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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 This paper shows how two seemingly unrelated results, the assertions of both of which end up having similar nature, are incarnations of a more general phenomenon. The first result is the Gross-Kohnen-Zagier Theorem, stating that the images of Heegner divisors inside Jacobians of modular curves behave like the coefficients of a modular form of weight \(\frac{3}{2}\). The second one is Zagier's result about traces of singular moduli (i.e., the traces of the values of the normalized \(j\)-invariant on CM points) becoming, up to the appropriate principal part, the coefficients of a weakly holomorphic modular form of weight \(\frac{3}{2}\). Both results have been generalized (the latter to other functions and higher levels, the former to other varieties like Shimura curves), but they were mainly still considered as independent phenomena.
The paper in question proves an analogue of the Gross-Kohnen-Zagier Theorem in quotients that are finer than the Jacobian (named generalized Jacobians). The resulting modular forms (of weight \(\frac{3}{2}\)) are now weakly holomorphic rather than holomorphic (the principal part vanishes when mapped to the Jacobian itself, whence the difference). Moreover, modular functions produce maps from these generalized Jacobians to the complex numbers, and using these maps the results of Zagier and others about singular moduli can also be proven.
In more detail, the Introduction in Section 1 mentions the basic notions and presents the main results. Section 2 introduces the generalized Jacobian \(J_{\mathfrak{m}}(X)\) associated with an effective divisor \(\mathfrak{m}\) on a smooth projective curve \(X\), which can be seen as the quotient of the divisors of degree 0 on \(X\) modulo the (principal) divisors of rational functions on \(X\) whose divisors, roughly speaking, satisfy a certain congruence condition modulo \(\mathfrak{m}\). These generalized Jacobians form an inverse system with respect to the associated effective divisors, all lying over the classical Jacobian \(J(X)\) of \(X\). The kernel of the map \(J_{\mathfrak{m}}(X) \to J(X)\) consists of an additive part with a nice basis and a multiplicative part modulo global constants. A canonical image of line bundles (with specific trivializations and a choice of base point) in \(J_{\mathfrak{m}}(X)\) is also presented. Section 3 skims through the construction of modular curves as the orthogonal Shimura varieties of isotropic even lattices of signature \((1,2)\), with congruence subgroups of \(\mathrm{SL}_{2}(\mathbb{Z})\) as discriminant kernels, and gives some basic information about the associated Weil representations of the metaplectic double cover of \(\mathrm{SL}_{2}(\mathbb{Z})\) and the corresponding spaces of vector-valued modular forms. The Heegner divisors \(Y(d,\varphi)\) are, in this case, sums of classes of positive definite binary quadratic forms of negative discriminant \(-d\) (normalized by the size of the stabilizer), which satisfy some congruence conditions given in terms of the Schwartz function \(\varphi\) on the (finite) discriminant group \(S_{L}\) of the lattice \(L\) with which one works. By choosing a cusp to as the base point, the divisors \(Y(d,\varphi)\) yield normalized divisors \(Z(d,\varphi)\) of degree 0. Note that \(d\) need not be an integer, but a rational number whose denominator is universally bounded in terms of \(L\).
Section 4 proves the main result of the paper for degree 0 divisors, which is the modularity of the degree 0 Heegner divisors \(Z(d,\varphi)\) in the generalized Jacobian associated with a cuspidal effective divisor, once the correct principal part (which one may consider as consisting of Heegner divisors with non-negative discriminants) is added. The terms with \(d=0\) are given in terms of the class of the line bundle of modular forms of weight \(-2\). The term for negative \(d\) can be non-trivial only if \(d\) is minus an integral square, where it is described explicitly in terms of appropriate isotropic lines in the lattice \(L\) and meromorphic functions resembling the ones appearing in the basis for the additive part of the kernel of the map \(J_{\mathfrak{m}}(X) \to J(X)\). In fact, the result is proved not in \(J_{\mathfrak{m}}(X)\) but in a quotient of it, denoted \(J_{\mathfrak{m}}^{\mathrm{add}}(X)\), the map from which to \(J(X)\) is just the additive part of the kernel from above (this is, in some sense, dual to replacing \(\mathfrak{m}\) by a smaller divisor). One also gets rid of the dependence on \(\varphi\) by tensoring \(J_{\mathfrak{m}}^{\mathrm{add}}(X)\) with the space dual to \(\mathbb{C}[S_{L}]\). The proof uses, like Borcherds' proof of the Gross-Kohnen-Zagier Theorem and its generalizations, Serre duality and the construction of Borcherds products. The classical Gross-Kohnen-Zagier Theorem is obtained either as the case \(\mathfrak{m}=0\) of that theorem, or after projecting all the coefficients to \(J(X)\), where all those with non-positive indices vanish. Note, however, that while the classical result only requires the weight and divisor of the Borcherds product (its product formula is just used as means of construction), the current theorem with the generalized Jacobian does require a deeper analysis of the formula defining the Borcherds product.
