Betti numbers; symplectic reduced spaces; intersection homology Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992) 151--184. Erratum: 68 (1992) 609. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The geometry and topology of quotient varieties of torus actions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 shows among other results the Deligne conjecture
\[
\text{Lef(Fr}^{n }\cdot b,K) =\sum_{D\in\Pi_{0}(\text{Fix Fr}^{n}\cdot b)} \text{ naive.loc}_{p}( \text{Fr}^{n}\cdot b,K)\tag{1}
\]
for \(U\) an open subset of a proper scheme \(X\), for \(b: V\to U\times_{k}U\) a correspondence (\(k\) the algebraic closure of a finite field) for \(K\) an element of the derived category \(D^{b}_{c}(U,\overline{\mathbb Q}_{l})\) of bounded complexes with constructible cohomology sheaves, for the corresponding Frobenius map Fr and if \(n\) is suitably large. Here Lef means the global trace, naive.loc means the naive local term (as an important property this term vanishes if the fiber of \(K\) is zero), and Fr means the geometric Frobenius over \(\mathbb{F}_q\). This result follows by upgrading in stages the Lefschetz-Verdier trace formula as found in Sémin. Géom. Algebr. 1965-66, SGA5, Lect. Notes Math. 589 [Exposé III, 73-137 (1977; Zbl 0355.14004) by \textit{A. Grothendieck}, and Exposé III B, 138-203 (1977; Zbl 0354.14006) by \textit{L. Illusie}]. For a proper statement of (1), it is necessary to use rigid geometry. Lefschetz-Verdier trace formula; Deligne conjecture; rigid geometry; proper scheme; constructible cohomology sheaves; Frobenius map; positive characteristic Fujiwara, K., \textit{rigid geometry, Lefschetz-verdier trace formula and deligne's conjecture}, Invent. Math., 127, 489-533, (1997) Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Finite ground fields in algebraic geometry Rigid geometry, Lefschetz-Verdier trace formula and Deligne's conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the non-cuspidal rational points on the curve \(X^+_ 0(N)\), which is the quotient of the modular curve \(X_ 0(N)\) by the action of the Atkin-Lehner involution \(w_ N\). From the modular interpretation of \(X_ 0(N)\) it is easy to see that there is a natural morphism \(f^+: X^+_ 0(N)\to J^-_ 0(p)=J_ 0(p)/(1+w_ p)\cdot J_ 0(p)\) when p is a divisor of N and \(J_ 0(p)\) the Jacobian of \(X_ 0(p)\). Using a result of \textit{B. Mazur} and \textit{M. Rapoport} [``Modular curves and the Eisenstein ideal'', Appendix, Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186 (1978; Zbl 0394.14008)] the author shows that if p is a prime divisor of N, and \(x\in X^+_ 0(N)\) is a non- cuspidal rational point, then \(f^+(x)| {\mathbb{F}}_ p\) is the unit section. Under the assumption that \(p=11\), or \(p\geq 17\) and \(J^-_ 0(p)({\mathbb{Q}})\) is finite one sees that \(f^+(x)\) is actually the unit section. This follows immediately from the fact that the image of the cuspidal divisor class group in \(J^-_ 0(p)\) is an étale group scheme, which is in fact the entire torsion subgroup of the Mordell-Weil group of \(J^-_ 0(p)\) (loc. cit. p. 143). Thus, \(f^+(x)\) generates an étale subgroup of \(J^-_ 0(p)\), and so \(f^+(x)\) is the unit section precisely when \(f^+(x)| {\mathbb{F}}_ p\) is so as well. When \(p\neq 37\) this implies that an elliptic curve that represents the \(point\quad x\) has complex multiplication. Hence, under the conditions on N and p given above, the only non-cuspidal rational points on \(X^+_ 0(N)\) are the C.M. points.
This paper extends earlier work of the author on \(X_{split}(p)\) [Compos. Math. 52, 115-137 (1984; Zbl 0574.14023)] and \(X^+_ 0(p^ r)\) [to appear]. non-cuspidal rational points; quotient of the modular curve; action of the Atkin-Lehner involution; Mordell-Weil group; complex multiplication Momose, F., Rational points on the modular curves \(X_0^+(N)\), J. Math. Soc. Jpn., 39, 269-286, (1978) Arithmetic ground fields for curves, Complex multiplication and abelian varieties, Rational points, Holomorphic modular forms of integral weight Rational points on the modular curves \(X^ +_ 0(N)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 algebraic geometry, embeddings of curves into projective space are defined via linear systems. A similar concept is known in tropical geometry, where the notion of an algebraic curve gets replaced by the notion of a \textit{tropical curve}, i.e. a finite metric graph. More concretely, given a tropical curve \(\Gamma\), a divisor \(D\), and a finite generating set \(F = \{f_0, \ldots, f_n\}\) of the space \(R(D)\) consisting of piecewise linear functions with pole orders bounded by \(D\), then we obtain a rational map
\begin{align*}
\phi_F : \Gamma &\longrightarrow \mathbb{T}\mathbb{P}^n \\
x &\longmapsto \big(f_0(x) : \ldots : f_n(x) \big).
\end{align*}
In the present article, the authors studies this situation under the additional information of an action of a finite group \(K\) on \(\Gamma\). Similar to the above, a finite generating set \(F = \{f_i\}_i\) of the space of \(K\)-invariant PL functions \(R(D)^K \subseteq R(D)\) gives a rational map \(\phi_F\) to tropical projective space. As a first theorem, the author shows that \(\phi_F\) exhibits \(\Gamma\) as a \(K\)-Galois cover of \(\phi_F(\Gamma)\) if and only if \(\phi_F\) is injective on the level of \(K\)-orbits. Here, a \textit{\(K\)-Galois cover} of tropical curves is a harmonic morphism \(\Gamma \to \Gamma'\) of degree \(|K|\), together with a \(K\)-action on \(\Gamma\), such that the morphism is the quotient map modulo \(K\). As a second theorem, the author shows that the required injectivity is actually automatic as long as \(D\) is effective, \(K\)-invariant, and of positive degree.
As an application, the authors studies tropical hyperelliptic curves, i.e. curves which posses a \(g^1_2\). In the classical setting, the canonical map \(\phi_{|K_C|}\) of an algebraic curve \(C\) which is given by the linear system of the canonical divisor \(K_C\) gives rise to a double cover \(C \to \mathbb{P}^1\). While tropically, a \(g^1_2\) gives rise to a harmonic double cover of a metric tree (the tropical analogue of \(\mathbb{P}^1\)), the canonical map will not always be of this form. In fact, the author shows that the canonical map is a double cover of a tree precisely when \(g(\Gamma) = 2\). Furthermore, the authors shows that if \(\Gamma\) is hyperelliptic and \(g(\Gamma) \geq 3\), then a double cover of the desired form is given by the part of the canonical linear system which is invariant under the hyperelliptic involution.
A noteworthy technical detail is that since \(\phi_F : \Gamma \to \phi_F(\Gamma)\) is always an isometry, the author has to introduce multiplicities on the edges of \(\Gamma\), given by the size of their stabilizer groups, in order to make the map harmonic. tropical curve; invariant linear subsystem; rational map; Galois covering; hyperelliptic tropical curve; canonical map Combinatorial aspects of tropical varieties, Geometric aspects of tropical varieties Galois quotients of tropical curves and invariant linear systems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an irreducible smooth complex projective curve. Let \(\mathcal{Q}(r,d)\) be the Quot scheme parameterizing all coherent subsheaves of \(\mathcal{O}_X^{\oplus r}\) of rank \(r\) and degree \(-d\). There are natural morphisms \(\mathcal{Q}(r,d) \longrightarrow \mathrm{Sym}^d (X)\) and \(\mathrm{Sym}^d(X)\longrightarrow \mathrm{Pic}^d(X)\). We prove that both these morphisms induce isomorphism of Brauer groups if \(d\geq 2\). Consequently, the Brauer group of \(\mathcal{Q}(r,d)\) is identified with the Brauer group of \(\mathrm{Pic}^d(X)\) if \(d\geq 2\). Brauer groups of schemes, Parametrization (Chow and Hilbert schemes), Stacks and moduli problems Brauer groups of 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 The paper under review provides new geometric examples where one can prove deep conjectures about \(K\)-theory and regulators of arithmetic schemes. The main result (denoted (1) below) is the proof of new cases of Belinsons's generalization of the Tate conjecture to higher \(K\)-theory (or higher Chow groups). Recall that Jannsen found counter-examples to the original formulation [\textit{U. Jannsen}, Mixed motives and algebraic \(K\)-theory. (Almost unchanged version of the author's habilitation at Univ. Regensburg 1988). Berlin etc.: Springer-Verlag (1990; Zbl 0691.14001)].
The geometric situation studied in the paper is the following. Let \(k\) be a base field, and consider a map \(\pi : X \to C\) over \(k\), from a smooth projective surface to a smooth projective curve, whose generic fiber is an elliptic curve. Let \(U\) be an open subscheme of \(X\) whose complement is a union of fibers of \(\pi\) (those of ``multiplicative typ'').
A motivating example is given by the universal family of elliptic curves \(U\) over a modular curve, and its canonical compactification \(X\). Regulators in this case were extensively studied by \textit{A. A. Beilinson} [Contemp. Math. 55, 1--34 (1986; Zbl 0609.14006)]. In several points the present paper can be seen as a generalization of Beilinson's work, although the techniques in the proofs are different (the use of \(p\)-adic Hodge theory here is essential).
The authors construct new examples where the following can be proved :
(1) (\(k\) a number field) The étale Chern class map
\[
c_{\mathrm{\'et}} : K_2 (U ) \otimes \mathbb Q_p \to H^2_{\mathrm{\'et}} (U_{\overline{k}} , \mathbb Q_p (2))^{\mathrm{Gal}(\overline{k}/k)}
\]
is surjective (Section 6.1).
(2) (\(k\) a \(p\)-adic local field) The \(p\)-adic regulator
\[
\rho : K_1 (X)^{(2)} \otimes \mathbb Q_p \to H^!_g (k, H^1_{\mathrm{\'et}} (X_{\overline{k}} , \mathbb Q_p (2)))
\]
is surjective (Theorem 7.0.3).
(3) (\(k\) a \(p\)-adic local field) The torsion subgroup \(\mathrm{CH}^2 (X)_{\mathrm{tors}}\) of the Chow group \(\mathrm{CH}^2 (X)\) is finite (Section 7.3).
The key ingredient is the introduction of the ``space of formal Eisenstein series'' (Section 5.2),
\[
E(\mathcal {X , D})_{\mathbb Z_p} \subset \Gamma(\mathcal{X }, \Omega^2_{\mathcal{X }/R} (\log \mathcal D)).
\]
(Here \(k\) is a \(p\)-adic local field, \(R\) its ring of integers, \(\mathcal X\) and \(\mathcal D\) are suitable models of \(X\) and \(X - U\) over \(R\).) This generalizes a construction due to Beilinson in the case of modular curves (see [loc. cit.]).
In 5.3 it is shown that the \(\mathbb Z_p\)-rank of \(E(\mathcal {X , D})_{\mathbb Z_p}\) bounds the dimension of the Galois invariant classes appearing in (1). Hence, if one can construct enough algebraic classes (reaching the bound), then one can deduce the surjectivity in (1). AS15 Masanori Asakura and Kanetomo Sato, \emph Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance, arXiv:1301.5829 (2013). Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory and homology; cyclic homology and cohomology, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Steinberg groups and \(K_2\), (Equivariant) Chow groups and rings; motives, Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry Syntomic cohomology and Beilinson's Tate conjecture for \(K_{2}\) | 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 G be a connected, reductive algebraic group defined over \(F_ q\), and let \(F: G\to G\) be a Frobenius morphism. Let \(G^ F\) be the finite group of F-fixed points of G. Corresponding to each F-stable maximal torus T, \textit{P. Deligne} and the author [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] defined functions \(Q^ G_ T\), known as Green functions, on the set of unipotent elements of \(G^ F\). Later the author gave a new definition of Green functions using intersection cohomology and in his work on character sheaves [Adv. Math. 61, 103-155 (1986; Zbl 0602.20036)] gave a method of computing these functions in principle. The coincidence of the two definitions of Green functions was known for large p (where q is a power of p) by work of Springer and Kazhdan, and for all ``good'' p in some cases by further work of the author [J. Algebra 104, 146-194 (1986; Zbl 0603.20037)]. In this paper the author proves that the two definitions coincide for all p, provided q is sufficiently large. More generally, he considers the generalized Green functions defined by him via intersection cohomology [in Adv. Math. 57, 226-265 (1985; Zbl 0586.20019)] and connects them with functions obtained via twisted induction from a Levi subgroup of G, when p is ``almost good''.
Let \({\mathcal D}G\) denote the bounded derived category of constructible \(\bar Q_{\ell}\)-sheaves on G (here \(\ell\) is a prime distinct from p). The abelian category \({\mathcal M}G\) of perverse sheaves on G is a full subcategory of \({\mathcal D}G\), and character sheaves are certain objects in \({\mathcal M}G\). In analogy with the Harish-Chandra theory for \(G^ F\), certain character sheaves are called cuspidal. If K is a complex in \({\mathcal D}G\) with a given isomorphism \(\phi\) : \(F^*K\overset \sim \rightarrow K\), then one gets a \(\bar Q_{\ell}\)-valued characteristic function \(\chi_{K,G}\) on \(G^ F\) by setting \(\chi_{K,\phi}(x)=\sum_{i\geq 0}(-1)^ iTr(\phi,H^ i_ xK)\) where \(H^ i_ xK\) is the stalk at x of the i-th cohomology sheaf \(H^ iK\) of K.
A cuspidal character sheaf on a Levi subgroup M of G gives rise to an induced complex in \({\mathcal D}G\) which is a direct sum of character sheaves on G. More precisely, suppose we have the following data. Let M be an F- stable Levi subgroup of a parabolic subgroup MV of G with unipotent radical V. Let \(\Sigma\) be the inverse image under \(M\to M/Z^ 0_ M\) (here \(Z_ M\) is the center of M) of a single, F-stable conjugacy class of \(M/Z^ 0_ M\). Let \({\mathcal E}\) be an M-equivariant (for the conjugation action of M) \(\bar Q_{\ell}\)-local system on \(\Sigma\) which gives rise, by extension first to the closure \({\bar \Sigma}\) of \(\Sigma\) as an intersection cohomology complex and then to M by 0 on \(M-{\bar \Sigma}\), to a complex \({\mathcal E}^{\#}\in {\mathcal D}M\) such that \({\mathcal E}^{\#}[\dim \Sigma]\) is a direct sum of cuspidal character sheaves on M. (Here [ ] denotes shift). Let \(\tau\) : \(F^*{\mathcal E}\overset \sim \rightarrow {\mathcal E}\) be an isomorphism. Then, to this data is associated an induced complex \(K\in {\mathcal D}G\) and an isomorphism \({\bar \tau}\): \(F^*K\overset \sim \rightarrow K\). Now assume further that \(\Sigma =CZ^ 0_ M\) where C is an F-stable unipotent conjugacy class of M, \({\mathcal E}={\mathcal F}\otimes \bar Q_{\ell}\) where \({\mathcal F}\) is an M-equivariant \(\bar Q_{\ell}\)-local system on C, and that \(\tau =\tau_ 0\otimes 1\) where \(\tau_ 0: F^*{\mathcal F}\overset \sim \rightarrow {\mathcal F}\) is an isomorphism. Then the restriction of the characteristic function \(\chi_{K,{\bar \tau}}\) to the set of unipotent elements of \(G^ F\) is called a generalized Green function and is denoted by \(Q^ G_{M,C,{\mathcal F},\tau_ 0}\). In particular, taking \(G=M\), \(K={\mathcal E}^{\#}[\dim \Sigma]\) we have \(Q^ M_{M,C,{\mathcal F},\tau_ 0}=(-1)^{\dim \Sigma}\chi_{{\mathcal E}^{\#},\tau}\). By applying the twisted induction map \(R^ G_{M,V}\) (which generalizes Deligne-Lusztig induction) to the function \(Q^ M_{M,C,{\mathcal F},\tau_ 0}\) we get another function \(\bar Q^ G_{M,V,C,{\mathcal F},\tau_ 0}\) on the set of unipotent elements of \(G^ F\). The main theorem of this paper (1.14) is then as follows. There is a constant \(q_ 0>1\) depending only on the Dynkin diagram of G, such that if \(q\geq q_ 0\) the following hold:
(a) If p is ``almost good'' for M, then \((-1)^{\dim \Sigma}\tilde Q^ G_{M,V,C,{\mathcal F},\tau_ 0}=Q^ G_{M,C,{\mathcal F},\tau_ 0}.\)
(b) Suppose p is ``almost good'' for G. Then the space \(F^ G_ G\) spanned by functions on \(G^ F\) of the form \(Q^ G_{G,C,{\mathcal F},\tau_ 0}\) (i.e. restrictions to the unipotent elements of the characteristic functions of cuspidal character sheaves of G, defined over \(F_ q)\) is precisely the space of \(\bar Q_{\ell}\)-valued functions on the unipotent elements of \(G^ F\) which are orthogonal to the functions of the form \(R^ G_{M,V}(f)\) for any \(M\neq G\) and any class function \(f: M^ F\to \bar Q_{\ell}\). In the case when M is a maximal torus T we can take \({\mathcal F}=\bar Q_{\ell}\), \(C=\{e\}\), and \(\tau =1.\)
In this case (Proposition 8.15) the assumptions of the theorem can be weakened and the conclusion is that the generalized Green function \(\tilde Q^ G_ T\) is equal to the Deligne-Lusztig Green function \(Q^ G_ T\). connected reductive algebraic groups; Frobenius morphism; unipotent elements; Green functions; intersection cohomology; character sheaves; twisted induction; Levi subgroup; derived category of constructible \(\bar Q_{\ell}\)-sheaves; perverse sheaves G. Lusztig, Green functions and character sheaves, Ann. of Math. 131 (1990), 355--408. Representation theory for linear algebraic groups, Ordinary representations and characters, Group actions on varieties or schemes (quotients), Cohomology theory for linear algebraic groups, \(p\)-adic cohomology, crystalline cohomology, Other algebraic groups (geometric aspects), Linear algebraic groups over finite fields Green functions and character 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 \(C\) be a smooth hyperbolic curve over a number field \(k\) with absolute Galois group \(G_ k\), and \(\varphi_ C:G_ k\to\text{Out}\pi_ 1\) the associated exterior Galois representation into the outer automorphism group of the geometric pro-\(\ell\) fundamental group \(\pi_ 1\) of \(C\) \((\ell:\) a fixed prime). The purpose of this paper is to show that the centralizer of the Galois image \(\varphi_ C(G_ k)\) in \(\text{Out}\pi_ 1\) (denoted \(E_ k^{(\ell)}(C))\) is a finite group approximating \(\text{Aut}_ kC\) when \(C\) is general in a certain sense.