Section 5 considers harmonic weak Maaß\ forms of weight 0, with meromorphic principal parts. If \(F\) is such a modular function, having no constant term at any cusp, and \(\mathfrak{m}\) is an effective cuspidal divisor that is strictly larger than the polar divisor of \(F\) at any cusp, then the paper constructs a trace map \(tr_{F}\) from \(J_{\mathfrak{m}}(X)\) to \(\mathbb{C}\), and shows that it factors through \(J_{\mathfrak{m}}^{\mathrm{add}}(X)\). Applying this map to the coefficients of the series from Section 4 thus produces a modularity result for singular moduli and its generalizations (of level 1, but for vector-valued generating series). Indeed, the terms with positive \(d\) reduce to the usual traces (sums of values at points) with values in the dual space of \(\mathbb{C}[S_{L}]\), and the non-positive index terms are also evaluated explicitly (an expression for the 0th term is proven only in the case where the modular curve maps to \(X(1)\), since the proof there uses the weight 12 modular form \(\Delta\)). A scalar-valued version, with higher level, is also given, which encompasses the known results about modularity of generating series of traces of modular functions. Finally, Section 6 investigates some generalization in several directions. The first one is concerned with discarding the condition that the divisors have degree 0. Then a similar modularity phenomenon for the divisors \(Y(d,\varphi)\) is shown, but involving a certain non-holomorphic Eisenstein series of weight \(\frac{3}{2}\). The second direction involves twisting the Heegner divisors with genus characters (which in particular annihilates the 0th coefficient), where modularity is obtained also in this setting, which in particular allows the authors to reproduce and generalize in this manner some known results about twisted traces of singular moduli. Applicability of this method for higher-dimensional Shimura varieties is also mentioned, with some details about the 2-dimensional case of \(X(1) \times X(1)\) being given explicitly. Heegner points; Generalized Jacobians; Singular Moduli; Borcherds Products; Harmonic Maass Forms Modular and Shimura varieties, Jacobians, Prym varieties, Theta series; Weil representation; theta correspondences, Fourier coefficients of automorphic forms Heegner divisors in generalized Jacobians and traces of singular moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a curve, an embeddable noetherian scheme of pure dimension 1. First introduced by Hartshorne, the \textit{generalized divisors} are non-degenerate fractional ideals of \(\mathcal{O}_X\)-modules. Generalized divisors up to linear equivalence are equivalent to \textit{generalized line bundles}, i.e. pure coherent
sheaves which are locally free of rank 1 at each generic point. Further generalization is the notion of \textit{torsion-free sheaves of rank 1}.
Now, let \(\pi: X\rightarrow Y\) be a finite, flat morphism between noetherian curves. The goal of the paper under review is to define and study direct and inverse image for generalized divisors and for generalized line bundles on \( X \) and on \(Y\). Moreover, in the cases where \( X \) and \(Y\) are projective curves over a field (possibly
reducible, non-reduced) and the codomain curve is smooth, the author discusses the same notions for families of effective generalized divisors, parametrized by the Hilbert scheme. The same assumptions are required to introduce the notion of compactified Jacobians parametrizing torsion-free rank-1 sheaves and to study the Norm and the inverse image maps between them. Finally, the author consider the fibers of the Norm map and introduces the Prym stack as the fiber over the trivial sheaf. generalized divisors; generalized line bundles; norm map; compactified Jacobians; Prym variety Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Algebraic moduli problems, moduli of vector bundles The direct image of generalized divisors and the norm map between 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 work represents the author's doctoral dissertation elaborated at the University of Münster, Germany, with C. Deninger as academic adviser. Its main object of study is A. Weil's classical explicit formula relating the zeros and the poles of the zeta function of an abelian variety over a finite field and its analytic interpretation in the framework of dynamical systems on foliated spaces. Such a connection has recently been predicted by \textit{C. Deninger} [Number theory and dynamical systems on foliated spaces, Jahresber. Dtsch. Math.-Ver. 103, No. 3, 79--100 (2001; Zbl 1003.11029)], in particular with a view to the analysis of so-called ``generalized solenoids with \(\mathbb{R}\)-action'', which appear as foliated spaces with a foliation by Kähler manifolds.
In this context, the author's central result is an analytical index theorem for the de Rham-Laplace operator along a certain complex foliation on a generalized solenoid. This is achieved by elaborating a transversal index theory for differential operators on generalized solenoids by analogy with \textit{M. F. Atiyah's} theory of elliptic operators with respect to actions of compact transformation groups [Elliptic operators and compact groups, Lect. Notes Math. 401. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0297.58009)], and applying then the Poisson sum formula. As the author demonstrates in the course of his work, his formula for the transversal index of the de Rham operator on a generalized solenoid may be regarded as analytic version of Weil's explicit formula for the zeta function of an abelian variety over a finite field, just as conjectured by C. Deninger a few years ago. All the necessary facts from the analytic theory of foliations, solenoids, and abelian varieties are carefully surveyed and appropriately woven into the text. finite ground fields; zeta functions; foliations; dynamical systems; differential operators; index theory Arithmetic ground fields for abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Abelian varieties of dimension \(> 1\), Foliations in differential topology; geometric theory, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) On abelian varieties and the transversal index theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper proves a relative Lefschetz formula for some equivariant arithmetic schemes. More specifically, let \(X\) and \(Y\) be arithmetic schemes whose generic fibers are smooth. Assume that \(X\) admits an automorphism of order \(n\), and let \(f:X\to Y\) be the composition of an \(\mu_n\)-equivariant closed immersion \(X\hookrightarrow Z\) and an \(\mu_n\)-equivariant morphism \(Z\to Y\), where \(Z\) is a regular arithmetic scheme and the \(\mu_n\)-action on \(Y\) is trivial. The main theorem (Theorem 6.1) is the Lefschetz formula for such \(f\). This answers a conjecture by \textit{V. Maillot} and \textit{D. Rössler} [in: From probability to geometry II. Volume in honor of the 60th birthday of Jean-Michel Bismut. Paris. Astérisque 328, 237--253 (2009; Zbl 1232.14016)], and it is an Arakelov-geometry analog of \textit{R. W. Thomason} [Duke Math. J. 68, No. 3, 447--462 (1992; Zbl 0813.19002)] as well as a generalization of an earlier work of the author [J. Reine Angew. Math. 665, 207--235 (2012; Zbl 1314.14049)] to a singular case. The formula takes place in the equivariant arithmetic Grothendieck groups \(\widehat {G_0}\), which are defined with respect to fixed wave front sets. The bulk of the paper consists of developing this \(\widehat {G_0}\)-theory and of proving two key results: the arithmetic concentration theorem (Theorem 5.5) and the vanishing theorem (Theorem 6.3). The paper has a thorough history of Lefschetz formula in various settings in Section 1, and Section 2 recalls necessary differential-geometric facts, such as equivariant Chern-Weil theory, equivariant analytic torsion forms, equivariant Bott-Chern singular currents, and Bismut--Ma immersion formula. fixed point formula; singular arithmetic schemes; Arakelov geometry Riemann-Roch theorems, Arithmetic varieties and schemes; Arakelov theory; heights, Group actions on varieties or schemes (quotients), Index theory and related fixed-point theorems on manifolds, Determinants and determinant bundles, analytic torsion A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author investigates the generalized Jacobian conjecture, i.e. whether an étale endomorphism \(\varphi\) of an algebraic variety \(X\) defined over an algebraically closed field of characteristic zero is an automorphism.