As a main tool for approaching this problem, in \S1, basic formation of the ``graded coordinate system'' on a pro-\(\ell\) mapping class group is established. Here the pro-\(\ell\) mapping class group of type \((g,n)\) means the group of (continuous) ``braid-like'' outer automorphisms of the pro-\(\ell\) fundamental group of an \(n\)-point punctured Riemann surface of genus \(g\). -- In \S2, Kaneko's pro-\(\ell\) analog of the Dehn-Nielsen theorem [\textit{M. Kaneko}, J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 2, 363-372 (1989; Zbl 0692.20018)] is generalized by introducing a certain new homomorphism. -- In \S3 and \S4, using the graded coordinate system of \S1, sufficient conditions for \(E_ k^{(\ell)}(C)\) to be finite and examples of certain special curves \(C\) supporting the conjecture \(\text{Aut}_ kC\cong E_ k^{(\ell)}(C)\) are given. -- In \S5, a formation of braid-like derivation algebras, partly generalizing methods of \textit{Y. Ihara} [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 99- 120 (1991; Zbl 0757.20007)] is presented, and as an application, it is shown that field towers arising from exterior Galois representations associated with suitable hyperelliptic curves are certainly ``infinite''. graded coordinate system; hyperbolic curve; exterior Galois representation; outer automorphism group; punctured Riemann surface; braid-like derivation algebras; exterior Galois representations Nakamura, H.; Tsunogai, H., Some finiteness theorems on Galois centralizers in pro-\textit{} mapping class groups, J. Reine Angew. Math., 441, 115-144, (1993) Coverings of curves, fundamental group, Riemann surfaces; Weierstrass points; gap sequences, Automorphisms of curves Some finiteness theorems on Galois centralizers in pro-\(l\) mapping class 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 The authors investigate the existence of a Poincaré sheaf for the moduli space of torsion free and locally free sheaves over a fixed real curve. Let \(Y\) be a geometrically irreducible reduced projective curve defined over the real numbers and let \(g=g(Y)\) be its arithmetic genus. Denote by \(U(n,d)\) the moduli space of torsion free stable sheaves of rank \(n\) and degree \(d\) over \(Y\) and by \(U'(n,d)\) the moduli space of locally free sheaves of rank \(n\) and degree \(d\) over \(Y\). The authors prove that, if the Euler characteristic \(\chi = d + n ( 1- g )\) of \(Y\) is coprime with \(2n\) then exists a Poincaré sheaf over \(U(n,d) \times Y\). Conversely, if exists a nonempty open subset \(U\) of \(U'(n,d)\) and a Poincaré sheaf over \(U \times Y\) then \(2n\) and \(\chi\) are coprime. Assuming that \(Y\) has a smooth real point, the authors prove the same result but replacing the condition that \(2n\) is coprime with \(\chi\) with the weaker one that \(n\) is coprime with \(\chi\). descent condition; Poincaré sheaf; singular curves; real curves; moduli space; existence Structure of families (Picard-Lefschetz, monodromy, etc.), Algebraic moduli problems, moduli of vector bundles, Real algebraic and real-analytic geometry Poincaré sheaves on the moduli spaces of torsionfree sheaves over an irreducible 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 Motivated by work of the second author on unlikely intersections [Some problems of unlikely intersections in arithmetic and geometry. With appendixes by David Masser. Annals of Mathematics Studies 181. Princeton,
NJ: Princeton University Press (2012; Zbl 1246.14003), (Chapter III)], the
authors proposed the following
Conjecture. Let \(\mathscr{S}\) be a
semiabelian scheme over a variety defined over \(\mathbb{C}\), and denote by
\(\mathscr{S}^{[c]}\) the union of its semiabelian subgroup schemes of codimension
at least \(c\). Let \(\mathscr{V}\) be an irreducible closed subvariety of
\(\mathscr{S}\). Then \(\mathscr{V}\cap \mathscr{S}^{[1+\dim(\mathscr{V})]}\) is
contained in a finite union of semiabelian subgroup schemes of \(\mathscr{S}\) of
positive codimension.
This conjecture is related to a conjecture of \textit{R.
Pink} [``A common generalization of the conjectures of André-Oort,
Manin-Mumford and Mordell-Lang'', Preprint] and to the conjectures of \textit{B.
Zilber} [J. Lond. Math. Soc., II. Ser. 65, No. 1, 27--44 (2002; Zbl
1030.11073)]. It is a relative form of the Manin-Mumford conjecture.
Recent
work of the authors [Am. J. Math. 132, No. 6, 1677--1691 (2010; Zbl 1225.11078)]
implies the conjecture when \(\mathscr{S}\) is isogenous to the fibred product of
two isogenous elliptic schemes and \(\mathscr{V}\) is a curve. In the paper under
review the authors prove the conjecture in the case where \(\mathscr{S}\) is a
fibred product of two arbitrary elliptic schemes and \(\mathscr{V}\) is a curve.
This gives the following impressive theorem towards the above conjecture in the
case where \(\mathscr{S}\) is proper and of relative dimension two.
Theorem. Let \(\mathscr{A}\) be a non-simple abelian surface scheme over a variety
defined over \(\mathbb{C}\), and let \(\mathscr{V}\) be an irreducible closed curve
in \(\mathscr{A}\). Then \(\mathscr{V}\cap \mathscr{A}^{[2]}\) is contained in a
finite union of abelian subgroup schemes of \(\mathscr{A}\) of positive
codimension.
Note that a recent preprint of the authors [``Torsion points
on families of simple abelian surfaces'', Preprint] settles the conjecture also
in the case where \(\mathscr{S}\) is a \textit{simple} abelian surface scheme and
\(\mathscr{V}\) is a curve.
It turns out, however, that some care is
necessary in connection with the above conjecture when \(\mathscr{S}\) is not
proper over the base (even if the relative dimension is still two): \textit{D.
Bertrand} [``Special points and Poincaré biextensions'', Preprint] discovered
a counterexample related to so-called Ribet sections in a situation where
\(\mathscr{S}\) is an extension of an elliptic scheme by \(\mathbb{G}_m\). On the
other, hand recent work of \textit{D. Bertrand}, \textit{A. Pillay} and the authors
[``Relative Manin-Mumford for semi-abelian surfaces'', Preprint,
\url{arXiv:1307.1008}] shows that Ribet sections are essentially the only
obstruction to the conjecture if \(\mathscr{S}\) is of relative dimension two over
a one-dimensional base. torsion; abelian scheme; unlikely intersections [14] D. Masser, U. Zannier, \(Torsion points on families of products of elliptic curves\) Advances in Maths 259, (2014), 116-133. &MR 31 Elliptic curves over global fields, Heights, Arithmetic ground fields for surfaces or higher-dimensional varieties, Algebraic theory of abelian varieties Torsion points on families of products of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We consider the following situation. Let \(X\) be a smooth projective surface and let \(B\) be a smooth projective curve, defined over an algebraically closed field \(k\) of characteristic zero. Let \(f:X \to B\) be a surjective morphism and let \(\omega_{X/B}\) denote the relatively canonical sheaf of differentials. Let us assume that the generic fibre is smooth of genus \(g\) and let us denote by \(\delta\) the number of singular points in the fibres. We write \(\Lambda_n\), for the determinant of \(f_*\omega^n_{X/B}\) and and \(\lambda_n\) for the degree of \(\Lambda_n\). Finally, let us assume that \(f\) is a relatively minimal model, which means that there are no exceptional curves among the fibres. -- In this situation, Iitaka's conjecture \(C_{2,1}\) is well-known:
Theorem 1.1. If, in the situation above, \(X_b\) denotes a general fibre, we have the subadditivity \(\kappa(X) \geq\kappa (B)+\kappa (X_b)\) of the Kodaira dimensions.
We immediately notice that, in order to give a proof for this, we may assume both \(B\) and \(X_b\) to have genus greater than zero. The result 1.1 contributes to the Enriques-Kodaira classification of surfaces, and Iitaka's conjecture \(C_{2,1}\) basically follows from:
Theorem 1.2. Keeping the assumptions made above and assuming in addition that \(f\) is non-isotrivial, we have \(\lambda_1 >0\).
There have been several kinds of proofs for theorem 1.2 but most of them used analytic methods. But in the special case of a familiy of curves over a curve, one can give both an elementary and a purely algebraic proof of theorem 1.2 based on positivity methods.
The aim of this paper is to present this proof, which exclusively uses methods from algebraic geometry and hence yields an algebraic proof of theorem 1.1. subadditivity of the Kodaira dimension Fibrations, degenerations in algebraic geometry, Families, moduli, classification: algebraic theory An algebraic proof of Iitaka's conjecture \(C_{2, 1}\) | 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 Consider a germ \(C\) of reduced curve on a smooth germ \(S\) of complex analytic surface. Assume that \(C\) contains a smooth branch \(L\). Using the Newton-Puiseux series of \(C\) relative to any coordinate system \((x, y)\) on \(S\) such that \(L\) is the \(y\)-axis, one may define the Eggers-Wall tree \(\Theta _L(C)\) of \(C\) relative to \(L\). Its ends are labeled by the branches of \(C\) and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically \(\Theta _L(C)\) into Favre and Jonsson's valuative tree \({\mathbb P}(\mathcal{V})\) of real-valued semivaluations of \(S\) up to scalar multiplication, and to show that this embedding identifies the three natural functions on \(\Theta _L(C)\) as pullbacks of other naturally defined functions on \({\mathbb P}(\mathcal{V})\). As a consequence, we generalize the well-known \textit{inversion theorem} for one branch: if \(L'\) is a second smooth branch of \(C\), then the valuative embeddings of the Eggers-Wall trees \(\Theta _{L'}(C)\) and \(\Theta _L(C)\) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space \({\mathbb P}(\mathcal{V})\) is the projective limit of Eggers-Wall trees over all choices of curves \(C\). As a supplementary result, we explain how to pass from \(\Theta _L(C)\) to an associated splice diagram. branch; characteristic exponent; contact; Eggers-Wall tree; Newton-Puiseux series; plane curve singularities; semivaluation; splice diagram; rooted tree; valuation; valuative tree Singularities in algebraic geometry, Complex surface and hypersurface singularities The valuative tree is the projective limit of Eggers-Wall trees | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 part of the literature on the geometric Langlands conjecture that seeks to prove the equivalence \(\text{IndCoh}_{\mathcal{N}}(\text{LocSys}_{\check{G}})\simeq \mathfrak{D}(\text{Bun}_G)\) (proposed in this form in [\textit{D. Arinkin} and \textit{D. Gaitsgory}, Sel. Math., New Ser. 21, No. 1, 1--199 (2015; Zbl 1423.14085)]) by embedding the DG-categories on either side of the equivalence into a larger ambient DG-category and matching a collection of generators for each category under these embeddings.
The primary contributions of this paper to this program are as follows: (1) A construction of this larger ambient category as the \textit{extended Whittaker category} \(\mathcal{W}h(G,\text{ext})\), (2) a construction of the \textit{functor of Whittaker coefficients} \(\text{coeff}_{G,\text{ext}}:\mathfrak{D}(\text{Bun}_G)\to\mathcal{W}h(G,\text{ext})\), and (3) a proof for \(G=\mathrm{GL}_n\) and \(G=\mathrm{PGL}_n\) of the conjecture of Gaitsgory that the functor of Whittaker coefficients is fully faithful (see [\textit{D. Gaitsgory}, Astérisque 370, 1--112 (2015; Zbl 1406.14008)] for this conjecture together with the proposed strategy of proof).
The Ran space of the curve \(X\) -- the moduli of finite non-empty subsets of \(X\) -- plays a central role in the constructions and proofs of this paper, many of which apply more generally than to the specific examples involved in the geometric Langlands conjecture. For instance, for any \(\mathfrak{D}(\text{Ran})\)-module category \(\mathcal{C}\) equipped with the action of a certain twisted group, the author constructs a Ran version of the extended Whittaker category for \(\mathcal{C}\) over \(\text{Ran}\). Additionally, the author constructs an `independent subcategory' functor that assigns to certain modules over \(\mathfrak{D}(\text{Ran})\) (called `naive unital') an underlying independent DG-category. \(\mathcal{W}h(G,\text{ext})\) is then built by taking the independent subcategory of the Ran version of the extended Whittaker category for \(\mathfrak{D}\)-modules on the \(\Omega_X\)-twisted Grassmannian of \(X\).
Similarly, the construction of the functor of Whittaker coefficients proceeds by first defining such a functor over the Ran space via a procedure of `averaging over an action of meromorphic jets of functions to a unipotent group, and then extracting an independent version of this functor. The proof that the coefficient functor is fully faithful for \(G=\mathrm{GL}_n\) or \(\mathrm{PGL}_n\) is presented in the final section and relies on what the author considers ``the main new idea of the present paper'', namely a ``blow-up trick'' that allows for a clever induction on \(n\).
Finally, I would be remiss not to mention that the introduction of this paper contains a marvellous discussion of the analogy between the contents of this paper and the function theoretic version of the Langlands program. The function theoretic version involves the analysis of vector space of automorphic functions via an attempted decomposition into Whittaker coefficients, and the appearance of certain geometric objects is demystified by the function theoretic analogy (e.g.\ the appearance of adèles and adèlic group points in the function theoretic program leads to the Ran space and jets of group valued meromorphic functions in the geometric program). geometric Langlands; Langlands duality; Whittaker Geometric Langlands program (algebro-geometric aspects), Vector bundles on curves and their moduli, Geometric Langlands program: representation-theoretic aspects On the extended Whittaker 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 In this very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks. smoothable; secant varieties; finite Gorenstein scheme; cactus variety; Veronese reembedding; Hilbert scheme Buczyński, J.; Jelisiejew, J., Finite schemes and secant varieties over arbitrary characteristic, Differential Geom. Appl., 55, 13-67, (2017) Determinantal varieties, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Homogeneous spaces and generalizations Finite schemes and secant varieties over arbitrary 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 Let \(f : S \rightarrow T\) be a relatively minimal projective morphism between a smooth projective surface \(S\) and a smooth curve \(T\) such that the genus \(g\) of the general fiber of \(f\) is positive. The slope of \(f\), \(\lambda_f\), is defined to be the ratio \(K_{S/T}^2/\deg (f_{\ast}K_{S/T})\), where \(K_{S/T}\) is the relative canonical sheaf of \(f\). Part of its significance is that if \(T=\mathbb{P}^1\), \(g \geq 2\), and \(f\) is semistable, then \(\lambda_f\) is related to the the number \(\sigma\) of the singular fibers of \(f\) by the relation \(K_{S/T}^2<(\sigma-2)(2g-2)\).
In this paper the authors study the slope of a morphism \(f : S \rightarrow T\) when \(S\) is a rational surface and \(T=\mathbb{P}^1\). In particular they obtain lower bounds for it under restrictions on the genus \(g\) and the gonality of the general fiber of \(f\). Their method is the following. It is known that in the case of the paper, \(\deg (f_{\ast}K_{S/T})=g\) and hence in order to bound the slope it suffices to bound \(K_{S/T}^2\). In order to do that the authors show that under certain restrictions on the genus \(g\) and the gonality of the general fiber \(C\) of \(f\), \(C+nK_S\) is effective for \(n=2,3\). Then there is a Zariski decomposition \(C+nK_S=P+N\) where \(P\) is a nef divisor and \(N\) an effective divisor. They calculate explicitely the divisors \(P\) and \(N\) and then the inequalities for \(K_{X/T}^2\) are obtained from the fact that \(P^2\geq 0\). fibration; minimal; slope Fibrations, degenerations in algebraic geometry, Rational and ruled surfaces On the slope of relatively minimal fibrations on rational complex surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book provides a profound introduction to some of the basic principles of both classical and modern algebraic geometry for graduate students or advanced undergraduates. Assuming only some previous knowledge of linear algebra and general topology, it also presents all the concepts, methods and results from commutative algebra, sheaf theory and cohomology as far as necessary to develop the foundations of algebraic geometry. These allied mathematical frameworks are treated separately in four appendices after the main text, thus making the textbook essentially self-contained, and therefore particularly suited for self-study by beginners or as an accompanying course book, respectively.
As to the precise contents, the text consists of six chapters and five appendices, where the latter amount to almost one half of the whole book. Chapter 1 is of motivating character and illustrates some ideas of classical algebraic geometry by means of Bézout's theorem on the intersection of projective plane curves, together with a discussion of the Sylvester matrix and the resultant of two polynomials.
Furthermore, an application of Bézout's theorem to Pascal's mystic hexagram in conic sections is described, and even the use of projective techniques in robotics is briefly sketched.
Chapter 2 develops the fundamental concepts of classical affine algebraic geometry, including Hilbert's Nullstellensatz, Gröbner bases in the computational algebra of polynomial rings, and applications of the latter to the robotics problems touched upon in the first chapter.
Chapter 3 is devoted to the local aspects of affine varieties, with particular emphasis on the Zariski tangent space, normal varieties, finite morphisms, and Serre's classification theorem for vector bundles on affine varieties.
Chapter 4 provides an introduction to the language of abstract algebraic varieties and schemes, thereby explaining (quasi-)coherent sheaves, subschemes, products, separated schemes, and their properties along the way.
Chapter 5 turns to projective varieties and their geometry, focussing here on invertible sheaves and divisors, morphisms and rational maps, products of projective varieties, graded ideals and modules, Noether normalization, and Bézout's theorem in the generalized intersection theory for projective varieties.