This generalized problem was first posted by \textit{M. Miyanishi} [Osaka J. Math. 22, No. 2, 345--364 (1985; Zbl 0614.14006)] who gave counterexamples and positive answers for certain varieties. Notably, these results seemed to underline the special role of \(\mathbb{C}^n\).
In the paper under review, the author treats the case of complements of affine plane curves
\(X={\mathbb{A}}^2\setminus C\) and shows that the generalized Jacobian conjecture is true except for
\(C=\{y^m-x^n =0\}\) with \(\gcd(m,n)=1\). In an appendix (written together with M. Miyanishi) it is proved that
for such curves there exist étale endomorphisms of arbirarily large degree. generalized Jacobian conjecture Aoki, Hisayo, Étale endomorphisms of smooth affine surfaces, J. Algebra, 226, 1, 15-52, (2000) Jacobian problem, Automorphisms of surfaces and higher-dimensional varieties, Coverings in algebraic geometry Étale endomorphisms of smooth affine 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 main purpose of the book under review is to introduce the basic concepts and methods of modern algebraic geometry to novices in the field, requiring just a solid knowledge of linear algebra as a prerequisite. More precisely, the author's intention is to explain some foundational principles of A. Grothendieck's theory of algebraic schemes and their morphisms, together with the necessary (and closely related) background material from commutative algebra. As the author points out in the preface, the present text grew out of his repeated courses and seminars on the subject, which usually consisted of one semester of commutative algebra and then continued with two semesters of scheme-theoretic algebraic geometry. In this approach, commutative algebra is treated as a separate first part, while the second part deals with four selected, general topics in algebraic geometry based on just that commutative-algebraic framework.
Accordingly, the book consists of two major parts, each of which is subdivided into several chapters and their respective sections.
Part A is titled ``Commutative Algebra'' and starts with an instructive and motivating introduction through examples from arithmetic.
Chapter 1 is devoted to the basic notions and results on commutative rings, their ideals, and their modules. Local rings, the localization principle, finiteness conditions for modules, and some fundamental exact sequences for modules are explained as well. Chapter 2 discusses rings with chain conditions, in particular the primary decomposition of ideals in Noetherian rings, Artinian rings and modules, the Artin-Rees Lemma, and the concept of Krull dimension of a ring. Chapter 3 presents the classical theory of integral ring extensions, including the Noether Normalization Lemma, Hilbert's Nullstellensatz, and the crucial Cohen-Seidenberg theorems. Chapter 4 turns to the process of coefficient extension for modules and its reverse, the so-called principle of descent. Apart from some foundational material on tensor products of modules, flat modules, and base change, the focus is here on the faithfully flat descent of modules and their homomorphisms, including a complete proof of Grothendieck's Fundamental Theorem on faithfully flat ring homomorphisms. In addition, the language of categories and functors is briefly explained in an extra section of this chapter. Homological methods in commutative algebra are the theme of Chapter 5, the last chapter of Part A. The main objects of study are the functors Ext and Tor, for the construction of which the general machinery of complexes, projective and injective resolutions, homology, and cohomology is developed.
Part B comes with the heading ``Algebraic Geometry'' and comprises four large chapters, each of which deals with its own separate basic topic. After an example-driven, motivating introduction to algebraic geometry via polynomial equations, algebraic sets, their morphisms and coordinate rings, finitely generated algebras, and their functorial properties, Chapter 6 is concerned with spectra of rings, general sheaf theory, inductive and projective limits, affine schemes and their quasi-coherent module sheaves, direct and inverse images of module sheaves on affine schemes, and the respective construction techniques and basic results. Chapter 7 is devoted to global schemes and their properties, including topics such as the construction of schemes by gluing, fiber products of schemes, subschemes and immersions, separated schemes, Noetherian schemes and their dimension, Čech cohomology of schemes, and Grothendieck cohomology of schemes.
Chapter 8 turns to the study of special morphisms of schemes. Using Kähler differential forms and sheaves of differential forms, morphisms of finite type and of finite presentation, unramified morphisms, étale morphisms, and smooth morphisms of schemes are described in greater detail. In particular, smooth morphisms are characterized by means of the Jacobian criterion in quite a novel fashion. Chapter 9 finally introduces the topic of projective schemes, invertible sheaves, divisors, proper morphisms, and projective embeddings.
Apart from the usual basic material on homogeneous prime spectra, Proj-schemes associated to graded rings and to quasi-coherent sheaves of algebras, respectively, Cartier and Weil divisors, invertible sheaves and Serre twists, global sections of invertible sheaves, finite morphisms, separated morphisms, and proper morphisms of schemes, there are some methodological extras in this chapter. Namely, ample invertible sheaves are defined via the use of quasi-affine schemes, on the one hand, and a proof of the projectivity of abelian varieties is given as an instructive application of ampleness on the other. Along the way, several fundamental theorems are stated and explained, as for example: Chow's Lemma, the Proper Mapping Theorem, and the Stein Factorization Theorem. The proofs of these further-leading results can easily be studied with the background knowledge provided by this last chapter of the book.
As a special feature of the entire exposition, each chapter comes with its own introduction, where the author motivates the respective contents by illustrating examples and spotlights the main aspects. Furthermore, each single section of the book ends with a carefully compiled selection of related exercises, most of which provide additional concepts, constructions, and results, thereby helping the reader develop both his understanding and his extended knowledge in a self-reliant way. Also, a useful glossary of notations, a comprehensive index, and a list of hints for further reading facilitate working with this textbook considerably.
Altogether, the current book is an excellent primer on the elements of modern algebraic geometry and commutative algebra. The lucid exposition bespeaks once more the author's didactic mastery, his rich teaching experience, and his user-friendly style of mathematical writing, as those are already well-known from his bestselling German textbooks [\textit{S. Bosch}, Algebra. (Algebra.) 7th revised ed. Springer-Lehrbuch. Berlin: Springer. viii, 376 p. (2009; Zbl 1163.00003) and Lineare Algebra. (Linear algebra). (Lineare Algebra.) 2nd revised ed. Springer-Lehrbuch. Berlin: Springer. x, 295 S. (2003; Zbl 1016.15001)]. algebraic geometry (textbook); commutative algebra (textbook); schemes; sheaves; sheaf cohomology; projective immersions; abelian varieties; Kähler differentials Bosch, S.: Algebraic Geometry and Commutative Algebra. Springer, London (2013) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Foundations of algebraic geometry, General commutative ring theory, Relevant commutative algebra, Schemes and morphisms, Divisors, linear systems, invertible sheaves, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local theory in algebraic geometry, Cycles and subschemes, Variation of Hodge structures (algebro-geometric aspects) Algebraic geometry and commutative algebra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [\textit{A. Grothendieck} (ed.), Documents Mathématiques (Paris) 3. Paris: Société Mathématique de France. (2003; Zbl 1039.14001)] SGA1, Exposé V, No. 4; an axiomatic approach to Galois theory was introduced. The motivation behind this approach was the desire to define the fundamental group of a scheme.
In topology, the fundamental group of a topological space \(X\) with given base point \(x_0\) can be defined as the automorphism group of a universal cover \(\pi: \widetilde{X} \to X\). Here, \(\widetilde{X}\) is connected, \(\pi\) is a covering map; and \(\pi\) is universal among the coverings of \(X\) in the following sense: If another cover \(\sigma: Y \to X\) is given (with \(Y\) connected), there exists a covering morphism \(\widetilde{X} \to Y\) commuting with the maps to \(X\).
However, when one considers schemes, this method is not directly applicable as it would then be impossible to define the universal cover of a given scheme \(X\). Instead, one works with finite étale covers of \(X\); which produce a Galois category with fiber functor given by fibers over a fixed base point. The automorphism group of the fiber functor gives a group called the étale fundamental group of \(X\), which has a natural structure as a profinite group. In the case of a complex variety, this is simply the profinite completion of the topological fundamental group. (This fact is further discussed in section 8.)
The paper under review is an exposition of this theory of Grothendieck and some of its applications. Let us describe the contents of each section.
In section 2, the author defines, following SGA1, the \textit{Galois category}. This is a category \(\mathcal{C}\) together with a covariant functor \(F\) from \(\mathcal{C}\) into the category of finite sets satisfying six axioms. After studying some properties of Galois categories, the author presents the main theorem of the section. The \textit{fundamental group} \(\pi_1(\mathcal{C}; F)\) of \(\mathcal{C}\) with base point \(F\) is defined as the automorphism group of \(F\); it is a profinite group. The main theorem states that there is an equivalence between \(\mathcal{C}\) and \(\mathcal{C}(\pi_1(\mathcal{C}; F))\) where the latter functor denotes the category of finite sets with a continuous left action of \(\pi_1(\mathcal{C}; F)\). Section 3 is devoted to the proof of this theorem. In section 4, the author proves that there is a natural equivalence between the category of Galois categories pointed with fiber functors and the category of profinite groups.
The remaining sections deal with applications to algebraic geometry. In section 5, the author discusses the étale covers of a scheme. After a subsection on the properties of \'{etale} maps of schemes, the main theorem of this section is proved. This theorem states that the category of étale covers of a connected scheme \(S\), along with the fiber functor defined by a geometric point of \(S\); is a Galois category. The associated profinite group is called the \textit{étale fundamental group} of \(S\) with the given point as base point.
In section 6, examples of the theory of the étale fundamental group are given. It is well known that the fundamental group of the spectrum of a field \(k\) is isomorphic to the absolute Galois group of \(k\); a proof is given. The author goes on to prove the first homotopy sequence and the product formula. In 6.3, a formula for the fundamental group of an abelian variety is given.
In section 7, the author presents the well-known short exact sequence relating the fundamental group of a scheme and the fundamental group of its extension to the separable closure of the base field. Let \(S\) be a scheme geometrically connected and of finite type over a field \(k\), and let \(s_0\) be a geometric point of \(S\). Let \(k^s\) denote the separable closure of \(k\). The canonical maps \((S_{k^s},s_0) \to (S,s_0) \to (\mathrm{spec}(k),s_0)\) induce a short exact sequence
\[
1 \to \pi_1(S_{k^s},s_0) \to \pi_1(S,s_0) \to \pi_1(\mathrm{spec}(k),s_0)=\mathrm{Gal}(k^s/k) \to 1.