Chapter 6 is centered on smooth projective curves, with a special view toward plane curves and M. Noether's \(AF+BG\) theorem, elliptic curves, and the Riemann-Roch theorem. Actually, the book culminates with two proofs of the Riemann-Roch theorem, where the first one is based on the classical Brill-Noether approach, while the second proof uses the modern conceptual framework of sheaf cohomology and Serre duality.
The main text, and the purely algebro-geometric part of the book end at this point.
Appendix A basically gives a fairly detailed and comprehensive introduction to commutative algebra as needed to understand the material in the previous chapters. This part, which virtually forms a book within the book, occupies about 150 pages. Containing also a glimpse of category theory as well as expositions on discrete valuation rings, Kähler differentials and multilinear algebra, among other topics, Appendix A can be used for a second course in abstract algebra likewise.
Appendix B delivers the basic facts from the theory of sheaves and ringed spaces, whereas Appendix C derives the categorical equivalence between vector bundles and locally free sheaves on locally ringed spaces.
Appendix D provides an introduction to the methods of homological algebra and their applications to the cohomology of sheaves. This incorporates chain complexes and their cohomology, resolutions and derived functors, \(\delta\)-functors, Ext and Tor for modules, Koszul complexes, Serre's construction of the cohomology of sheaves, Čech cohomology, and a full proof of the Serre duality theorem (as used in Chapter 6).
Each subsection comes with a collection of related exercises, which are very carefully thought out, highly instructive and utmost helpful for a deeper understanding of the material. Answers to roughly half of these working problems are collected in Appendix E, while the other half is available from an instructor's manual supplementing the present textbook.
Altogether, this profound introduction to algebraic geometry and its allied theories distinguishes itself by its high degree of lucidity, comprehensiveness, mathematical rigor, versality, and student-friendliness likewise. Chapters 1 and 2, together with parts of Appendix A could be adopted for a one-semester introductory course in algebraic geometry, while the other parts may be suitable for subsequent semesters. Also, as already mentioned earlier, Appendix A could make up the text for a basic course in commutative, categorical and multilinear algebra. In fact, the entire book is an excellent source for divers course designs, giving instructors a wide range of options and much flexibility. As for self-study, readers can easily choose their path through the wealth of presented material, according to their interest or mathematical basic knowledge.
No doubt, this fine textbook is a welcome addition to the already existing variety of primers on algebraic geometry and commutative algebra. textbook (algebraic geometry); varieties and schemes; sheaves; sheaf cohomology; vector bundles; algebraic curves; commutative algebra; intersection theory Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Relevant commutative algebra, Varieties and morphisms, Schemes and morphisms, Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Curves in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra Introduction to algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This textbook originated from notes distributed to the participants of a course on arithmetic algebraic surfaces for graduate students. The aim of that course was to describe the foundations of the geometry of arithmetic surfaces as presented in the works of \textit{I. R. Shafarevich} [``Lectures on minimal models and birational transformations of two-dimensional schemes'', Tata Institute of Fundamental Research. 37 (1966; Zbl 0164.51704)] and \textit{S. Lichtenbaum} [Am. J. Math. 90, 380-405 (1968; Zbl 0194.22101)], and also the theory of stable reduction of algebraic curves as developed in the celebrated paper by \textit{P. Deligne} and \textit{M. Mumford} on the irreducibility of the moduli spaces of algebraic curves [Publ. Math., Inst. Hautes Étud. Sci. 36, 75-100 (1969; Zbl 0181.48803)].
Alas, in writing up those lecture notes, the author had to face the amazing fact that, in spite of the importance of recent developments in this field, there does not exist any suitable book in the literature that treats these subjects in a comprehensive and systematic manner, and at a level that is accessible to students or other non-specialists in the field.
With a view to this difficulty in teaching advanced arithmetic algebraic geometry, the author decided to let his notes grow into such a still missing, systematic and comprehensive textbook. Thus the aim of the book under review is to gather together all the relevant concepts, methods and results, now classical and absolutely indispensable in arithmetic geometry, in order to make them more easily accessible to a larger audience.
The outcome is a thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting, that is from A. Grothendieck's modern and most general point of view, followed by a second part on the arithmetic theory of algebraic curves and surfaces.
As to the first, purely algebro-geometric part of the book, it seems fair to say that this is, after A. Grothendieck's voluminous treatise ``Éléments de géométrie algébrique. I--IV'' (EGA I--IV), the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form, whereas the second, merely arithmetic part provides the very first systematic and coherent introduction to the advanced theory of arithmetic curves and surfaces at all. Moreover, the entire text is arranged in such exhaustive a way that the book is essentially self-contained, keeping the prerequisites at a minimum, and perfectly suitable for seasoned graduate students. Another feature of this highly valuable book on algebraic and arithmetic geometry is provided by the vast amount of illustrating, theoretically important examples as well as by the approximately six hundred included exercises. Now, as to the concrete contents of the book, the text is divided into ten chapters, where the first seven chapters discuss the basic theory of algebraic schemes and their morphisms (à la Grothendieck), while the remaining three chapters are devoted to the arithmetic theory of algebraic curves and surfaces.
Chapter 1 presents some relevant material from higher commutative algebra that is frequently used throughout the entire text. This includes, amongst other topics, a detailed discussion of the concept of flatness and formal completion of a ring.
Chapter 2 gives an introduction to schemes, their general properties, and their dimension theory, whilst chapter 3 turns to the study of morphisms of schemes and their behavior under base change. As for applications of the general theory of schemes, the special case of algebraic varieties over a field is discussed in a separate section of the third chapter. Chapter 4 is devoted to some important local properties of schemes such as normality, smoothness, flat and smooth morphisms, étale morphisms, and Zariski's Main Theorem. Coherent sheaves and their Čech cohomology is the topic of chapter 5, with a special emphasis on projective schemes. This includes those fundamental results like the behavior of higher direct images of sheaves under flat base change, the connectedness principle (Zariski), and the cohomology of the fibers of a projective morphism. Chapter 6 treats the differential calculus on schemes via Kähler differentials, sheaves of relative differentials, canonical sheaves on smooth schemes, and the according duality theory (à la Grothendieck). In this context, local complete intersections and regular immersions are also discussed. Chapter 7 deals with divisors (Cartier divisors and Weil divisors), including Van der Waerden's purity theorem, and turns then to the specific theory of algebraic curves over a field. This encompasses the basic standard material: the Riemann-Roch theorem, the Hurwitz formula, hyperelliptic curves, the classification of curves of small genus, singular curves, and Picard varieties of curves.
After this basic course on algebraic geometry, provided by chapters 1-7, the author takes up his original goal of the course, which was to give an introduction to the advanced theory of arithmetic curves and surfaces. To this end, he discusses, in chapter 8, the fundamental framework for the birational geometry of algebraic surfaces. This includes the technique of blowing-up, the notion of excellent schemes, catenary schemes, and the study of fibered surfaces. The latter topic is crucial for the heart piece of the book, namely the study of relative curves over a Dedekind scheme. Chapter 9 is devoted to regular fibered surfaces, i.e., to regular fibered, connected noetherian schemes of dimension two. The author develops the relevant intersection theory on regular surfaces, starting from the local definition, studies then the relation between morphisms of fibered surfaces and intersection numbers, discusses Castelnuovo's criterion and minimal surfaces in the sequel, and ends this chapter with investigating minimal regular fibered surfaces and canonical models for minimal arithmetic surfaces, including Artin's contractibility criterion for regular fibered surfaces and Weierstrass models of arithmetic elliptic curves.
The concluding chapter 10 is entitled ``Reduction of algebraic curves''. After discussing general properties that essentially follow from the study of arithmetic surfaces conducted before, the author describes the different types of reduction of arithmetic curves in detail (models and reductions, reduction of elliptic curves, stable curves, stable reduction, and stable models). The end of this final chapter is devoted to the stable reduction theorem of Deligne-Mumford and its beautiful proof by \textit{H. Artin} and \textit{G. Winters} [Topology 10, 373-383 (1971; Zbl 0221.14018)].
The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research.
With arithmetic geometry in mind, the author has kept the outset of the text as general as possible. In particular, it is almost never supposed that the ground field is algebraically closed, nor of characteristic zero, nor even perfect. The advantage of this approach, which to this extent cannot be found somewhere else in the textbook literature, is that the reader acquires the right (general) reflexes from the beginning on. As for the study of algebraic varieties, there are many other excellent (specific) textbooks that can be consulted. As stated before, this book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive, in addition to all the other mentioned advantages of the book. schemes; coherent sheaves; cohomology of schemes; duality theory; algebraic curves; birational geometry of surfaces; arithmetic algebraic curves; arithmetic algebraic surfaces; stable reduction of curves; arithmetic algebraic geometry; birational geometry of algebraic surfaces Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press, 2002, Translated from the French by Reinie Erné, Oxford Science Publications Birational geometry, Arithmetic varieties and schemes; Arakelov theory; heights, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Schemes and morphisms Algebraic geometry and arithmetic 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 authors construct modular Deligne-Mumford stacks \({P}_{d,g}\) representable over \(\overline{M}_{g}\) parametrizing Néron models of Jacobians as follows. Let \(B\) be a smooth curve and \(K\) its function field, let \(\chi_{K}\) be a smooth genus\(-g\) curve over \(K\) admitting stable minimal model over \(B.\) The Néron model \(N(Pic^{d} \chi_{K}) \rightarrow B\) is then the base change of \({P}_{d,g}\) via the moduli map \(B \rightarrow \overline{{M}_{g}}\) of \(f,\) i.e. \(N(Pic^{d} \chi_{K}) \cong {P}_{d,g} \times_{\overline{{M}_{g}}} B.\) Moreover \({P}_{d,g}\) is compactified by a Deligne-Mumford stacks over \(\overline{{M}_{g}},\) giving a completion of Néron models naturally stratified in terms of Néron models.
Contents: 1. Introduction. 2. Notation and terminology. 3. The Néron model for the degree\(-d\) Picard scheme. 4. The balanced Picard functor. 5. Balanced Picard schemes and stacks. 6. Néron models and balanced Picard schemes. 7. Completing Néron models via Néron models. 8. The compactification as a quotient. 9. Appendix. Delinge-Mumford stacks; Néron models of Jacobians; Picard scheme Caporaso, L., Néron models and compactified Picard schemes over the moduli stack of stable curves, Amer. J. Math., 130, 1-47, (2008) Families, moduli of curves (algebraic), Jacobians, Prym varieties Néron models and compactified Picard schemes over the moduli stack 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 We show that the natural morphism \(\phi :\pi _{ 1 }(X_{ \eta },x_{ \eta })\rightarrow \pi _{ 1 }(X,x)_{ \eta }\) between the fundamental group scheme of the generic fiber \(X_{ \eta }\) of a scheme \(X\) over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of \(X\) is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed \(G\)-torsor over \(X_{ \eta }\) to be extended over \(X\). We finally provide examples where \(\phi :\pi _{ 1 }(X_{ \eta },x_{ \eta })\rightarrow \pi _{ 1 }(X,x)_{ \eta }\) is an isomorphism. Antei, M.: Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber, Journal de théorie des nombres de Bordeaux 22, No. 3, 525-543 (2010) Homotopy theory and fundamental groups in algebraic geometry, Group schemes, Generalizations (algebraic spaces, stacks) Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0575.00008.]
For an elliptic surface \(f: X\to C,\) C a smooth curve, over the complex numbers, Kodaira's logarithmic transform constructs a compact complex surface X' and an analytic map \(g: X'\to C\) with multiple elliptic fibres of given order over prescribed points \(P_ 1,...,P_ n\) on C such that \(g^{-1}(C-\{P_ 1,...,P_ n\})\) is isomorphic to \(f^{-1}(C-\{P_ 1,...,P_ n\})\). In the present note, an analogue of this construction in characteristic \(p\) is described provided that X is rational. The technical result states that the étale cohomology groups \(H^ i(C,Pic^ 0(X/C))\), \(i=1,2\), are zero and torsion-free, respectively. It is interpreted as the absence of an obstruction to constructing global torsors with prescribed multiple fibres. elliptic surface; global torsors with prescribed multiple fibres Special surfaces, Finite ground fields in algebraic geometry, Families, moduli, classification: algebraic theory, Homogeneous spaces and generalizations, Arithmetic ground fields for surfaces or higher-dimensional varieties An analogue of the logarithmic transform in characteristic p | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Usually, moduli spaces in geometry are singular varieties. In order to avoid the difficulties related to their singular nature, the recently invented ``Derived Deformation Theory Program (DDT)'' aims at developing appropriate versions of the (non-abelian) derived functor of the respective moduli functor. Rather than ordinary varieties or schemes, the resulting geometric objects are sought to be ``dg-schemes'', i.e., geometric objects whose algebras of functions are commutative differential graded algebras, and which are considered up to quasi-isomorphisms. While the DDT program appears to be well-established in the formal case, mainly in view of the recent fundamental work of M. Kontsevich, S. Barannikov, V. Hinich, M. Manetti, and others, the structure of global derived moduli spaces is much less understood.
In this vein, the aim of the paper under review is to provide a comprehensive DDT-type construction of the derived Hilbert scheme. In a foregoing paper [cf.: \textit{I. Ciocan-Fontanine} and \textit{M. Kapranov}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 34, 403--440 (2001; Zbl 1050.14042)], the authors have already constructed a derived version of a first global algebro-geometric moduli space, namely of Grothendieck's wellknown ``Quot scheme''. Using a somewhat similar but nevertheless different approach, the authors are now investigating another important global moduli space in the context of the DDT program. While in the usual algebraic geometry, the Hilbert scheme is a particular case of the Quot scheme, the two constructions turn out to diverge considerably when passing to the framework of derived categories.
More precisely, let \(k\) be a field of characteristic zero, \(X\) a smooth projective variety over \(k\), and \({\mathcal O}_X(1)\) a very ample line bundle on \(X\) defining a projective embedding. For a given polynomial \(h\), the authors construct a dg-manifold \(\text{RHilb}^{\text{LCI}}_h(X)\) as the derived version of the classical geometric Hilbert scheme \(\text{Hilb}_h(X)\) of closed subschemes of \(X\) with Hilbert polynomial \(h\) relative to the polarization \({\mathcal O}_X(1)\). However, when the polynomial \(h\) is identically 1, then the derived Hilbert scheme turns out to coincide with the variety \(X\) whereas the derived Quot scheme \(\text{RQuot}({\mathcal O}_X)\) is known to be different from \(X\).
As for applications of these DDT-type constructions, the earlier constructed dg-manifolds RQuot are suitable for describing the derived moduli spaces of vector bundles on a fixed variety \(X\). In contrast, the dg-schemes \(\text{RHilb\,}h(X)\) established here are expected to play a similar rôle with regard to the derived moduli spaces of projective varieties themselves, which the authors corroborate by two striking examples. Namely, they use the explicit structure of the dg-schemes \(\text{RHilb\,}h(X)\) to construct two types of geometric derived moduli spaces:
(1) the derived space of maps \(\text{RMap}(C,Y)\) from a fixed projective scheme \(C\) to a fixed smooth projective variety \(Y\) and
(2) the derived stack of stable degree-\(d\) maps \(R\overline M_{g,n}(Y,d)\) from \(n\)-pointed nodal curves of genus \(g\) to a given smooth projective variety \(Y\). The latter example completes some earlier work of \textit{M. Kontsevich} [in: The moduli spaces of curves, Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)] and others (Behrend-Manin, Fulton-Pandharipande), thereby contributing to the mathematical theory of Gromov-Witten invariants. All in all, this is a very comprehensive paper of fundamental importance in derived moduli theory. The conceptual ingredients and refined techniques for the construction of derived Hilbert schemes include cotangent complexes, Harrison homology, derived moduli of operad algebras, derived schemes of ideals in finite-dimensional commutative algebras, and the theory of algebraic stacks. In spite of its highly advanced character, the exposition is very detailed, systematic and clear. moduli spaces; derived categories; operad algebras; Gromov-Witten invariants Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail M., Derived Hilbert schemes, J. Amer. Math. Soc., 15, 4, 787-815, (2002) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Supervarieties, Derived categories, triangulated categories, Nonabelian homological algebra (category-theoretic aspects) Derived Hilbert 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 paper gives a very good systematic presentation of the equivalence between the algebraic function fields in one variable over the field \(\mathbb{R}\) of real numbers and the Klein surfaces. In section 1 Klein surfaces and morphisms between them are defined, and example as well as the basic facts about them are given. The double covering of a Klein surface and the quotient of a Riemann surface under an antianalytic involution is described, and it is noted that these two constructions are mutually inverse. Section 2 is devoted to the notion of a meromorphic function of a compact Klein surface. It is shown that the field of meromorphic functions of a compact Klein surface is an algebraic function field in one variable over \(\mathbb{R}\). Also there exists a functor of the category \(\mathcal K\) of compact Klein surfaces to the category \({\mathcal F}_ \mathbb{R}\) of the algebraic function fields in one variable over \(\mathbb{R}\). An intensive study of the set \(S(E\mid\mathbb{R})\) of proper valuation rings \(V\) of \(E\in{\mathcal F}_ \mathbb{R}\) with \(V\supset\mathbb{R}\) is the object of section 3. The main results of this section are:
(a) The residue field of \(V\in S(E\mid\mathbb{R})\) is \(\mathbb{R}\) iff \(E\) admits some ordering with respect to which \(V\) is convex.
(b) The Riemann theorem about the dimension \(\ell(L)\) of the space \(L(D)\) associated to the divisor \(D\) of \(E\mid \mathbb{R}\).
With these notations it is proved in section 4 that \(S(E\mid\mathbb{R})\) admits a unique structure of a Klein surface for which \(p:S(E(\sqrt{- 1})(\mathbb{C})\to S(E\mid\mathbb{R})\) is a morphism of Klein surfaces and \(M(S(E\mid\mathbb{R}))=E\). Further it is shown that every compact Klein surface \(S\) is isomorphic to \(S(M(S)\mid\mathbb{R})\). Also: \(S\mapsto M(S)\) and \(E\mapsto S(E\mid\mathbb{R})\) give an equivalence between \(\mathcal K\) and \({\mathcal F}_ \mathbb{R}\). Here the Klein surfaces with non empty boundary correspond to the formally real fields. Among a series of interesting comments and remarks we mention merely two:
(i) There are several non homeomorphic curves with the same field of rational functions, but there is a unique one among them which is irreducible, compact, non-singular and affine.