\]
In other words, the fundamental group can be said to have two parts: a ``geometric'' part, and an ``arithmetic'' part. We note that the geometric part can sometimes be calculated, in characteristic 0, using the fact that the étale fundamental group is the profinite completion of the topological fundamental group (section 8). Section 7 ends with a discussion on the section conjecture. Section 8 includes a discussion of Serre's GAGA theorem and its well-known consequence that for a scheme that is locally of finite type over the complex numbers, the étale fundamental group is the profinite completion of the topological fundamental group.
Section 9 contains statements on the semicontinuity of fundamental groups, the first and the second homotopy sequences. In section 10, the étale fundamental group is proved to be a birational invariant of proper regular schemes over a field using the Zariski-Nagata purity theorem. In section 11, it is proved that the étale fundamental group of a proper connected scheme over an algebraically closed field is topologically finitely generated (that is, as a profinite group). There is also an appendix on descent theory.
In the opinion of the reviewer, this is a very good survey article for those interested in Galois theory, the étale fundamental group and related topics; and it can be used in conjunction with standard texts on this topic such as SGA1, or the recent book by [\textit{T. Szamuely}, Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics 117. Cambridge: Cambridge University Press. (2009; Zbl 1189.14002)]. Galois categories; algebraic geometry; étale fundamental group; arithmetic geometry Cadoret, Anna, Galois categories. Galois categoriesArithmetic and Geometry around {G}alois Theory, Progr. Math., 304, 171-246, (2013) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory, Generalizations (algebraic spaces, stacks), Homotopy theory and fundamental groups in algebraic geometry Galois 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 In the paper under review, the author studies the Betti numbers and Poincaré polynomial of the moduli space of certain parabolic bundles over a curve. The main result is a closed form for the Poincaré polynomial. The main ideas are to use Weil conjectures (Deligne's theorem) and to solve a recursive formula by using a method of Zagier. In section two, the author reviews basic facts about parabolic bundles, parabolic stability, and parabolic moduli spaces. The quasi-parabolic Siegel formula for parabolic bundles on a curve over a finite field \(\mathbb{F}_q\) is also recalled. In section three, it is assumed that the parabolic semistability implies the parabolic stability. Under this condition, by using the quasi-parabolic Siegel formula, the author obtains a recursive formula for the Poincaré polynomial of the moduli space of parabolic stable bundles over a curve. In section four, substitutions \(\omega_i \to -t^{-1}\) and \(q \to t^{-2}\) are justified. These substitions provide a recipe to compute the Poincaré polynomial directly from the computation of the \(\mathbb{F}_q\)-rational points. In section five, the author solves the recursive formula obtained in section three and obtains a closed formula for the Poincaré polynomial. The idea is to generalize a method of Zagier to the parabolic setting. Finally, the last section (section six) contains some explicit examples. Betti numbers; parabolic vector bundles; moduli space; Poincaré polynomial; Weil conjectures Holla, Y.I.: Poincaré polynomial of the moduli spaces of parabolic bundles. Proc. Indian Acad. Sci. Math. Sci. 110 (3), 233-261 (2000) Vector bundles on curves and their moduli, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Algebraic moduli problems, moduli of vector bundles Poincaré polynomial of the moduli spaces of parabolic bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review the author investigates Grothendieck rings appearing in real geometry, most notably for arc-symmetric sets, and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by \textit{C. McCrory} and \textit{A. Parusiński} [Math. Sci. Res. Inst. Publ. 58, 121--160 (2011; Zbl 1240.14012)]. Let \(S \subseteq \mathbb{R}^n\) be a semialgebraic set. A semialgebraically constructible function on \(S\) is an integer valued function that can be written as a finite sum \(\sum_{i \in I} m_i 1_{S_i}\), where for each \(i \in I\), \(m_i\) is an integer and \(1_{S_i}\) is the characteristic function of a semialgebraic subset \(S_i\) of \(S\). Semialgebraically constructible functions form a commutative ring that will be denoted by \(F(S)\). Dealing with real algebraic sets rather than semialgebraic ones, the push-forward along a regular mapping of the characteristic function of a real algebraic set can not be expressed in general as a linear combination of characteristic functions of real algebraic sets. Nevertheless, the set of all such push-forwards forms a subring \(A(S)\) of \(F(S)\), which is endowed with the same operations. \textit{R. Cluckers} and \textit{F. Loeser} noticed in the introduction of [Invent. Math. 173, No. 1, 23--121 (2008; Zbl 1179.14011)] that \(F(S)\) is isomorphic to the relative Grothendieck ring of semialgebraic sets over \(S\), the push-forward corresponding to the composition with a semialgebraic mapping. The aim of the paper is to continue the analogy further in order to relate the rings of algebraically constructible functions and Nash constructible functions to Grothendieck rings appearing in real geometry. If \(S\) is a real algebraic variety, the author shows that the relative Grothendieck ring \(K_0(\mathbb{R}Var_S)\) of real algebraic varieties over \(S\) maps surjectively to the ring \(A(S)\) of algebraically constructible functions, together with an analogous statement for the ring of Nash constructible functions \(N(S)\). applications of methods of K-theory; constructible functions; semialgebraic sets Applications of methods of algebraic \(K\)-theory in algebraic geometry On Grothendieck rings and algebraically constructible 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 The article targets the following problem: given a scheme \(X\), find a proper birational morphism \(Y\to X\) with the property that the geometry of \(Y\) is simpler. Simplification ususally is formulated in terms of some special property of schemes. E.g., one version of this problem (when we ask for the smoothness) requires the cosntruction of the resolution. Weaker versions ask about weaker particular properties, some of them formulated in terms of local cohomology. E.g., one requires that \(Y\) must satisfy Serre's \(S_2\) condition, or (version introduced by Faltings) \(Y\) required to be Cohen-Macaulay. In this sense the existence of the `Macaulayfication' was proved by \textit{T. Kawasaki} [Trans. Am. Math. Soc. 352, No. 6, 2517--2552, appendix 2541--2547, 2548--2552 (2000; Zbl 0954.14032)]. In the present paper the authors consider generalized Serre conditions \(S_\rho\), which includes the classical Serre \(S_r\) and Cohen-Macaulay conditions as well. Under certain hypotheses the authors characterize those schemes which admit canonical finite \(S_\rho\)-ifications. perverse coherent sheaves; local sohomology; Serre condition; Macaulayfication; \(S_2\)-ification Local cohomology and algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Generalized Serre conditions and perverse 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 Let \(C\) be an integral projective curve, defined over an algebraically closed field of arbitrary characteristic, and denote by \(J:= \text{Pic}^0_C\) its generalized Jacobian. Then, even in the singular case, \(J\) has a natural compactification \(\overline J\) which is the fine moduli space of torsion-free sheaves of rank 1 and degree 0 over \(C\) This so-called compactified Jacobian \(\overline J\) was introduced by \textit{A. Altman} and \textit{S. Kleiman} in the late 1970s [Bull. Am. Math. Soc. 82, 947--949 (1976; Zbl 0336.14008); Adv. Math. 35, 50--112 (1980; Zbl 0427.14015); Am. J. Math. 101, 10--41 (1979; Zbl 0427.14016)], and it has been an object of intensive study ever since.
In the paper under review, the authors generalize the classical autoduality theorem for Jacobians of nonsingular curves in the following sense:
Given an invertible sheaf \(L\) of degree 1 over \(C\), form the corresponding Abel map \(A_L: C\to\overline J\) and its pull-back map \(A^*_L: \text{Pic}^0_{\overline J}\to J\). Then, if the curve \(C\) has, at worst, points of multiplicity 2, the the map \(A^*_L\) is an isomorphism.
Moreover, it is shown that, if \(C\) has only singularities of embedding dimension 2, then \(A^*_L\) is independent of the choice of the line bundle \(L\). This establishes a natural autoduality between \(J\) and \(\text{Pic}^0_{\overline J}\), which commutes with specializing the curve \(C\). The fine analysis leading to this main result is valid, more generally, for families of such curves, and the autoduality theorem is indeed proved in its relative version. algebraic curves; generalized Jacobians; Abel map; Picard schemes; singularities Esteves, E.; Gagné, M.; Kleiman, S. L., Autoduality of the compactified Jacobian, J. Lond. Math. Soc. (2), 65, 3, 591-610, (2002), MR 1895735 (2003d:14038) Jacobians, Prym varieties, Picard schemes, higher Jacobians, Singularities of curves, local rings, Families, moduli of curves (algebraic) Autoduality of the compactified Jacobian. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 seems that the first linkage between topological and algebraic characteristic classes has been discovered by Serre, who related the Hasse-Witt invariants of the trace form of a Galois extension to the Stiefel-Whitney classes of the corresponding orthogonal representation. In the present highly original book many more deep examples are given for the powerful application of topological methods to Galois representation theory, mainly based on the author's own research in this area.
Chapter 1 and 2 give an introduction to abelian and non-abelian group cohomology with an emphasis on the actual calculation of the \(\mod 2\) cohomology rings of cyclic, dihedral and generalized quaternion groups and the integral cohomology ring of the dihedral group of order 8. In chapter 3 Serre's formula and its extension to arbitrary orthogonal representations due to Frohlich are proved. Chapter 4 develops the Koslowski transfer in the context of topological spaces, which is then converted into the algebraic setting and yields Kahn's higher-dimensional analogs of the formula of Serre and Frohlich.
The last 3 chapters introduce and apply a method christened ``explicit Brauer induction'', since it expresses a finite-dimensional representation of a finite group in a canonical way as a linear combination of monomial representations. This canonical form is derived from results in stable homotopy theory, which assumes more familiarity with topology than the rest of the book. The major application is a new and essentially local construction of the local root numbers related to Artin L-functions.
The book is certainly very useful both for topologists who are looking for applications in other areas of mathematics and for people working in algebra and number theory, since the methods developed without any doubt will have many more interesting applications in the future. characteristic classes; Galois representation theory; explicit Brauer induction; stable homotopy theory; Artin L-functions Snaith, V. P.: Topological methods in Galois representation theory. Canad. math. Soc. monograph (1989) Galois cohomology, Research exposition (monographs, survey articles) pertaining to field theory, Quaternion and other division algebras: arithmetic, zeta functions, Characteristic classes and numbers in differential topology, Galois cohomology, Brauer groups of schemes, Galois theory, Stable homotopy theory, spectra Topological methods in Galois representation 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 In this paper, it is proved that the isomorphism class of the tame fundamental group of a smooth, connected curve over an algebraically closed field \(k\) of characteristic \(p>0\) determines the genus \(g\) and the number \(n\) of punctures of the curve, unless \((g,n)=(0,0), (0,1)\). Moreover, in the case \(g=0\), \(n>1\) and \(k=\) the algebraic closure of the prime field \(\mathbb F_p\), it is shown that the isomorphism class of the tame fundamental group even completely determines the isomorphism class of the curve as a scheme (though not necessarily as a \(k\)-scheme). As a key tool for the proofs, the author invents a generalization of \textit{M. Raynaud}'s theory of theta divisors [Bull. Soc. Math. Fr. 110, 103--125 (1982; Zbl 0505.14011)]. algebraic curves; positive characteristic; theta divisor Akio Tamagawa, On the tame fundamental groups of curves over algebraically closed fields of characteristic >0, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., vol. 41, Cambridge Univ. Press, Cambridge, 2003, pp. 47 -- 105. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves of arbitrary genus or genus \(\ne 1\) over global fields On the tame fundamental groups of curves over algebraically closed fields of characteristic \(>0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper deals with a natural generalization of the classical theory of the Jacobian of an algebraic curve. More precisely, viewing the Jacobian of a curve as a parameter space for line bundles with fixed Chern class, the author generalizes this to higher dimensional algebraic varieties as follows. Fix the Chern classes for a holomorphic rank \(n\) vector bundles over a complex projective variety of dimension \(n\), and construct a ``canonical'' family of such bundles. The author introduces a parameter space of such a family, and call it a ``nonabelian Jacobian''.