(ii) The Klein surface \(S\) with empty boundary is orientable iff \(M(S)\) contains \(\mathbb{C}\).
This carefully written paper is very interesting and recommended even for specialists. algebraic function fields; Klein surfaces; formally real fields Gamboa, JM, Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves, Mem. Real Acad. Cienc. Exact. Fís. Nat. Madr., 27, iv+96, (1991) Arithmetic ground fields for curves, Klein surfaces, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Arithmetic theory of algebraic function fields Compact Klein surfaces with boundary viewed as real compact smooth 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 As the authors point out in the preface, the present text is an expanded version of the lecture notes for a course on Riemann surfaces and A. Grothendieck's ``dessins d'enfants'' (French for ``children's drawings''), which has been taught for several years to students of the masters degree in mathematics at Universidad Autónoma de Madrid, Spain. Accordingly, the text is geared toward seasoned students who have taken courses in linear algebra, general topology, field theory, and basic complex analysis. Otherwise, the prerequisites are kept to a minimum. As the title of the book indicates, the material is divided into two major parts, consisting of two chapters each. Whilst the first part provides an elementary introduction to the classical theory of compact Riemann surfaces, the second part is devoted to the much more recent topic of Bely theory and the allied framework of ``dessins d'enfants''. In this regard, the book under review is among the very few primers on Riemann surfaces that also cover the comparatively modern aspects of the latter topic. Actually, the excellent textbook ``Algèbre et théories galoisiennes'' by \textit{R. Douady} and \textit{A. Douady} [Algèbre et théories galoisiennes. 2ème éd., revue et augmentée. Paris: Cassini (2005; Zbl 1076.12004)] covers much related material, also in an introductory, however by far more abstract fashion, and may therefore be seen as a useful companion to the present representation.
As for the precise contents, the first chapter of the book gives the basic definitions concerning Riemann surfaces, including holomorphic maps, meromorphic functions, differentials, automorphisms, algebraic curves, and numerous concrete examples. Furthermore, the topology of Riemann surfaces, coverings, the Riemann-Hurwitz formula for ramified coverings, the function field of a compact Riemann surface, and the functorial equivalences between compact Riemann surfaces, function fields in one variable, and irreducible algebraic curves, respectively, are discussed in great detail.
Chapter 2 extends the study of compact Riemann surfaces by using the uniformization theorem as a basic tool, thereby explaining the existence of sufficiently many meromorphic functions, Fuchsian groups and their fundamental domains, hyperbolic geometry and Fuchsian triangle groups, automorphism groups of compact Riemann surfaces, monodromy groups, Galois coverings, and the normalization of a covering of \(\mathbb{P}^1_{\mathbb{C}}\).
In the remaining two chapters of the book, this approach to compact Riemann surfaces (or algebraic curves, respectively) is applied to give an introduction to the Grothendieck-Belyi theory of ``dessins d'enfants'', and its connection to Riemann surfaces definable over Chapter 3 is devoted to a proof of \textit{G. V. Belyi's} celebrated theorem from [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267--276 (1979; Zbl 0409.12012)], which is stated here in the following form: Let \(C\) be a compact Riemann surface. Then the algebraic curve associated to \(C\) is given by a polynomial \(F(X,Y)\) with coefficients in \(\overline{\mathbb{Q}}\subset\mathbb{C}\) if and only if there is a morphism \(f:C\to\mathbb{P}^1_{\mathbb{C}}\) with at most three branch points.
The proof given here is a novel, tailor-made and more elementary variant of the original proof, which entirely remains within the scope of the present text. The authors' criterion for definability over \(\overline{\mathbb{Q}}\) was first published by the second author in [Q. J. Math. 57, No. 3, 339--354 (2006; Zbl 1123.14016)], and this is the crucial ingredient of the proof of Belyi's theorem as presented in this chapter of the book.
Chapter 4 introduces Grothendieck's ``dessins d'enfants'', that is, a type of graph drawing suitable for the study of Riemann surfaces and their combinatorial invariants with respect to the action of the Galois group \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on certain coverings (Belyi pairs).
The main goal, in this context, is to give a proof of the so-called Grothendieck correspondence. This fascinating result from the 1980s establishes a one-to-one correspondence between graphs dividing an orientable surface into a disjoint union of cells on the one hand, and algebraic curves \(C\) endowed with a function \(f:C\to P^1_{\mathbb{C}}\) ramified over three points, both with coefficients in \(\overline{\mathbb{Q}}\), on the other. These particular graphs are called ``dessins d'enfants'', whereas the above data \((C,f)\) are commonly known as ``Belyi pairs''. The authors give a very detailed description of Grothendieck's correspondence, including the study of Belyi pairs via Fuchsian groups and some properties of the action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on ``dessins d'enfants'', thereby instructively-illustrating this beautiful relationship between topology, graph theory, complex analysis, algebra, and arithmetic geometry. In particular, this chapter is of very topical character and leads the reader to the forefront of current research in Grothendieck-Belyi theory.
Apart from the overall utmost lucid, detailed, and careful presentation of the material, another outstanding feature of the book under review is the great wealth of concrete, instructive examples illustrating the respective theoretical topics. In fact, about one third of the entire text is devoted to completely worked out examples and clarifying remarks, which certainly helps the reader understand the various abstract concepts, methods, and interrelations discussed in the course of the text. On the other hand, there are no exercises accompanying the main text. Nevertheless, the ambitious reader might try to work out many of the given examples independently, which would provide a large number of exercises, too.
All in all, this textbook gives a very profound first introduction to compact Riemann surfaces and the related Grothendieck-Belyi theory via ``dessins d'enfants'', and it should be seen as a solid, highly valuable basis for the study of the current research literature in the field simultaneously. compact Riemann surface; algebraic curves; Fuchsian groups; uniformization; Galois coverings; Belyi theorem; Belyi function; Belyi pair; definability over \(\mathbb{Q}\) Girondo, E.; González-Diez, G., Introduction to compact Riemann surfaces and dessins D'enfants, London Mathematical Society Student Texts, vol. 79, (2012), Cambridge University Press Cambridge Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Families, moduli of curves (analytic), Coverings of curves, fundamental group Introduction to compact Riemann surfaces and dessins d'enfants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 establishes several fundamental theorems of the theory of integral transforms on derived categories of possibly singular varieties. Let me remark that this paper is written in the language of the derived algebraic geometry, so that a certain amount of knowledge is required.
The first main theorem (Theorem 1.1.3, Theorem 3.0.2) states that we have a one-to-one correspondence between coherent kernels with functors taking perfect complexes over \(X\) to coherent complexes over \(Y\), where \(X\) is a proper relative algebraic space over \(S\) and \(Y\) is a locally Noether \(S\)-stack with \(S\) a perfect stack. It is an analogue of the classical Schwartz kernel theorem in functional analysis.
The second main theorem (Theorem 1.2.4, Theorem 5.0.2) is a relative version of the first one. It establishes the equivalence between kernels which are coherent relative to the source with functors taking coherent complexes to coherent complexes. Here the relative coherence means finite Tor-dimension (Definition 1.2.3).
The proofs of both theorems, in particular of the second one, is based on the analysis of the shrink integral transforms (\S 4). Such a transform do not give an equivalence between the categories of shrink quasi-coherent sheaves. In Theorem 1.3.1 (Theorem 4.0.5) states that it gives an equivalence between categories of ''bounded-above'' sheaves, where the growth is estimated by the usage of t-structures. Fourier-Mukai transforms; coherent sheaves; derived categories; derived algebraic geometry Ben-Zvi, D., Nadler, D., Preygel, A.: Integral transforms for coherent sheaves. arXiv:1312.7164 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories and commutative rings Integral transforms for coherent 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 This work is devoted to study orientation theory in arithmetic geometry within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \textit{arithmetic cohomology theory}, either represented by a Cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a new one for residue morphisms. They are applied to rational motivic cohomology and étale rational \(\ell\)-adic cohomology, as expected by Grothendieck in [\textit{P. Berthelot} (ed.) et al., Théorie des intersections et théorème de Riemann-Roch. Lect. Notes Math. 225. Berlin-Heidelberg-New York: Springer (1971; Zbl 0218.14001)]. Motivic cohomology; motivic homotopy theory, Arithmetic varieties and schemes; Arakelov theory; heights, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, \(K\)-theory in geometry, Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) Orientation theory in arithmetic 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 This memoir generalizes many important results in birational Arakelov geometry about arithmetic divisors to \textit{adelic} arithmetic divisors. An arithmetic divisor on an arithmetic variety is a pair of an \(\mathbb R\)-Cartier divisor \(\mathscr D\) with a single Green function which is a real-valued function (with logarithmic poles on \(|\mathscr D|\)) on the \(\mathbb C\)-valued points. An adelic arithmetic divisor on a projective variety \(X\) over a number field \(K\) is \((D,\{g_{\mathfrak p}\}, g_\infty)\), where \(D\) is an \(\mathbb R\)-Cartier divisor on \(X\), \(g_{\mathfrak p}\) is a Green function on the Berkovich analytification of \(X\times_{\mathrm{Spec }K} \mathrm {Spec }K_{\mathfrak p}\) for each non-archimedean place \(\mathfrak p\), and \(g\) is a Green function on \(X(\mathbb C)\), satisfying some compatibility conditions. The author defines the notion of volume of such divisors as the growth rate of small sections, and derives the adelic-arithmetic-divisor versions of the following results: (1) continuity of the volume function, (2) generalized Hodge index theorem, (3) Fujita's approximation theorem, (4) Zariski decomposition on curves, (5) numerical characterization of nefness on curves, (6) a partial result toward a generalization of Dirichlet's unit theorem. A key is approximating an adelic arithmetic divisor with arithmetic divisors. This follows from the denseness of model functions in the space of continuous functions on the Berkovich space, which in turn is proved similarly as in [\textit{S. Boucksom} et al., J. Algebr. Geom. 25, No. 1, 77--139 (2016; Zbl 1346.14065)]. adelic Arakelov geometry; green functions; Berkovich space; Hodge index theorem; Zariski decomposition; Fujita's approximation; numerical criterion of neffness A. Moriwaki, Adelic divisors on arithmetic varieties, Mem. Amer. Math. Soc. 242, no. 1144, American Mathematical Society, Providence, R.I., 2016 Arithmetic varieties and schemes; Arakelov theory; heights, Heights Adelic divisors on arithmetic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a family of genus \(g\) algebraic curves, with the equation \(f(x,y, \Lambda)=0\), we consider two fiber-bundles \mathsf{U} and \mathsf{X} over the space of parameters \(\Lambda\). A fiber of \mathsf{U} is the Jacobi variety of the curve. \mathsf{U} is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber of \mathsf{X} is the \(g\)-th symmetric power of the curve. We describe the algebraic groupoid structure on \mathsf{X} using the Weierstrass gap theorem to define the `addition law' on its fiber. The addition theorems that are the subject of the present study are represented by the formulas, mostly explicit, determining the isomorphism of groupoids \mathsf{U}\(\to\)\mathsf{X}. At \(\mathrm{g}=1\) this gives the classic addition formulas for the elliptic Weierstrass \(\wp\) and \(\wp'\) functions. To illustrate the efficiency of our approach the hyperelliptic curves of the form \(y^2=x^{2g+1}+\Sigma^{2g-1}_{i=0}\,\lambda_{4g+2-2i}x^i\) are considered. We construct the explicit form of the addition law for hyperelliptic Abelian vector functions \(\wp\) and \(\wp'\) (the functions \(\wp\) and \(\wp'\) form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of \(\wp\) and \(\wp'\)). Addition formulas for the higher genera zetafunctions are discussed. The genus 2 result is written in a Hirota-like trilinear form for the sigma-function. We propose a conjecture to describe the general formula in these terms. Buchstaber, V. M.; Leykin, D. V.: Addition laws on Jacobian variety of plane algebraic curves, Proceedings of the Steklov institute of mathematics 251, 1 (2005) Elliptic functions and integrals, Elliptic curves Hyperelliptic addition law | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 classic monograph originally grew out of a series of lectures delivered by J.-P. Serre at McGill University (Montreal) in 1967 (see the first edition 1968; Zbl 0186.25701). Ever since, over the past thirty years, it provided a standard text on \(\ell\)-adic representation theory and elliptic curves in algebraic number theory and arithmetic geometry. Due to its fundamental depth and literary brilliance, which are of timelessly exemplary splendour, and in regard of the spectacular recent developments concerning the Taniyama-Weil conjecture and Fermat's Last Theorem, which are closely related to the topics treated in this classic standard text, Serre's book has maintained both mathematical actuality and outstanding significance in the existing literature.
In the first reprint of his treatise, which appeared in 1989 (Zbl 0709.14002), the author had enhanced the (otherwise unchanged) text by a few short remarks referring to new results obtained between 1967 and 1988, as well as by a supplementary bibliography listing up a selection of the most important works that appeared during this period. The present book under review is an accurate reproduction of that reprint from 1989. arithmetic ground fields; Hodge-Tate modules; Tate conjecture; \(\ell\)-adic representation; elliptic curves; Taniyama-Weil conjecture Serre, Jean-Pierre, Abelian \(l\)-adic representations and elliptic curves, Research Notes in Mathematics 7, 199 pp., (1998), A K Peters, Ltd., Wellesley, MA Global ground fields in algebraic geometry, Elliptic curves, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local Lie groups, Elliptic curves over global fields Abelian \(\ell\)-adic representations and elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians More than thirty years ago, \textit{L. Illusie} developed his highly abstract and sophisticated theory of cotangent complexes for morphisms of schemes [Complexe cotangent et deformations. I. Lect. Notes Math. 239 (1971; Zbl 0224.13014); Complexe cotangent et deformations. II, Lect. Notes Math. 283 (1972; Zbl 0224.13014)], thereby establishing an utmost general and powerful cohomological framework in algebraic deformation theory, which has found far-reaching applications ever since.
In the meantime, the appearance of logarithmic structures in algebraic geometry, together with their most recent applications to both Gromov-Witten theory and intersection theory on degenerations of algebraic varieties, has brought about the natural question of to what extent L. Illusie's theory of cotangent complexes can be generalized to logarithmic geometry, and how this logarithmic version can be used to understand more complicated deformation-theoretic problems arising there. The paper under review is devoted to exactly this important question. As the author points out, his interest in the development of a logarithmic version of the theory of cotangent complexes for morphisms of fine log-schemes comes from two sources. First, there is some strong demand to generalize \textit{F. Kato}'s so-called ``log smooth deformation theory'' [Tôhoku Math. J., II. Ser. 48, No. 3, 317--354 (1996; Zbl 0876.14007)] in order to tackle specific deformation problems in the logarithmic category. Secondly, the first steps into such a theory of the log cotangent complex have recently been made by \textit{K. Kato} and \textit{T. Saito} [Publ. Math., Inst. Hautes Étud. Sci. 100, 5--151 (2004; Zbl 1099.14009)], and a systematic elaboration of this approach seems to be just as worthwile as promising.
In this vein, the author provides a construction that associates to every morphism of fine log schemes \(f:X\to Y\) a projective system \(L_{X/Y}=(\dots\to L_{X/Y}^{\geq-n-1}\to L_{X/Y}^{\geq -n}\to\cdots\to L_{X/Y}^{\geq 0})\), where \(L_{X/Y}^{-n}\) is an essentially constant ind-object in the derived category of sheaves of \({\mathcal O}_X\)-modules with support in \([-n,0]\). The system \(L_{X/Y}\) is then called the log cotangent complex of the morphism \(f\), and the author shows that this object admits a number of nice functorial properties analoguous to those of L. Illusie's classical cotangent complex for morphisms of ordinary schemes. The construction is based upon the author's stack-theoretic approach to logarithmic geometry [\textit{M. C. Olsson}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 36, No. 5, 747--791 (2003; Zbl 1069.14022)], together with some suitable generalizations of the log stacks introduced there. As an application of his log cotangent complex, the author explains how to compute crystalline cohomology of so-called log complete intersections in terms this log cotangent complex. Modelled on L. Illusie's original approach, the author's logarithmic method leads to a description of the relationship between logarithmic cotangent complexes and deformations of fine log schemes, just as desired. In the sequel, the problem of the existence of a reasonable theory of cotangent complexes for log schemes admitting a distinguished triangle is analyzed. Finally, the author discusses an alternate approach to defining a log cotangent complex for morphisms of log schemes due to O. Gabber (unpublished), including a careful comparison between the two constructions (and their properties) delivered in the present paper. As the author points out in the acknowledgements, this extensive paper grew out of his Ph.D. thesis written under A. Ogus as academic supervisor. simplicial monoids; deformation theory; logarithmic geometry; derived category of sheaves; algebraic stacks; crystalline cohomology M. C. Olsson, ''The logarithmic cotangent complex,'' Math. Ann., vol. 333, iss. 4, pp. 859-931, 2005. \(p\)-adic cohomology, crystalline cohomology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Generalizations (algebraic spaces, stacks), Formal methods and deformations in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Derived categories, triangulated categories The logarithmic cotangent complex | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians These expository notes consist of three chapters and three appendices. The first chapter reviews varieties, curves, divisors, and the Riemann-Roch Theorem. The second chapter deals with the zeta function of a curve over a finite field. The third chapter gives the Bombieri-Stepanov proof of the Riemann hypothesis for curves over finite fields, as one may find in the books of \textit{O. Moreno} [Algebraic curves over finite fields. Cambridge Tracts in Mathematics, 97. Cambridge etc.: Cambridge University Press (1991; Zbl 0733.14025)] and \textit{S. Stepanov} [Codes on algebraic curves. New York, NY: Kluwer Academic/Plenum Publishers (1999; Zbl 0997.94027)]. The first appendix gives scheme-theoretic formulations of the ideas in Chapter 1 and includes the usual proof of the Riemann-Roch Theorem using sheaf cohomology. The second appendix presents Weil's explicit formulas and Oesterlé's use of them to obtain better bounds for the maximum number of points on a curve of a given genus over a finite field. Examples using a Maple program are also given. The third appendix consists of Weil's original proof of the Hasse-Weil bound. curves over finite fields; zeta function; rational point; Riemann hypothesis Curves over finite and local fields, Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Arithmetic ground fields for curves, Rational points, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry Rational points on curves over 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 develop the theory of ``cuspidalizations'' of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute \(p\)-adic version of the Grothendieck conjecture holds for hyperbolic curves ``of Belyi type''. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck conjecture that was shown in [\textit{A. Tamagawa}, Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)]. S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), 451-539. Coverings of curves, fundamental group Absolute anabelian cuspidalization of proper hyperbolic 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 purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme:
(1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).