This kind of object was introduced for \(n=2\) by the author [J. Differ. Geom. 74, No. 3, 425--505 (2006; Zbl 1106.14030)], and the present paper provides a generalization of that paper to an arbitrary dimension \(n\geq 2\).
A careful study, made by the author, of the geometric structure of this nonabelian Jacobian of an \(n\)-dimensional projective variety \(X\) (with \(n\geq 2\)), gives rise to many structures on \(X\), as the following author's abstract suggests:
Author's abstract: Let \(X\) be a smooth projective variety of dimension \(n\geq 2\). It is shown that a finite configuration of points on \(X\) subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations: (i) Lie algebras and their representations (ii) Fano toric variety whose hyperplane sections are Calabi-Yau varieties. These features imply that the points cease to be 0-dimensional objects and acquire dynamics of linear operators ``propagating'' along the paths of a particular trivalent graph. Furthermore following particular linear operators along the ``shortest'' paths of the graph one creates, for every point of the configuration, a distinguished hyperplane section of the Fano variety in (ii), i.e. the points ``open up'' to become Calabi-Yau varieties. Thus one is led to a picture which is very suggestive of quantum gravity according to string theory. higher dimensional projective varieties; vector bundles; zero-cycles; configurations of points; Lie algebras; Higgs bundles I. Reider, Configurations of points and strings , J. Geom. Phys. 61 (2011), 1158-1180. Configurations and arrangements of linear subspaces, Relationships between algebraic curves and integrable systems Configurations of points and strings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Overview: this is an excellent introduction to the moduli space of stable quotients, which is a compactification of the space of maps from curves to Grassmannians. The focus here is on sheaf theoretic compactifications via the Quot scheme, geometry of such compactifications, and the comparison of related geometric invariants.
Section 3.1: Consider the space of maps from a fixed domain curve to Grassmannians. Two compactifications are described, via stable maps and via the Quot scheme, respectively. In general, the two compactifications may have different boundaries, but they both carry compatible virtual fundamental classes. As a result, related counting invariants are defined.
Section 3.2: Consider the space of maps when the domain curve varies in moduli. Using a modification of the Quot scheme over nodal curves, the Quot scheme compactification provides the so-called moduli space of stable quotients, originally due to Marian, Pandharipande and the author [\textit{A. Marian} et al., Geom. Topol. 15, No. 3, 1651--1706 (2011; Zbl 1256.14057)]. Properties of the stable quotient space are discussed, including its properness over the Deligne-Mumford moduli space of curves, its obstruction theory and virtual fundamental class, and geometric invariants.
Section 3.3: The author compares the stable quotient and stable map invariants arising from local geometry and hypersurface geometry.
Section 3.4: Extensions and more general geometries are considered here, including \(\epsilon\)-stable quotients, wall-crossing formulas, certain GIT quotients, and the comparison between related geometric invariants. stable quotients; stable maps; Quot scheme; Gromov-Witten theory; virtual fundamental class; obstruction theory Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Notes on the moduli space of stable quotients | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians From the introduction: ``In this paper we formulate and prove a version of the Grothendieck Section Conjecture [GSC for short] . For function fields of algebraic varieties over algebraically closed ground fields, this conjecture states, roughly, that the existence of group-theoretic sections of homomorphisms of their absolute Galois groups implies the existence of geometric sections of morphisms of models of these fields (...) This raises the question of functoriality, i.e., the reconstruction of rational morphisms between algebraic varieties from continuous homomorphisms of absolute Galois groups of their function fields. This general fundamental question was proposed by Grothendieck and lies at the core of the Anabelian Geometry Program.''
The main open problem concerns a Galois-theoretic criterium for the existence of rational sections of fibrations. More precisely, let $\pi:X \to Y$ be a fibration of integral algebraic varieties over an algebraically closed field $k$, with geometrically irreducible generic fiber of dimension at least $1$ over a base $Y$ of dimension $\geq 2$. This defines a field embedding $\pi^*:K := k(Y) \hookrightarrow L := k(X)$ such that the image of $L$ is algebraically closed in $K$. Dually, this induces a surjective restriction homomorphism of absolute Galois groups $G_K \to G_L$. Fix a prime $\ell \neq char(k)$ and write $\mathcal{G}$ for the maximal pro-$\ell$-quotient of $G$. The previous restriction then induces homomorphisms $\mathcal{G}_K^a \to \mathcal{G}_L^a$ and $\mathcal{G}_K^c \to \mathcal{G}_L^c$, where $\mathcal{G}^a$ (resp. $\mathcal{G}^c$) denotes the first quotient in the derived (resp. central) descending series of $\mathcal{G}$. The authors present here what they call a ``minimalistic'' version of GSC. Let $\pi_a:\mathcal{G}_K^c \to \mathcal{G}_L^a$ be the natural surjection, and $\Sigma_K = \Sigma(\mathcal{G}_K^c)$ the set of topologically non cyclic subgroups $\sigma \subset \mathcal{G}_K^a$ whose preimages $\pi_a^{-1}(\sigma) \subset \mathcal{G}_K^c$ are abelian. Assume that $\pi_a$ admits a section $\xi_a:\mathcal{G}_L^a \to \mathcal{G}_K^a$ such that $\xi_a(\Sigma_L) \subset \Sigma_K$. Then there exists a finite purely inseparable extension $\iota^* : L \hookrightarrow L' =k(Y')$ and a rational map $\xi:Y' \to X$ such that $\iota^*.\pi^*(L) = \iota^*(L)\subset L'$. Thus $\xi(Y')$ is a section over $Y$, modulo purely inseparable extensions.