(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).
One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem.
Let \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.
The number \(g\) is the genus of the ``curve \(C\) upstairs'' of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). -- Note that although in the statement of the theorem we spoke of ``the'' Hurwitz ``scheme,'' there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are ``more important'' than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.
Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:
Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\). finite Picard group; Hurwitz scheme; Hurwitz coverings; admissible coverings S. Mochizuki, ''The geometry of the compactification of the Hurwitz scheme,'' Publ. Res. Inst. Math. Sci., vol. 31, iss. 3, pp. 355-441, 1995. Families, moduli of curves (algebraic), Picard groups, Coverings in algebraic geometry The geometry of the compactification of the Hurwitz scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension 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 Cohomology theories for varieties over a finite field \(k\) of characteristic \(p\) play a crucial role in arithmetic geometry. For example, the Weil conjectures were proved by developing cohomological machinery for such varieties. There are two ``flavors'' of cohomology theories for such varieties: \(p\)-adic, and \(\ell\)-adic for \(\ell \neq p\). Grothendieck introduced \(\ell\)-adic étale cohomology in 1960 in order to solve the Weil conjectures [\textit{A. Grothendieck}, in: Proc. Int. Congr. Math. 1958, 103--118 (1960; Zbl 0119.36902)]. The theory quickly reached maturity, and Deligne completed a proof of the Weil conjectures using \(\ell\)-adic étale cohomology in [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 43, 273--307 (1973; Zbl 0287.14001)]. The \(p\)-adic theory slightly predates the \(\ell\)-adic one: Dwork gave the first proof of the rationality of the zeta function, and his proof made use of \(p\)-adic methods that can now be understood in cohomological terms [\textit{B. Dwork}, Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)]. But the theory has been slower to develop; a complete \(p\)-adic proof of the Weil conjectures was not completed until 2006 [\textit{K. S. Kedlaya}, Compos. Math. 142, No. 6, 1426--1450 (2006; Zbl 1119.14014)].
However, \(p\)-adic cohomological techniques are now reaching a stage of maturity in which they can even be used to prove results that likely cannot be proved using \(\ell\)-adic methods. In this context, the paper under review serves an important role in the mathematical literature: it is a very readable introduction to the theory of convergent and overconvergent \(F\)-isocrystals, key objects appearing in \(p\)-adic cohomology that are analogous to \(\ell\)-adic étale local systems (ie, lisse \(\ell\)-adic sheaves). The paper begins with accessible definitions of isocrystals and then proceeds to collect numerous statements of many important and foundational results about isocrystals. The paper also includes expositions of the theories of slopes and weights (the former of which has no direct \(\ell\)-adic analog). The relevant references for these results are often scattered in papers by many authors and spanning several decades, and often involve long and very technical proofs, so this succinct consolidation is very welcome development for those approaching \(p\)-adic cohomology de novo. Readers interested in working with isocrystals are encouraged to read this paper for details. isocrystals; rigid cohomology; crystalline cohomology; Weil cohomology \(p\)-adic cohomology, crystalline cohomology, \(p\)-adic differential equations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local ground fields in algebraic geometry, Rigid analytic geometry Notes on isocrystals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complex hypersurface in a \(\mathbb{P}^n\)-bundle over a curve \(C\). Let \(C'\to C\) be a Galois cover with group \(G\). In this paper we describe the \(\mathbb{C}[G]\)-structure of \(H^{p,q}(X\times_C C')\) provided that \(X\times_C C'\) is either smooth or \(n=3\) and \(X\times_C C'\) has at most ADE singularities. As an application we obtain a geometric proof for an upper bound by Pál for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. elliptic surfaces; Mordell-Weil rank under base change Elliptic surfaces, elliptic or Calabi-Yau fibrations Chevalley-Weil formula for hypersurfaces in \(\mathbb{P}^n\) over curves and Mordell-Weil ranks in function field towers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(C=V(f_1,\dots,f_{n-1})\) be a reduced curve in the affine space \({\mathbb A}^n\). Pick a polynomial \(f_n\) such that \(C\cap V(f_n)\) is reduced. For any polynomial \(h\) define inductively \(R_0=J^{-1}h\) and \(R_i=J^{- 1}\text{Jac}(f_1,\dots,f_{n-1},R_{i-1})\), where \(J\) is the Jacobian of \(f_1,\dots,f_n\). These objects, computed at the points of \(C\cap V(f_n)\), define a generalization of classical Jacobi residues. The case \(i=1\) is the one considered by B. Segre.
The author shows that, when the degree of \(h\) is bounded by a constant \(N(i)\), then the \(i\)-th residue formula \(\sum_{P\in C\cap V(f_n)} R_i(P)=0\) holds. The number \(N(i)\) depends on the intersection of the projective closure \(C'\) of \(C\), \(V(f_n)\) and the hyperplane \(H\) at infinity. The proof is achieved using the residue theorem on \(C'\). For \(i=1\), the formula extends Segre's result to the case \(C'\cap V(f_n)\cap H\neq\emptyset\). curve in the affine space; Jacobi residues Jacobian problem, Polynomial rings and ideals; rings of integer-valued polynomials, Plane and space curves On a formula of Beniamino Segre | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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{D. Mumford} [Compos. Math. 24, 239--272 (1972; Zbl 0241.14020)] constructed degenerations of abelian varieties to split algebraic tori. \textit{G. Faltings} and \textit{C.-L. Chai} [``Degeneration of abelian varieties'' (1990; Zbl 0744.14031)] generalized Mumford's techniques to describe degenerations of abelian varieties to semi-abelian varieties. As a consequence, they could derive an arithmetic version of a smooth toroidal compactification of the moduli space of principally polarized abelian varieties.
In the first part of the present thesis, the construction of Mumford, Faltings and Chai is generalized to include the case of abelian varieties together with a certain endomorphism structure. It is then applied to study degenerations of abelian 3-folds with complex multiplication in the ring of integers of an imaginary quadratic field, i.e. the modular problem associated to Picard modular surfaces. As a consequence, the author obtains arithmetic compactifications of Picard modular surfaces. One application is the ability to define integral automorphic forms corresponding to unitary groups of signature \((1,2)\). This leads to arithmetic versions of minimal compactifications of Picard modular surfaces. The second part of the thesis gives a description of the relative complete models of Picard modular surfaces and determines their quotients by period groups. Here the main work consists in determining the polarization of the degenerate fibre. Modular and Shimura varieties, Other groups and their modular and automorphic forms (several variables), Arithmetic aspects of modular and Shimura varieties Relatively complete models of Picard modular 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 [For an overview over the entire collection see Zbl 0811.14019.]
These article gives an overview of some results from algebro-geometric intersection theory. This results will be used, in the later chapters, for the main purpose of this collection of lecture notes, i.e., for a complete presentation of G. Faltings' theorem confirming the Mordell conjecture.
Basically, the author provides a brief but coherent account of those methods in algebraic geometry, which are centered around the concepts of ample line bundles on projective varieties, intersection numbers of closed subvarieties with line bundles, numerical equivalence of curves in projective varieties, and the Chow group \(A^ 1(X)\) of a variety \(X\). At the end, two concrete lemmas on intersection numbers on abelian varieties are proved. They are explicitly used in the proof of Faltings' theorem [cf. \textit{C. Faber}, ``Geometric part of Faltings' proof'', in these Proc., chapter IX, 83-91 (1993; Zbl 0811.14023)].
The author's survey is easily accessible for non-experts in algebraic geometry, and most of the material covered here can be found in \textit{R. Hartshorne's} lecture notes ``Ample subvarieties of algebraic varieties'', Lect. Notes Math. 156 (1970; Zbl 0208.489)]. intersection theory; Mordell conjecture; Chow group A. J. de Jong, ``Ample line bundles and intersection theory'' in Diophantine Approximation and Abelian Varieties (Soesterberg, Netherlands, 1992) , ed. B. Edixhoven and J.-H. Evertse, Lecture Notes in Math. 1566 , Springer, Berlin, 1993, 69--76. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves Ample line bundles and intersection theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is a survey of the known results concerning the following problem: Given a smooth projective surface \(X\) and a generic projection \(\pi:X\to {\mathbb{P}}^2\), compute the fundamental group \(\Pi\) of the complement of the branch curve of \(\pi\). The group \(\Pi\) does not change if \(X\) varies in a flat family, therefore if one can degenerate \(X\) to a union of planes such that no three of them have a common line, then the computation is reduced to the case where the branch curve is a union of lines with certain combinatorial properties. In order to describe an algorithm for computing the fundamental group of the complement of a plane curve, the author introduces braid groups and braid monodromy and a suitable form of the Van Kampen theorem and gives some explicit examples.
Notice that one of the main motivations for this kind of computations was the question whether the connected components of the moduli spaces of surfaces of general type correspond to diffeomorphism classes of the underlying \(4\)-manifolds: recently \textit{M. Manetti} [``On the moduli space of diffeomorphic surfaces of general type'' (preprint S.N.S., Pisa 1998)] seems to have given a negative answer to the above question. fundamental group; braid group; algebraic surface; complement of the branch curve; braid monodromy; Van Kampen theorem; diffeomorphism classes M. Teicher, Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 127 -- 150. Homotopy theory and fundamental groups in algebraic geometry, Ramification problems in algebraic geometry, Coverings in algebraic geometry, Families, moduli, classification: algebraic theory, Braid groups; Artin groups Braid groups, algebraic surfaces and fundamental groups of complements of branch curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a field. The Grothendieck ring \(K_0(\text{Var}_k)\) of varieties over \(k\) is the abelian group generated by the isomorphism classes \([X]\) of separated \(k\)-schemes of finite type over \(k\) subject to the relations \([X] = [Y] + [X\backslash Y]\) for \(Y \subset X\) a closed subscheme; multiplication is given by \([X_1] \cdot [X_2] = [X_1 \times X_2]\).
The author gives sufficient conditions for the classes of given varieties over \(k\) to be algebraically independent in \(K_0(\text{Var}_k)\). To this end, he constructs refined versions (so-called motivic measures) of the natural ring homomorphism from \(K_0(\text{Var}_k)\) to the ring of virtual \({\mathbb Q}_\ell\)-representations of \(\text{Gal}(\bar{k}| k)\) given by \(\ell\)-adic cohomology with compact support. The algebraic independence of virtual \(\ell\)-adic representations is then reduced to a problem about representations of (possibly non-connected) reductive groups or is approached using a lemma of Skolem.
If \(k\) is a number field the author finally shows that the classes in \(K_0(\text{Var}_k)\) of elliptic curves \(E_i\) over \(k\) are algebraically independent provided the \(E_i\) are pairwise non-isogenous and satisfy \(\text{End}_{\bar{k}}(\bar{E_i}) = {\mathbb Z}\). If \(k\) is a finite field, the author constructs an infinite sequence of proper smooth and geometrically connected curves over \(k\) whose classes in \(K_0(\text{Var}_k)\) are algebraically independent; furthermore he shows that the class of a single variety is algebraically dependent if and only if its dimension is 0; this leads to the result that \(K_0(\text{Var}_k)\) contains infinitely many zero divisors. Grothendieck ring of varieties; motivic measure; \(\ell\)-adic Galois representations; weight filtration; zeta function; non-isogenous elliptic curves Naumann N.: Algebraic independence in the Grothendieck ring of varieties. Trans. Am. Math. Soc. 359(4), 1653--1683 (2007) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Varieties and morphisms, Motivic cohomology; motivic homotopy theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields Algebraic independence in the Grothendieck ring 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 Let \(K\) be a number field, \(\overline K\) an algebraic closure for \(K\) and \(X\) an algebraic curve over \(K\) of genus \(g\geq 2\). Assume that \(X\) is embedded in its Jacobian \(J\) via a \(K\)-rational Albanese map \(i: X\to J\). The Manin-Mumford conjecture states that the set \(X(\overline K)\cap J(\overline K)^{\text{tors}}\) is finite.
The first proof of this conjecture was provided by \textit{M. Raynaud} [Invent. Math. 74, 207--233 (1983; Zbl 0564.14020) and some years later a second by \textit{R. F. Coleman} [Duke Math J. 541, 615--640 (1987; Zbl 0626.14022)]. In case where \(X= X_0(p)\), with \(p\) a prime \(\geq 23\), the Coleman-Kaskel-Ribet conjecture states that the set of torsion points on \(X\) in the embedding \(i_\infty:X\to J\) is precisely \(\{0,\infty\}\cup H\), where \(H\) is the set of hyperelliptic branch points on \(X\) when \(X\) is hyperelliptic and \(p\neq 37\) and \(H=\varnothing\) otherwise.
In the paper under review, the authors deal with Galois-theoretic techniques for studying torsion points on curves and give new proofs of the above results. modular curve; torsion point; abelian variety; hyperelliptic curve; cuspidal subgroup Baker, MH; Ribet, KA, Galois theory and torsion points on curves, J. Théor. Nombres Bordx., 15, 11-32, (2003) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic aspects of modular and Shimura varieties, Arithmetic ground fields for curves Galois theory and torsion 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 Suppose \(X\) is a finite type scheme, faithfully flat over a complete discrete valuation ring \(R\) with field of fraction \(K\). Let \(X_K\) denote its generic fibre and let \(f: Y \to X_K\) be a \(G\)-torsor, with \(G\) an affine, finite, and algebraic \(K\)-group scheme. In this paper, the authors provide a criterion to extend \(f\) to a torsor over \(X\) whose generic fibre is isomorphic to \(f\). More specifically, they prove the following theorem.
Theorem. Let \(R\) be a complete DVR. Suppose there exist a finite field extension \(K'/K\) and a finite and faithfully flat morphism \(\varphi: Z \to X\) such that \(\varphi^*\varphi_*\mathcal{O}_Z\) is a free \(\mathcal{O}_Z\)-module and \(\varphi_K = \lambda \circ f_{K'}\) where \(\lambda: X_{K'} \to X_K\) and \(F_{K'} : Y_{K'} \to X_{K'}\) are the natural pullback morphisms. If moreover \(\mathcal{O}_Z(Z) = R'\) where \(R'\) is the integral closure of \(R\) in \(K'\), then there is a \(M\)-torsor \(f_1: Y_1 \to X\), for some quasi-finite affine and flat group scheme \(M\) over \(R\), whose generic fibre equals the \(G\)-torsor \(f:Y \to X_K\).
As an application of the main theorem, the authors provide a result to the base change behavior of the fundamental group scheme.
Corollary. Let \(R\) be a (not necessarily complete) DVR and \(X\), \(Y\) as in the last theorem. Assume \(Y \to X_K\) can be extended to some finite and faithfully flat morphism \(\varphi: Z \to X\) such that \(\varphi^*\varphi_*\mathcal{O}_Z\) is a free \(\mathcal{O}_Z\)-module. Then there exists a \(M\)-torsor \(f_1: Y_1 \to X\) for some quasi-finite affine and flat group scheme \(M\) over \(R\), whose generic fibre equals the \(G\)-torsor \(f: Y \to X_K\). torsors; affine group schemes; models; prime to \(p\) torsors Group actions on varieties or schemes (quotients), Group schemes, Arithmetic algebraic geometry (Diophantine geometry) Extension of torsors and prime to \(p\) fundamental group scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a (connected) nodal curve \(X\) defined over an algebraically closed field and a polarisation \(\phi\) of \(X\), one can define a concept of semistability for torsion-free rank \(1\) sheaves on \(X\) and hence, using GIT, construct a coarse moduli space \(\overline{J}(X)=\overline{J}_\phi(X)\) for such sheaves. Any such \(\overline{J}(X)\) is called in this paper a compactified Jacobian of \(X\). (As the authors point out, there are other possible compactifications of the Jacobian of \(X\).) Moreover, by results of \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] and \textit{R. Pandharipande} [J. Am. Math. Soc. 9, No. 2, 425--471 (1996; Zbl 0886.14002)], there exists, for any \(d\), a universal compactified Jacobian \(\overline{J}_{d,g}\) over the moduli space \(\overline{M}_g\) of stable curves of genus \(g\geq2\). At least in characteristic \(0\), the fibre of \(\overline{J}_{d,g}\) over the point corresponding to a curve \(X\) is isomorphic to a compactified Jacobian of \(X\) modulo its automorphism group. The main theorem (Theorem A) of the present paper gives an explicit description of the completed local rings of \(\overline{J}(X)\) at any point and of \(\overline{J}_{d,g}\) at any point corresponding to an automorphism-free stable curve \(X\). The description identifies these local rings with rings of invariants described in terms of the dual graph of \(X\). A consequence of this theorem (Theorem B(i)) states that \(\overline{J}(X)\) has Gorenstein, semi-log-canonical singularities and, in particular, is seminormal. Moreover (Theorem B(ii)), if \(I\) is a polystable torsion-free rank \(1\) sheaf on \(X\), then the corresponding point \([I]\) of \(\overline{J}(X)\) lies in the smooth locus if and only if \(I\) fails to be locally free only at separating nodes. The theorems are proved using deformation theory. nodal curves; semistable sheaves; compactified Jacobians Casalaina-Martin, S; Kass, JL; Viviani, F, The local structure of compactified Jacobians, Proc. Lond. Math. Soc. (3), 110, 510-542, (2015) Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties, Formal methods and deformations in algebraic geometry, Singularities of curves, local rings The local structure of compactified 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 This paper is of interest to those studying number theory and algebraic geometry. The main objects of study are overconvergent isocrystals on a scheme over a field of characteristic \(p >0\). Overconvergent isocrystals can be roughly thought of as modules which are vector bundles with flat connection on a smooth dense open subset and have certain analytic behavior on the boundary of the open subset. The use of analytic methods on a scheme over a field of characteristic \(p >0\) is permitted by realizing the scheme as the special fiber of a scheme over characteristic \(0\). Let \((V,m)\) be a complete discrete valuation ring with \(\mathrm{char}(V/m)=p >0\) and \(\mathrm{char}(\mathrm{Quot}(V))=0\). If \(Z\) is a well-behaved scheme of finite type over \(V/m\) then (locally) there is a well-behaved scheme \(\mathfrak{Z}\) of finite type over \(V\) with special fiber \(Z = \mathfrak{Z} \times_{\mathrm{Spec}(V)} \mathrm{Spec}(V/m)\). Since \(V\) is complete, it is possible to ask analytic questions about functions on \(\mathfrak{Z}\) and its generic fiber. A theorem of Berthelot states that the definition of the category of (convergent) isocrystals for different choices of \(\mathfrak{Z}\) give equivalent categories and naturally respect passing to open subsets.
Let \(X \rightarrow Y\) be a dense open immersion of schemes of finite type over \(V/m\). Define \(\mathrm{Isoc}^{\dagger}(X,Y)\) to be the category of isocrystals on \(X\) overconvergent with respect to \(Y \setminus X\) and \(\mathrm{Isoc}(X)\) the category of (convergent) isocrystals on \(X\). An interesting question to ask is if the restriction on the analytic behavior on the boundary allows morphisms defined on \(X\) to extend uniquely to all of \(Y\). In categorical terms, is the pullback functor
\[
j^*: \mathrm{Isoc}^{\dagger}(X,Y) \rightarrow \mathrm{Isoc}(X)
\]
fully faithful? This conjecture was posed by \textit{N. Tsuzuki} as 6.2.1 in [Duke Math. J. 111, No. 3, 385--418 (2002; Zbl 1055.14022)] where a proof for the unit-root case was provided. Tsuzuki also posed a related version of this conjecture where \(\mathcal{O}_Y\) is replaced by the bounded Robba ring and \(\mathcal{O}_X\) by the Amice ring [2.3.1, loc. cit.]. Tsuzuki proved that Conjecture 2.3.1 [loc. cit.] implies Conjecture 6.2.1 [loc. cit.].
The reviewed paper contains two main results. The first result is a counterexample to Conjecture 2.3.1 [loc. cit.]. The paper proposes that Conjecture 2.3.1 [loc. cit.] may be true if the pullback functor is restricted to the subcategory of modules satisfying a certain condition called the DNL condition. The second result is that this new conjecture implies that the pullback functor is fully faithful when restricted to the category of overconvergent isocrystals satisfying the DNL condition, the analogous version of Conjecture 6.2.1 [loc. cit.]. It is worth noting that \textit{K. S. Kedlaya} [Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter. 819--835 (2004; Zbl 1087.14018)] has shown \(j^*\) is fully faithful if one requires that isocrystals also carry a Frobenius structure. Overconvergent isocrystals with a Frobenius structure satisfy the DNL condition. isocrystals; full faithfulness; counterexample; Tsuzuki's full faithfulness Rigid analytic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Varieties over finite and local fields, \(p\)-adic cohomology, crystalline cohomology Some notes on Tsuzuki's full faithfulness conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the present paper, we study the ordinariness of coverings of stable curves. Let \(f : Y \rightarrow X\) be a morphism of stable curves over a discrete valuation ring \(R\) with algebraically closed residue field of characteristic \(p > 0\). Write \(S\) for Spec\(R\) and \(\eta\) (resp. \(s\)) for the generic point (resp. closed point) of \(S\). Suppose that the generic fiber \(X_\eta\) of \(X\) is smooth over \(\eta\), that the morphism \(f_\eta : Y_\eta \rightarrow X_\eta\) over \(\eta\) on the generic fiber induced by \(f\) is a Galois étale covering (hence \(Y_\eta\) is smooth over \(\eta\) too) whose Galois group is a solvable group \(G\), that the genus of the normalization of each irreducible component of the special fiber \(X_s\) is \(\geq 2\), and that \(Y_s\) is ordinary. Then we have that the morphism \(f_s : Y_s \rightarrow X_s\) over \(s\) induced by \(f\) is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where \(X_s\) is a stable curve. If, moreover, we suppose that \(G\) is a \(p\)-group, and that the \(p\)-rank of the normalization of each irreducible component of \(X_s\) is \(\geq 2\), we can give a numerical criterion for the admissibility of \(f_s\). Coverings of curves, fundamental group, Positive characteristic ground fields in algebraic geometry On the ordinariness of coverings 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 the first place, the present text is the English translation of the author's lecture notes ``Residuen und Dualität auf projektiven algebraischen Varietäten'', the German original of which was published in 1986 at the University of Regensburg, Germany (Zbl 0597.14013). These notes were based on an advanced course for students with a solid background in basic commutative algebra and algebraic geometry, where the main goal of this underlying course was to develop the framework of local and global duality theory in the special case of irreducible algebraic varieties over an algebraically closed ground field \(k\) in terms of sheaves of differential forms and their residues.
In modern algebraic geometry, duality theory was introduced by A. Grothendieck in the 1960s as a conceptual approach to generalize the classical residue theorem and Serre's duality theorem for algebraic curves with a view toward higher-dimensional, possibly singular algebraic schemes. Grothendieck's very general, essentially abstract cohomological duality theory was developed further, in the sequel, by P. Deligne, R. Hartshorne, S. Kleiman, J.-L. Verdier, and others. This general approach is still best described in \textit{R. Hartshorne}'s standard text ``Residues and Duality'' (Zbl 0212.26101) published in 1966. In the following decades, various authors have tried to give concrete interpretations and applications of Grothendieck's abstract duality theory in special mainly down-to-earth cases, thereby relating Grothendieck's ideas to the classical origin in the theory of differential forms and their residue calculus. A crucial breakthrough, in this respect, was \textit{J. Lipman}'s monograph ``Dualizing Sheaves, Differentials, and Residues on Algebraic Varieties'' (Zbl 0562.14003) from 1984, in which the author developed an alternative approach to residues and duality for algebraic varieties in a more adapted and concrete way, mainly by using classical sheaf cohomology. In E. Kunz's course notes, the English translation of which are now under review again, J. Lipman's approach is further ``specified in terms of concrete Kähler differential forms and their (appropriately defined) residues. In view of the rapidly growing current interest in computational algebraic geometry, the author's very explicit description of duality theory for protective algebraic varieties appears to be as topical and useful as it did twenty-two years ago. As for the original German version, we may refer to \textit{R. Berger}'s review of it (Zbl 0597.14013) from 1986, but it should be emphasized that the current English translation has been considerably extended, updated, and enhanced by two additional contributions provided by David A. Cox and by A. Dickenstein.
Now as before, the text is introductory in nature and chiefly geared toward graduate students in algebraic geometry. In this vein, it provides a highly profound, comprehensive, detailed and inspiring introduction to the more advanced literature on the subject, in particular to the related work of J. Lipman and his collaborators in the last two decades.
The new English edition of the book consists of fourteen sections, out of which the first twelve ones form the faithful translation of the German original text. Accordingly, Section 1 provides the basic material on local cohomology functors with values in abelian sheaves, whereas Section 2 discusses some additional prerequisites from the the theory of noetherian affine schemes and their local cohomology. Section 3 describes the canonical map from local to global cohomology in terms of Čech cohomology, where the latter is explained in full detail. Section 4 is devoted to Koszul complexes and their basic properties, with a special focus on the link to both Čech complexes and local cohomology. This approach leads to an explicit calculus for special local cohomology classes, which is then effectively used in the later sections.
After the foregoing preparations, Section 5 introduces residues of local cohomology classes of differential forms for formal power series rings. This construction of residues, which only uses basic properties of Kähler differential forms, appears as a concrete version of Grothendieck's much more general approach in a special case. As a first step in this context, it makes the connection to the classical concept of residues clearly recognizable and, moreover, transpires a first version of the local duality theorem, which is used later to prove a more general duality theorem for protective algebraic varieties. Section 6 recalls some classical facts concerning the cohomology of quasi-coherent sheaves on protective schemes and shows then, as a fundamental tool in the author's approach, how this kind of cohomology is related to the local cohomology in the vertex of the associated affine cone. This is used, in the following Section 7, to derive specific duality and residue theorems for protective spaces, thereby preparing the ground for generalizations to arbitrary protective algebraic varieties and their intersection theory. Section 8 discusses some more background material from differetial algebra, including traces, complementary modules, and differents. This ring-theoretic framework is used in Section 9 to construct the sheaf of regular \(d\)-forms on an integral algebraic variety of dimension \(d\) over an algebraically closed field \(k\). Actually, the author's construction provided here is a simplified, particularly tailor-made variant of his and R. Waldi's much more general theory of regular differential forms developed about twenty years ago [cf.: \textit{E. Kunz} and \textit{R. Waldi}, Regular Differential Forms, Contemp. Math. 79, Amer. Math. Soc., Providence, RI (1988; Zbl 0658.13019)]. Section 10 turns then to the explicit construction of residues of regular differential forms on algebraic varieties. This section culminates in a generalization of the local duality for protective spaces (Section 7) and in further applications of the residue theorem for \(\mathbb{P}^d_k\). Section 11 finally derives the announced duality and residue theorems for arbitrary integral protective varieties by reduction to the previously discussed special case of \(\mathbb{P}_k^d\), thereby exploiting a general method due to J. Lipman. Section 12 adds some related material on complete duality for coherent sheaves and their cohomology on Cohen-Macaulay varieties. Section 13, a contribution to the original text by A. Dickenstein, describes a number of explicit applications of residues and duality in algebraic geometry, including some recent work on the interpolation problem and on the membership problem for polynomials, among other effective methods in concrete algebra and geometry. Section 14, written by D. A. Cox, discusses the more recent concept of toric residues, which was introduced by the contributor himself in 1996 [cf. \textit{D.~A.~Cox}, Ark. Math. 34, 73--96 (1996; Zbl 0904.14029) This is done by giving several versions of the definition of toric residues in order to show the close connections with the earlier sections of the current book. This section leads the reader into the realm of toric geometry, of which a basic knowledge on the reader's side is required, and it demonstrates the significance of E. Kunz's concrete approach to residues and duality by pointing out some toric generalizations of it. The book comes with a very rich and up-to-date bibliography, together with numerous hints for further reading throughout the entire text. Also, there are an utmost carefully compiled index and an exhaustive glossary of notation, which certainly help the reader work efficiently with these excellent lecture notes.
This work, like the numerous other well-known textbooks by E. Kunz, stands out by its exemplary lucidity, thoroughness, detailedness, rigor, and determination. It represents another masterpiece of expository writing in algebra and geometry provided by a renowned researcher and teacher, whose expertise in combining modern developments and classical topics, abstract theories and concrete applications as well as advanced mathematics and didactic principles is truly admirable. algebraic varieties; Cech cohomology; local cohomology; residues; duality theorems; Koszul complex; differential forms; toric varieties; toric residues Kunz, E., Residues and duality for projective algebraic varieties, Univ. lect. ser., vol. 47, (2008), AMS Providence, RI Research exposition (monographs, survey articles) pertaining to algebraic geometry, Projective techniques in algebraic geometry, Birational geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local cohomology and algebraic geometry, Complete intersections, Toric varieties, Newton polyhedra, Okounkov bodies, Residues for several complex variables, Modules of differentials, Computational aspects in algebraic geometry Residues and duality for projective algebraic varieties. With the assistance of and contributions by David A. Cox and Alicia Dickenstein | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By a Picard curve the authors understand a projective plane curve with affine equation \(y^3=p(x)\), where \(p\) is a square-free polynomial of degree 4. Picard curves have genus 3 (being non-singular plane quartics) and a unique point \(\infty\) at infinity; hence any divisor \(D\) of degree 0 on a Picard curve is linearly equivalent to a divisor \(D^*-d^*\infty\), where \(d^*=\deg D^* \leq 3\), and \(D^*\) is an effective divisor supported on finite points, non-collinear if \(d^*=3\). The first part of the paper sketches an algorithm for effecting this reduction, i.e. for finding \(D^*\) given \(D\). The second part obtains an explicit description of the Jacobian of a Picard curve, using the divisors \(D^*-d^*\infty\) as representatives of classes of divisors of degree zero, analogous to the description of the Jacobian of a hyperelliptic curve given by \textit{D. Mumford} [``Tata Lectures on Theta. II: Jacobian theta functions and differential equations'', Prog. Math. 43 (1984; Zbl 0549.14014)].
The paper by \textit{E. R. Barreiro, J. Estrada Sarlabous} and \textit{J.-P. Cherdieu} [in: Coding theory, cryptography and related areas, Proc. Int. Conf., Guanajuato 1998, 13-28 (2000)] is a sequel to the paper under review, in which some improvements are made and some topics clarified. Picard curve; Jacobian Sarlabous, J. Estrada; Barreiro, E. Reinaldo; Barceló, J. A. Piñeiro: On the Jacobian varieties of Picard curves: explicit addition law and algebraic structure. Math. nachr. 208, 149-166 (1999) Computational aspects of algebraic curves, Jacobians, Prym varieties, Computational aspects of higher-dimensional varieties On the Jacobian varieties of Picard curves: Explicit addition law and algebraic structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves an `arithmetic general theorem' which refines a result from an earlier preprint of Ru and Vojta. To formulate the author's result, we introduce the necessary notation. Let \(K\) be a number field or a function field of characteristic \(0\), \(F\) a finite extension of \(K\), \(X\) a geometrically integral irreducible projective variety of \(K\) of dimension \(n\) and \(D_1,\ldots , D_q\) non-zero effective Cartier divisors on \(X\), defined over \(F\), that intersect properly. Further, let \(L\) be a big line bundle on \(X\), defined over \(K\) and \(h_L(\cdot)\) a logarithmic height associated with \(L\). Let \(S\) be a finite set of places of \(K\), \(\lambda_{D_j,v}(\cdot )\) a Weil function on \(X\) associated with \(D_j\) and \(v\in S\) normalized with respect to \(K\) and \(m_S(\cdot , D_j)=\sum_{v\in S}\lambda_{D_j,v}(\cdot )\) the corresponding proximity function. For \(j=1,\ldots ,q\), define the quantity
\[
\beta (L,D_j)=\liminf_{m\to\infty}\frac{\sum_{\ell\geq 1} h^0(X,mL-D_j)}{mh^0(X,mL)}.
\]
In his paper, the author gives various other expressions for \(\beta (L,D_j)\). In particular, he shows that \(\beta (L,D_j)=\int_0^{\infty}\frac{\mathrm{Vol}(L-tD_j)}{\mathrm{Vol}(L)}dt\), where for a line bundle \(M\), \(\mathrm{Vol}(M)=n!\lim_{m\to\infty}h^0(mM)/m^n\).
Then the author's main arithmetic theorem is as follows:
for every \(\epsilon >0\), the inequality
\[
\Big(\min_{1\leq j\leq q} \beta (L,D_j)\Big) \sum_{j=1}^q m_S(x,D_j)\leq (1+\epsilon )h_L(x)\tag{1}
\]
holds for all \(x\in X(K)\) except for a proper Zariski closed subset of \(X\).
In an earlier preprint of theirs, on which the author apparently based his paper, Ru and Vojta proved something similar with smaller quantities instead of the \(\beta (L,D_j)\). In the revised version which was eventually published [Am. J. Math. 142, No. 3, 957--991 (2020; Zbl 1457.32044)], \textit{M. Ru} and \textit{P. Vojta} proved a result similar to that of the author but with instead of (1) the inequality
\[
\sum_{j=1}^q \beta (L,D_j)m_S(D_j, x)\leq (1+\epsilon )h_L(x).
\]
Ru and Vojta proved their result only over number fields and for divisors \(D_j\) defined over \(K\).
The author follows the approach of Ru and Vojta. He makes a reduction to Schmidt's subspace theorem using the ideas of \textit{P. Corvaja} and \textit{U. Zannier} [Am. J. Math. 126, No. 5, 1033--1055 (2004; Zbl 1125.11022); Ann. Math. (2) 160, No. 2, 705--726 (2004; Zbl 1146.11035)] and \textit{P. Autissier} [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 2, 221--239 (2009; Zbl 1173.14016); Duke Math. J. 158, No. 1, 13--27 (2011; Zbl 1217.14020)]. rational points; Diophantine approximation; Schmidt's subspace theorem; filtered linear series; Vojta's conjecture Rational points, Arithmetic varieties and schemes; Arakelov theory; heights, Heights, Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.), Divisors, linear systems, invertible sheaves On arithmetic general theorems for polarized 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 author studies the cohomological Brauer group for product of schemes. The main result for smooth projective \(X\) and \(Y\) over a ground field \(k\) is the following: The canonical sequence
\[
0\longrightarrow \text{Br}'(k)\longrightarrow \text{Br}'(X)\times \text{Br}'(Y) \longrightarrow \text{Br}'(X\times Y)
\]
is exact, provided the following conditions hold: The base-changes \(X^s=X\otimes k^s\) and \(Y^s=Y\otimes k^s\) to the separable closure have Néron-Severi groups without torsion, at least one of the cohomology groups \(H^1(X,\mathscr{O}_X)\) and \(H^1(Y,\mathscr{O}_Y)\) vanishes, and the product \(X\times Y\) contains a zero-cycle of degree one.
This result is a special case of a more general theorem for fiber products \(X\times_SY\), where \(S\) is a locally noetherian base scheme and \(f:X\to S\) and \(g:Y\to S\) are faithfully flat and locally of finite type. Here the assumptions and the result would be more complicated to state, and involve the étale index for the structure morphism \(X\times_SY\to S\). This number \(I(f)\geq 1\) is defined for morphisms \(f:X\to S\) admitting finite étale quasisections of constant degree; the number is then the greatest common divisor of the degrees of such quasisections.
The main idea for the proofs is to use suitable truncated complexes in derived categories of abelian sheaves. Brauer group Brauer groups of schemes The units-Picard complex and the Brauer group of a product | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, the author considers all degree \(n\in\mathbb{N}^*\) flat generically étale covers \(p:\Gamma\longrightarrow X\), marked at a subset \(D\) of cardinality \(d\in\mathbb{N}\), satisfying a natural tangency condition inside \(\mathrm{Jac}(\Gamma)\), where \((X,q)\) is a curve of arithmetic genus \(g>0\), marked at a smooth point and defined over an algebraic closed field. He focuses mainly in the case \(D\subset p^{-1}(q)\), with \(q\) a non-Weierstrass point and \(g\leq d<n\). All such covers are zero divisors of so-called \(d\)-tangential polynomials and factor through the same ruled surface over \(X\). Conversely, any generic \(d\)-tangential polynomial gives back such a cover. When \(X\) is a smooth complex curve, he generates all \(d\)-tangential polynomials at once, in terms of the Baker-Akhiezer function of \((X,q)\). At last, he considers new phenomena in positive characteristic, namely, infinite towers of Artin-Schreier \(1\)-tangential covers wildly ramified at a unique point. This paper is organized as follows : the first section is an introduction to the subject. In Section 2, the author defines (minimal/indecomposable) \(D\)-tangential covers and present a \(D\)-tangential criterion characterizing them by the existence of a meromorphic, so-called \(D\)-tangential function. He restricts to \(d\)-tangential ones and give examples of decomposable ones. In Section 3, the author studies the characteristic polynomials of the tangential functions of flat \(d\)-tangential covers. He shows that their coefficients are holomorphic outside \(q\in X\) and satisfy affine conditions defining the subvariety of \(d\)-tangential polynomials of degree \(n\). He also find conditions under which this subvariety is not empty, calculates its dimension and proves its generic element to be irreducible. Sections 4 and 5 deal with tangential covers as divisors of a ruled surface and with tangential polynomials via the Baker-Akhiezer function, respectively. Section 6 deals with towers of Artin-Schreier \(1\)-tangential covers (\(p>0\)). The author restricts at last to the positive characteristic case and obtains new phenomena. The paper is supported by an appendix where applications to \(d\times d\)-matrix elliptic (as well as rational and trigonometric) KP solitons are given. The author considers a complex rational curve \(X\) with a node or a cusp and obtain polynomial equations for the spectral curves associated to \(d\times d\) matrix KP trigonometric and rational solitons. coverings; curves; singularities, integrable systems; KP solitons Coverings of curves, fundamental group, Coverings in algebraic geometry, Singularities of curves, local rings, Relationships between algebraic curves and integrable systems Tangential covers and polynomials over higher genus 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 nonhyperelliptic complete smooth curve of genus 4. The canonical morphism \(C\to {\mathbb{P}}^ 3\) is an embedding and the image is defined as the intersection of a quadric and a cubic. In this paper we will discuss a geometric way of constructing such equations from the theta divisor on the Jacobian of C.
Recall that the singularities of the theta divisor correspond to linear systems \(| D|\) on C of dimension one and degree 3. For a general curve there are two such systems \(| D|\) and \(| K-D|\), for some special curves there is only one \(| D| =| K-D|\). We will give a construction of the equations of the canonical curve when the theta divisor has two singular points. This construction was suggested by the work of \textit{L.Ehrenpreis} and \textit{H. M. Farkas} [Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, Ann. Math. Stud. 79, 105- 120 (1974; Zbl 0365.30009)], where they show that it works for an unknown but sufficiently general Riemann surface.
We will indicate the construction in the Riemann surface case. Let \({\mathbb{C}}^ 4\) be the universal covering space of the Jacobian. Let x in \({\mathbb{C}}^ 4\) be a point corresponding to a \(g^ 1_ 3\), say \(| D|\), on C. We make the power series expansion \(\theta (y)=f_ 2+f_ 3+..\). of the theta function at x where \(f_ i\) is a homogeneous polynomial of degree i in y-x. The polynomials \(f_ 2\) and \(f_ 3\) may be identified with forms on the canonical space \({\mathbb{P}}^ 3\). As such their zeroes contain the canonical curve. We can prove that if \(| 2D| \neq | K|\), \(f_ 2\) and \(f_ 3\) define the canonical curve. (If \(| 2D| =K\), \(f_ 3=0\) and the method fails.) We will formulate our results algebraically so that they hold for a curve over any field of characteristic not equal to two. theta divisor on the Jacobian of nonhyperelliptic complete smooth; curve; linear systems Kempf, GR, The equations defining a curve of genus \(4\), Proc. Am. Math. Soc., 97, 219-225, (1986) Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Theta functions and abelian varieties, Jacobians, Prym varieties, Arithmetic ground fields for curves, Compact Riemann surfaces and uniformization The equations defining a curve of genus 4 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 curve of genus g defined over an algebraically closed field K, and let \(D=\{D_ t\}_{t\in T}\) be an algebraic family of divisors on C which is parametrized by a curve T. In order to be able to decide whether such a family is ''contained in a linear series'', i.e. whether all the divisors \(D_ t\) are mutually linearly equivalent, \textit{G. Castelnuovo} [R. Accad. Lincei, Rend., Ser. V 15, 337-344 (1906)] introduced a remarkable invariant \((D)\geq 0\) which has the property that \(z(D)=0 \Leftrightarrow\) all the \(D_ t\) are linearly equivalent ( \(\Leftrightarrow D\equiv 0\), i.e. \(D\sim a\times C+T\times A,\) with \(a\in Div(T)\) and \(A\in Div(C));\) for this reason, \textit{C. Torelli} [Rend. Circ. Mat. Palermo 37, 27-46 (1914)] later called z(D) the ''equivalence defect'' of D. Castelnuovo's principal result is that this invariant may be calculated from the fundamental invariants attached to the family by the formula
\[
(*)\quad z(D)=frac{1}{2}\sigma(D,D),
\]
where \(\sigma(D,D)=2n(D)\nu(D)-\gamma(D)\) is the ''Severi-Weil metric'' on Di\(v(T\times C)\), and \(n(D)=(D\cdot t\times C)\) (for \(t\in T)\), \(\nu(D)=(D\cdot T\times P)\) (for \(P\in C)\), and \(\gamma(D)=(D.D).\) From (*) one obtains the inequality \(\sigma(D,D)\geq 0,\) which is often referred to as the ''Inequality of Castelnuovo-Severi'' [cf. \textit{A. Mattuck} and \textit{J. Tate}, Abh. Math. Semin. Univ. Hamburg 22, 295-299 (1958; Zbl 0081.376)].
The main purpose of the paper under review is to investigate the formula (*). As it turns out, if one uses Castelnuovo's original definition of z(D), then (*) is no longer valid if char(K)\(\neq 0\); one therefore has to redefine the invariant z(D) in a suitable way. Two such methods are known: (1) via intersection numbers involving the \(\theta\)-divisor on the Jacobian variety \(J_ C\) of C [due to \textit{G. Castelnuovo}, Atti R. Accad. Lincei, Rend., Cl. Sci. Fis. Mat. Nat. Ser. V 30, 50-55, 99-103, 195-200, 355-359 (1921)] and also \textit{A. Weil} [''Variétés abéliennes et courbes algébriques'' (Paris 1948; Zbl 0037.162)]; and (2) via intersection theory on symmetric products of curves and Schubert's formula (in char(K)\(\neq 0\), due to \textit{J. Igusa} [J. Math. Soc. Japan 1, 147-197 (1949; Zbl 0039.032)]). - In this paper another method is presented, one that does not leave the surface \(X=T\times C\); it is as follows. For any family \(D=\{D_ t\}_{t\in T}\) let \(\delta(D)=\min_{t} \ell(D_ t)\) denote the ''generic dimension'' of D (here, \(\ell(D_ t)=\dim H^ 0(C,{\mathcal L}(D_ t))),\) and let \(var(D)=\sum(\ell(D_ t)-\delta(D_ t))\) denote the ''total variation'' of the family. Then one has: Theorem: Let \(D\in Div(X/T)^+,\) i.e. let D be an effective divisor on \(X=T\times C\) which is flat over T. If \(\delta(D)=1\), then
\[
(**)\quad \nu(D)+var(D)\leq frac{1}{2}\delta(D,D).
\]
Moreover, if in addition we have \(n(D)=g\) and \(var(D)=0,\) then equality holds in (**); i.e. we have \(\nu(D)=frac{1}{2}\sigma(D,D).\) Thus, if we put \(z'(D)=\max(\nu(D')+var(D')),\) where the maximum extends over all divisors \(D'\in Div(X/T)^+\) with \(\delta(D')=1\) and \(D'\equiv D,\) then on the one hand it is easy to see that \(z'(D)=0 \Leftrightarrow\) \(D\equiv 0\) (of course one has \(z'(D)\geq 0),\) and on the other hand we have ''Castelnuovo's formula'': \(z'(D)=frac{1}{2}\sigma(D,D).\) (Also, it is easy to see that in characteristic 0 one has \(z'(D)=z(D)\).) Moreover, from the above theorem it is possible to deduce ''Castelnuovo's formula on Jacobians'', viz.
\[
(***)\quad(T\cdot \alpha_ D^*(\Theta))=frac{1}{2}\sigma(D,D),
\]
where \(\Theta\) denotes the theta- divisor on the Jacobian variety \(J_ C\) of C, \(\alpha_ D:J_ T\to J_ C\) denotes the homomorphism induced by D, and ( \(\cdot)\) denotes the intersection number of the Jacobian \(J_ T\) of T. By a more careful analysis of the above proof one also obtains a formula for the local intersection number \((T\cdot \alpha_ D^*(\Theta))_ x\) (provided that this intersection number is defined), which after summation over \(x\in J_ T\) yields the ''global formula'' (***). algebraic family of divisors on a curve of genus g; Castelnuovo-Severi inequality; finite characteristic; linear series; equivalence defect; theta-divisor on the Jacobian variety Kani E.: Castelnuovo's equivalence defect. J. Reine Angew. Math. \textbf{352}, 24-70 (1984). Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Theta functions and abelian varieties, Projective techniques in algebraic geometry On Castelnuovo's equivalence defect | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 denote by \({\mathcal M}^ 0_ g\) the moduli space of smooth curves of genus \(g\) \((g\geq 3)\) without automorphisms, and by \({\mathcal C}_ g@>\pi>>{\mathcal M}^ 0_ g\) the universal curve over \({\mathcal M}^ 0_ g\). For any integer \(d\), we denote by \(\psi_ d:{\mathcal T}^ d_ g\to{\mathcal M}^ 0_ g\) the universal Picard (Jacobian) variety of degree \(d\); the fiber \(J^ d(C)\) over a point \([C]\) of \({\mathcal M}^ 0_ g\) parametrizes line bundles on \(C\) of degree \(d\), modulo isomorphism. We describe a group \({\mathcal N}({\mathcal T}^ d_ g)\) (which we call the relative Néron-Severi group of \({\mathcal T}^ d_ g)\) defined to be the group of line bundles on \({\mathcal T}^ d_ g\), modulo the relation that two line bundles are equivalent if their restrictions to the fibers of the map \(\psi_ d\) are algebraically equivalent.
Lemma: The Néron-Severi group of the Jacobian of a curve \(C\) with general moduli is generated by the class \(\theta\) of its theta divisor.
We can define an embedding of groups \(\varphi_ d:{\mathcal N}({\mathcal T}^ d_ g)\hookrightarrow\mathbb{Z}\). To describe the group \({\mathcal N}({\mathcal T}^ d_ g)\) is equivalent to finding the generator \(k^ d_ g\) of the image of the map \(\varphi_ d\).
Theorem: For \(d=0,\ldots,g-1\) the numbers \(k^ d_ g\) are given by the following formula: \(k^ d_ g=(2g-2)/\text{g.c.d}(2g-2,g+d-1)\). universal Picard variety; universal Jacobian variety; moduli space of smooth curves; relative Néron-Severi group Kouvidakis, A., The Picard group of the universal Picard varieties over the moduli space of curves, J. Differential Geom., 34, 3, 839-850, (1991) Picard groups, Jacobians, Prym varieties, Families, moduli of curves (algebraic) The Picard group of the universal Picard varieties over the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X_0\) be a projective non-singular curve over \(\mathbb F_q\), \(S_0\) a finite set of closed points, and let \((X,S)\) be obtained from \((X_0,S_0)\) by extension of scalars to an algebraic closure of \(\mathbb F_q\). The relation between cuspidal automorphic representations (for \(\text{GL}(n)\)), and \(n\)-dimensional irreducible smooth \(\overline{\mathbb Q}_\ell\)-sheaves on \(X_0-S_0\). So, shows that the number of isomorphism classes of \(n\)-dimensional irreducible smooth \(\overline{\mathbb Q}_\ell\)-sheaves on \(X-S\), fixed by Frobenius, and with given ramification at \(S\), is finite. The trace formula gives tools to compute it.
In all known cases, it is given by formula resembling a Lefschetz fixed point formula. We give examples of this, and conjecture which cohomology should appear in the hoped for Lefschetz formula. Deligne, P., Comptage de faisceaux \textit{\(\mathcal{l}\)}-adiques, Astérisque, 369, 285-312, (2015) Étale and other Grothendieck topologies and (co)homologies, Algebraic moduli of abelian varieties, classification, Representation-theoretic methods; automorphic representations over local and global fields Counting \(\ell\)-adic 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 Given a cohomological field theory (CohFT) \(\Omega_{g,n}:(V^*)^{\otimes n}\to H^*(\overline M_{g,n})\), under convergence assumptions, it is possible to realize \(V\) as the tangent space of a point of a Frobenius manifold, and \(\Omega_{g,n}\) can be extended to a family of CohFTs over this Frobenius manifold. For such CohFTs (called convergent CohFTs) the reconstruction theorem of Givental-Teleman reconstructs \(\Omega_{g,n}\) from its underlying Frobenius manifold near a semisimple point (up to choice of integration constants), and also extends a neighborhood of this point to a convergent CohFT.
By a result of Hertling, the germ of an \(N\)-dimensional semisimple Frobenius manifold near a smooth point \(p\) on the discriminant locus is of the form \(I_2(m)\times A_1^{N-2}\) for some integer \(m\geq 3\), so in particular all but two idempotent vector fields extend to \(p\).
The main result of the paper under review combines the Givental-Teleman reconstruction theorem with Herting's result to prove that, for \(m=3\), a neighborhood of a smooth point on the discriminant of a semisimple Frobenius manifold can be extended to a convergent CohFT.
As an application the paper shows that a large set of tautological relations obtained from the Givental-Teleman classification for semisimple CohFTs follow from Pixton's generalized Faber-Zagier relations. Frobenius manifolds; discriminant; cohomological field theories; tautological ring Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic) Frobenius manifolds near the discriminant and relations in the tautological ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Many articles on Grothendieck dessins, Belyi functions, hypermaps on Riemann surfaces etc. have at least in part the character of survey articles or give new access to old material. The present article again belongs to this category of papers and tries to shed new light on some old subjects and to make their connection visible. The first subject is the well-known result of \textit{G. V. Belyi} [Math. USSR, Izv. 14, 247-256 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267-276 (1979; Zbl 0409.12012)]:
Theorem 1. A compact Riemann surface \(X\) is isomorphic to the Riemann surface \(C(\mathbb{C})\) consisting of the complex points of a nonsingular projective algebraic curve \(C\) defined over a number field if and only if there is a nonconstant meromorphic function \(\beta\) on \(X\) ramified over at most three points.
Such functions will be called Belyi functions. Of course we may assume that they are ramified over the three points 0, 1 and \(\infty\). The surprisingly simple algorithm found by Belyi to prove the `only if' part of the theorem is reproduced in many later papers and will not be discussed here, we will care about the `if' part only. For this proof Belyi refers to Weil's criteria [\textit{A. Weil}, Am. J. Math. 78, 509-524 (1956; Zbl 0072.16001)]. As a variant finally coming from the same source one may use the explicit knowledge of the Galois groups of the maximal extensions of \(\mathbb{C}(x)\) and \(\overline \mathbb{Q}(x)\) unramified outside three points [\textit{B. H. Matzat}, ``Konstruktive Galoistheorie'', Lect. Notes Math. 1284 (1987; Zbl 0634.12011)]. Both proofs rely on a heavy machinery and it is far from being obvious how to fit the problem precisely to the hypotheses of Weil's criteria, and whether or not such a powerful tool is needed. I hope therefore that another version of this proof will be of some use, but the reader will recognize that we do not leave the neighbourhood of Weil's ideas. In this part of the proof -- lemmas 3 and 4 -- old-fashioned and elementary but still vital and useful algebraic geometry is needed: Zariski topology, generic points and a specialization argument. Similar ideas often have been used in the literature: Shimura's and Taniyama's proof that abelian varieties with complex multiplication may be defined over a number field [\textit{G. Shimura} and \textit{Y. Taniyama}, ``Complex multiplication of abelian varieties and its applications to number theory'' (1961; Zbl 0112.03501); proposition 26] is an early example. In the present paper, certain coverings of \(X\) are the main topological tools. This part (up to lemma 2) is based on the rigidity of triangle groups.
As a last subject, the methods explained in this note give a new proof for the essential part of Popp's result on a conjecture of Rauch about Riemann surfaces with many automorphisms (theorem 5) showing again the close connection of Belyi's theorem to moduli problems. These Riemann surfaces with many automorphisms turn out to be of particular interest for Galois actions (theorem 7). esquisse d'un programme; dessins des enfants; Grothendieck dessins; hypermaps on Riemann surfaces; Belyi functions Wolfart, Jürgen, The ``obvious'' part of Belyi's theorem and Riemann surfaces with many automorphisms. Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser. 242, 97-112, (1997), Cambridge Univ. Press, Cambridge Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Coverings of curves, fundamental group The `obvious' part of Belyi's theorem and Riemann surfaces with many automorphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(C_0\) be an irreducible, projective algebraic curve over a field of characteristic \(0\). Assume that \(C_0\) has one node and let \(s:C\to C_0\) be a normalization. It is natural to expect that the theory of vector bundles on \(C_0\) is related, via the pull-back, to the theory of vector bundles on \(C\) plus some local data on the two points \(P_1\) and \(P_2\) which map to the node \(P\). The author studies the generalized theta divisors on a compactification of the moduli stack on \(C_0\).
In particular, the author considers the Gieseker compactification, obtained by using vector bundles on some modification of \(C_0\). The corresponding stack GVB of Gieseker vector bundles carries a generalized theta divisor \(\Theta\), which is proven to factorize on some natural extension of the moduli VB of vector bundles on \(C\). This factorization turns out to be canonical.
Namely, if VB is the moduli stack of rank \(n\) vector bundles on \(C\), then there is a natural map \(\text{KGL}\to\text{VB}\), where KGL is an extension of GVB, obtained (roughly speaking) by adding to a vector bundle on \(C_0\) the datum of an isomorphism between the fibers of \(s^*(E)\) at \(P_1,P_2\). The author then considers the variety \(\text{PB}=\text{Fl}(E) \times_{\text{VB}} \text{Fl}(F)\), where Fl is the variety of the full flags in a vector space and \(E,F\) are the fibers of the universal bundle on VB over the sections \(P_1\) and \(P_2\). The relations between KGL an PB yields a canonical decomposition:
\[
H^0(\text{GVB}, \Theta^k) \simeq \oplus_{(a,b)\in A\subset{\mathbb Z}^n\times {\mathbb Z}^n} H^0(\text{PB}, \Theta_{\text{PB}}^k(a,b)).
\]
Kausz, I, A canonical decomposition of generalized theta functions on the moduli stack of gieseker vector bundles, J. Algebraic Geom., 14, 439-480, (2005) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians After having fixed two positive integers \(g\geq 3\) and \(d\), one can consider the functor of isomorphism classes of smooth, proper and connected complex curves of genus \(g\), admitting a finite morphism of degree \(d\) to \(\mathbb{P}_{\mathbb{C}}^1\), whose branch divisor is supported at \(2g+2d-2\) distinct points. This moduli problem is known to have a coarse moduli space, denoted by \(\mathcal{H}_{d,g}\), which is a normal, \(\mathbb{Q}\)-factorial and irreducible quasi-projective complex variety of dimension \(2g+2d-5\). The Picard rank conjecture predicts that \(\mathrm{Pic}(\mathcal{H}_{d,g})\otimes \mathbb{Q}=0\). One consequence of this conjecture is the expectation that a certain partial compactification \(\widetilde{\mathcal{H}_{d,g}}\) of \(\mathcal{H}_{d,g}\), obtained by allowing nodal, irreducible curves and non simply branches, should have rational Picard group generated by boundary classes. It is known that the boundary classes can be expressed in terms of ``tautological classes''. In particular the expectation is that \(\mathrm{Pic}(\widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) is generated by tautological classes. The validity of the conjecture was known before the paper under review for \(d=2,3\) and for large \(d\), namely \(d>2g-2\). In the paper under review the authors prove this conjecture under the assumption that \(3\leq d\leq 5\) (Theorem A). The strategy of the proof consists in first showing that for \(d\geq 3\) (resp. \(d\geq 4\)) there are at least two (resp. at least three) divisorial components supported on \(\widetilde{\mathcal{H}_{d,g}}\setminus\mathcal{H}_{d,g}\) whose classes are linearly independent in \(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) (Proposition 2.15). The most delicate part is then to show that \(\mathrm{rk}(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q})\leq 2\) (resp. \(\leq 3\)) for \(d=3\) (resp. for \(d=4,5\)). In order to produce these upper bounds the authors find a suitable open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) which can be expressed as successive quotients of an open in a projective space by the action of explicit linear algebraic groups and whose number of divisorial components in \(\widetilde{\mathcal{H}_{d,g}}\setminus U\) can be explicitly computed. The authors associate to each smooth curve \(C\) of genus \(g\), equipped with a degree \(d\) morphism \(C\rightarrow \mathbb{P}_{\mathbb{C}}^1\) an embedding of \(C\) into the projectification of the associated Tschirnhausen bundle over \(\mathbb{P}_{\mathbb{C}}^1\). A resolution of the structure sheaf of the curve via this embedding is computed by using theorem 2.1 in [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)]. The open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) is obtained via a suitable mixing (depending on \(d\)) of the loci of curves where the associated Tschirnausen bundle and the first bundle in the Casnati-Ekedahl resolution are most generic. The loci are compared with certain Severi varieties as considered in [\textit{A. Ohbuchi}, J. Math., Tokushima Univ. 31, 7--10 (1997; Zbl 0938.14011)]. These Severi varieties are defined as follows. Fix \(m\) a positive integer, consider the Hirzebruch surface \(\mathbb{F}_m\) and \(\tau\subset \mathbb{F}_m\) a section with self-intersection equal to \(m\). Define \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\) as the closure in the linear series \(|d\tau|\) of the locus \(\mathcal{U}_g(\mathbb{F}_m,d\tau)\) parametrizing irreducible, nodal curves of genus \(g\) in \(|d\tau|\). Ohbuchi gives a way to produce from a smooth curve of genus \(g\) equipped with a degree \(d\) morphism to \(\mathbb{P}_{\mathbb{C}}^1\) and a positive integer \(m\) a point in the Severi variety \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\).
\noindent As consequence of this association the authors can show that for any \(d\) and \(m\) larger than an explicit number (depending on \(d\) and \(g\)) the Picard rank conjecture is equivalent to \(\mathrm{Pic}(\mathcal{U}_g(\mathbb{F}_m,d\tau))\otimes\mathbb{Q}=0\) (Theorem B). Hurwitz space; Picard group A. Deopurkar and A. Patel, The Picard rank conjecture for the Hurwitz spaces of degree up to five. Available at http://arxiv.org/pdf/1402.1439v2, 2014. Families, moduli of curves (algebraic), Picard groups The Picard rank conjecture for the Hurwitz spaces of degree up to five | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper is concerned with the computation of the degree of a ``generalized Plücker embedding'' of the Quot scheme \(R_d\) of quotients of degree \(d\), rank \(2\) quotients of \(\mathcal O_{\mathbb P^1}^4\).
This Quot scheme is a fine moduli space and it yields a natural (smooth and irreducible) compactification of the moduli space parametrizing maps of degree \(d\) from \(\mathbb P^1\) to the Grassmannian of lines \(G(2,4)\). The embeddings considered by the author are given by natural line bundles on the Quot scheme, and it is important from the point of view of enumerative geometry to compute its degrees. Such numbers were computed via Quantum Cohomology by \textit{M. S. Ravi, J. Rosenthal} and \textit{X. Wang} [Math. Ann. 311, No. 1, 11--26 (1998; Zbl 0902.14036)] and can be obtained also via the Vafa and Intriligator formula from the work of \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)].
In the paper under review the author computes these numbers using Bott's residues formula by means of a \(\mathbb C^*\) action. The explicit computation in the case of \(d=3\) is carried out in an appendix. Plücker degree; intersection theory; Bott's residues formula; Quot scheme Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives Intersection numbers on the compact variety of rational ruled 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 We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called \textit{jumps}. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property. Jacobians, Prym varieties, Arcs and motivic integration Jumps and motivic invariants of semiabelian 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 \(\pi:X\rightarrow S\) be a flat morphism of algebraic varieties of relative dimension \(n\). The Deligne product is a symmetric multilinear map \((L_0,\dots,L_n)\mapsto\langle L_0,\dots,L_n\rangle\) from \(\mathrm{Pic}(X)^{n+1}\) to \(\mathrm{Pic}(S)\) which can be defined recursively by choosing sections of \(L_0,\dots,L_n\). In the article under review, the authors aim to compute the Deligne product for the morphism \(\pi_{N,m}:\mathcal M_{g,N+m}\rightarrow\mathcal M_{g,N}\), where for any integer \(M\geqslant 0\), \(\mathcal M_{g,M}\) denotes the moduli space of curves of genus \(g\) with \(M\) marked points, and \(\pi_{N,m}\) is defined by forgetting the last \(m\) marked points.
Let \((\mathcal C_{g,N+m},P_1,\dots,P_{N+m})\) be the universal object of the representable functor \(\mathcal M_{g,N+m}\). The Picard group of \(\mathcal M_{g,N+m}\) is a free abelian group generated by \(\widetilde{\ell}_i=P_i^*(K_{N+m})\) (\(i\in\{1,\dots,N+m\}\)) and the Mumford class \(\widetilde{\lambda}\), where \(K_{N+m}\) is the relative dualizing sheaf of \(C_{g,N+m}\). The authors define
\[
T_{N,m}(a_1,\dots,a_{N+m}):=\langle\widetilde{\ell}_1,\dots, \widetilde{\ell}_1,\dots,\widetilde{\ell}_{N+m},\dots,\widetilde{\ell}_{N+m}\rangle,
\]
where \(a_1,\dots,a_{N+m}\) are non-negative integers with \(a_1+\dots+a_{N+m}=m+1\), and each \(\widetilde{\ell}_i\) appears exactly \(a_i\) times in the Deligne product. They then prove two recursive formulae:
\[
T_{N,m+1}(a_1,\dots,a_{N+m},0)=\sum_{i=1}^{N+m}T_{N,m}(a_1,\dots ,a_{i}-1,\dots,a_{N+m}),
\]
\[
T_{N,m+1}(a_1,\dots,a_{N+m},1)=(N+m+2g-2)T_{N,m}(a_1,\dots,a_{N+m}),
\]
which enable them to compute \(T_{N,m}(a_1,\dots,a_N,d_1,\dots,d_m)\) for non-negative integers \(a_1,\dots,a_N,d_1,\dots,d_m\) with sum \(m+1\), and hence to compute the Deligne product \(\langle L_1,\dots,L_{m+1}\rangle\) for \(L_i\) in the subgroup of \(\mathrm{Pic}(\mathcal M_{g,N+m})\) generated by \(\widetilde{\ell}_1,\dots,\widetilde{\ell}_{N+m}\) (\(i\in\{1,\dots,m+1\}\)). Deligne product; moduli spaces of curves; line bundles Families, moduli of curves (algebraic) Deligne products of line bundles over moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper concerns Grothendieck's section conjecture in anabelian geometry: let \(X\) be a smooth, proper, geometrically connected algebraic curve of genus at least 2 defined over a field \(k\) of characteristic zero. Denote by \(G_k\) the absolute Galois group of \(k\), by \(\pi_1(X)\) the étale fundamental group of \(X\), and by \(\pi_1(\overline{X})\) the étale fundamental group of \(\overline{X}:=X\times_k\overline{k}\). These groups naturally sit in an exact sequence
\[
1\rightarrow \pi_1(\overline{X})\rightarrow \pi_1(X) \rightarrow G_k\rightarrow 1.
\]
Every \(k\)-rational point \(x\in X(k)\) defines a continuous group-theoretic section \(s_x:G_k\rightarrow \pi_1(X)\) of the projection map \(\mathrm{pr}:\pi_1(X)\twoheadrightarrow G_k\), determined up to conjugation by the geometric fundamental group \(\pi_1(\overline{X})\). Grothendieck's Anabelian Section Conjecture (GASC) states that if \(k\) is finitely generated over \(\mathbb{Q}\), then this association yields a bijection between \(X(k)\) and the set of conjugacy classes of sections of \(\mathrm{pr}:\pi_1(X)\twoheadrightarrow G_k\). There is also a birational version of the conjecture (BGASC), in which the fundamental groups \(\pi_1(X)\) and \(\pi_1(\overline{X})\) are replaced by the absolute Galois groups \(G_X:=\text{Gal}(K_X^{\mathrm{sep}}/K_X)\) and \(\overline{G_X}:=\text{Gal}(K_X^{\mathrm{sep}}/K_X\overline{k})\) (here \(K_X\) is the function field of the curve \(X\)). Finally, there are \(p\)-adic versions of both the section conjecture and its birational analogue, where \(k\) is taken to be a finite extension of \(\mathbb{Q}_p\) for some prime \(p\). The \(p\)-adic BGASC is now a theorem due to \textit{J. Koenigsmann} [J. Reine Angew. Math. 588, 221--235 (2005; Zbl 1108.14021)] and (in strengthened form) \textit{F. Pop} [Compos. Math. 146, No. 3, 621--637 (2010; Zbl 1210.11072)].
By work of \textit{S. Mochizuki} [Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019)], distinct rational points of \(X\) yield distinct conjugacy classes of sections of \(\mathrm{pr}:\pi_1(X)\rightarrow G_k\) (in both the original and \(p\)-adic settings). But the remaining question of whether every conjugacy class of sections is point-theoretic (i.e. arises from a \(k\)-rational point) is wide open. The idea of the present paper is to attempt a reduction of the GASC and \(p\)-adic GASC to their respective birational versions. To that end, observe that there is a natural surjection \(G_X\twoheadrightarrow\pi_1(X)\), so one may ask whether a given section \(s:G_k\rightarrow\pi_1(X)\) may be lifted to a section \(\tilde{s}:G_k\rightarrow G_X\). This is called the \textit{cuspidalisation problem} for the section \(s\), and if \(s\) is point-theoretic, then such a lifting \(\tilde{s}\) always exists. In this paper, the author investigates a variant of the cuspidalization problem, where the full absolute Galois group \(G_X\) is replaced by a certain quotient \(G_X^{c-ab}\) called the \textit{maximal cuspidally abelian quotient} of \(G_X\).
Section 1 of the paper is devoted to specifying a necessary cohomological condition for a section \(s\) to be point-theoretic: the étale cycle class of \(s\) must be uniformly orthogonal to \(\text{Pic}^\wedge\). The main result of the paper (Theorem 2.3.5) says that if the field \(k\) satisfies a certain cohomological finiteness condition, then a section \(s:G_k\rightarrow \pi_1(X)\) has cycle class uniformly orthogonal to \(\text{Pic}^\wedge\) if and only if it can be lifted to a section \(\tilde{s}:G_k\rightarrow G_X^{c-ab}\). In fact, the author works more generally over fields \(k\) of arbitrary characteristic, and with geometrically pro-\(\Sigma\) fundamental groups, where \(\Sigma\) is a nonempty set of prime numbers not containing \(\text{char}(k)\). The third and final section of the paper presents examples of non-point-theoretic sections of geometric pro-\(\Sigma\) fundamental groups in the case of a \(p\)-adic field \(k\), provided that \(p\not\in\Sigma\). arithmetic fundamental groups; sections; cuspidally abelian fundamental groups Saïdi, The cuspidalisation of sections of arithmetic fundamental groups, Adv. Math. 230 (4-6) pp 1931-- (2012) Coverings of curves, fundamental group, Galois theory, Galois theory The cuspidalisation of sections of arithmetic 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 A theorem of \textit{M. Green} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 4, No. 1, 87--103 (1991; Zbl 0735.14004)] and \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, No.3, 361--401 (1993; Zbl 0798.14005)] (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines Galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the Galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic. Picard varieties; translates of abelian subvarieties; components Pink, Richard; Roessler, Damian, A conjecture of Beauville and Catanese revisited, Math. Ann., 0025-5831, 330, 2, 293-308, (2004) Picard schemes, higher Jacobians, Picard groups, Divisors, linear systems, invertible sheaves A conjecture of Beauville and Catanese revisited | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It was known that a projective, smooth, connected curve \(\bar{Y}\) defined over an algebraically closed field \(k\) of positive characteristic \(p\) can be lifted to characteristic zero, i.e. there is a discrete valuation ring \(R\) of characteristic zero with residue field \(k\) and a relative \(R\) curve \(Y\) with special fibre \(\bar{Y}\).
\textit{F. Oort} [``Some questions in algebraic geometry'' (1995), \url{http://www.math.uu.nl/~oort0109/A-Qnew.ps}] proposed a similar problem, on whether a Galois cover \(\bar{f}: \bar{Y} \rightarrow \bar{X}\) with Galois group \(\Gamma\) can be lifted in the same way to a Galois cover of relative \(R\)-curves. For a general group \(\Gamma\) this is not possible, there are obstructions based on the size of \(\Gamma\) with respect to the genus, the Bertin obstruction [\textit{J. Bertin}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 55--58 (1998; Zbl 0952.14018)] and the Hurzitz tree obstruction [\textit{L. H. Brewis} and \textit{S. Wewers}, Math. Ann. 345, No. 3, 711--730 (2009; Zbl 1222.14045)]. But for cyclic groups all these obstructions vanish. The Oort conjecture claims that a cyclic group cover of order \(p^n\) can always be lifted to characteristic zero.
The breakthrough result of the authors completes the result of \textit{F. Pop} [Ann. Math. (2) 180, No. 1, 285--322 (2014; Zbl 1311.12003)] and the two articles together provide a full proof of the Oort conjecture.
The proof uses the local nature of the lifting problem expressed in terms of the local-global lifting property and restates the problem in terms of a lifting problem of formal power series in terms of rigid analytic geometry. This problem is restated in the language of characters, i.e. elements in \(H^1(\mathbb{K},\mathbb{Z}/p^n \mathbb{Z})\), where \(\mathbb{K}\) is the function field of the curve \(X\) in the generic fibre. Such a character corresponds to a branched cover \(Y \rightarrow X\) and several invariants are attached to a character, like three types of Swan conductors, which measure how bad is the reduction of a cover. The proof uses an induction process based on a detailed study of \(\mathbb{Z}/p\mathbb{Z}\)-extensions which are the building blocks of the induction. branched cover; Galois group; lifting; Oort conjecture Obus, A.; Wewers, S., Cyclic extensions and the local lifting problem, Annals of Mathematics, 180, 233-284, (2014) Automorphisms of curves, Separable extensions, Galois theory, Curves over finite and local fields, Inseparable field extensions, Galois theory and commutative ring extensions, Witt vectors and related rings, Rigid analytic geometry, Coverings of curves, fundamental group Cyclic extensions and the local lifting problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the function field of a smooth and proper curve \(S\) over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(A\) be an ordinary abelian variety over \(K\). Suppose that the Néron model \(\mathcal A\) of \(A\) over \(S\) has some closed fibre \(\mathcal A_s\), which is an abelian variety of \(p\)-rank \(0\).
We show that in this situation the group \(A(K^{\mathrm{perf}})\) is finitely generated (thus generalizing a special case of the Lang-Néron theorem). Here \(K^{\mathrm{perf}}=K^{p^{-\infty}}\) is the maximal purely inseparable extension of \(K\). This result implies in particular that the ``full'' Mordell-Lang conjecture is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai. abelian variety; purely inseparable; strongly semistable; rational point; function field; Harder-Narashima filtration Rössler, D.: On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic, Comment. math. Helv. 90, No. 1, 23-32 (2015) Abelian varieties of dimension \(> 1\), Rational points, Varieties over finite and local fields On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic | 0 |
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