The main result of this paper establishes a link between the minimalistic GSC above and homomorphisms of multiplicative groups of fields preserving algebraic independence (as announced in the title). Note that $\mathcal{G}_K^a \cong \mathrm{Hom}(K^\times, \mathbb{Z}_l(1))$ by Kummer theory, $\xi_a$ induces the dual homomorphism $\hat{\psi}:\hat{K}^\times \to \hat{L}^\times$ of pro-$\ell$-completions, and the inclusion $\xi_a(\Sigma_L) \subset \Sigma_K$ says that $\hat{\psi}$ respects the skew-symmetric pairings on $\hat{K}^\times$ and $\hat{L}^\times$, with values in the second Galois cohomology groups of the corresponding fields (with $\ell$-torsion coefficients). If the restriction $\psi$ of $\hat{\psi}$ to $K^\times /k^\times$ satisfies $\psi : K^\times /k^\times \subseteq L^\times /k^\times \subset \hat{L}^\times$, then $\psi$ respects algebraic dependence, i.e. it maps algebraically dependent elements in $K^\times$ to algebraically dependent elements of $L^\times$ modulo $k^\times$ (the latter notion is intuitively clear, even if its formal definition can be cumbersome).
The authors' central theorem reads as follows.
From now on, $K$ will be an arbitrary field, and $v$ a non archimedean valuation on $K$, with the usual pertaining notations (valuation ring $\mathcal{O}_{K,v}$, maximal ideal $\mathfrak{m}_{K,v}$, residue field $\mathbf{K}_v$, etc.) Let us consider a tower of extensions $k \subseteq \tilde{k} \subseteq \tilde{k}_a \subset K$, where $k$ is the prime subfield of $K$, $\tilde{k}_a$ the algebraic closure of $\tilde{k}$ in $K$, and a second tower $l \subseteq \tilde{l} = \tilde{l}_a \subset L$ extending the first one.
Give a homomorphism $\psi:K^\times /k^\times \to L^\times /\tilde{l}^\times$ such that: (i) $\psi$ preserves algebraic dependence w.r.t. $\tilde{k}$ and $\tilde{l}$; (ii) there exist $y_1,y_2 \in \psi(K^\times /k^\times)$ such that $y_1,y_2$ are algebraically independent modulo $\tilde{l}^\times$. Suppose moreover that $\psi$ satisfies a certain technical assumption called (AD) (which holds in non null characteristic but is conjecturally superfluous). Then three cases and only three can occur:
(P) there exists a field $F \subset K$ such that $\psi$ factors through $K^\times /k^\times \twoheadrightarrow K^\times /F^\times$;
(V) there exists a non trivial $v$ on $K$ such that the restriction of $\psi$ to $\mathcal{O}_{K,v}^\times\mathcal{O}_{k,v}^\times$ is trivial on $(1+\mathfrak{m}_{K,v})^\times/\mathcal{O}_{k,v}^\times$ and it factors through the reduction map $\mathcal{O}_{K,v}^\times\mathcal{O}_{k,v}^\times \twoheadrightarrow \mathbf{K}_v^\times /\mathbf{k}_v^\times \to L^\times /\tilde{l}^\times$;
(VP) there exists a non trivial valuation $v$ on $K$ and a field $\mathbb{F}_v \subset \mathbf{K}_v$ such that the restriction of $\psi$ to $\mathcal{O}_{K,v}^\times\mathcal{O}_{k,v}^\times$ factors through $\mathcal{O}_{K,v}^\times\mathcal{O}_{k,v}^\times \twoheadrightarrow K \times\mathbf{K}_v^\times /\mathbf{k}_v^\times \to L^\times /\tilde{l}^\times$.
The idea of the proof of the main theorem is to reduce the problem to a question in plane projective geometry over the prime subfield $k$, viewing $\mathbb{P}(K) := K \times /k \times$ as a projective space over $k$ and describing the homomorphisms $\psi:\mathbb{P}(K) \to \mathbb{P}(L)$ which preserve algebraic dependence. Indeed it suffices to show the existence of a subgroup $\mathfrak{U} \subset \mathbb{P}(K)$ such that, for every projective line $\mathfrak{l} \subset \mathbb{P}(K)$, $\mathfrak{U} \cap \mathfrak{l}$ is either:
(i) the line $\mathfrak{l}$;
(ii) a point $\mathfrak{q} \in \mathfrak{l}$;
(iii) the affine line $\mathfrak{l} \setminus \mathfrak{q}$;
(iv) if $k = \mathbb{Q}$, a set projectively equivalent to $\mathbb{Z}_{(p)}$ in $\mathbb{A}_1(\mathbb{Q}) \subset \mathbb{P}^1(\mathbb{Q})$.
Actually such a subgroup is necessarily either $F^\times / k^\times$ for some subfield $F$ of $K$, or some $\mathcal{O}_{K,v}$ associated to a valuation $v$ of $K$. In case (V) (resp. (P)), $\psi$ factors through a valuation (resp. a subfield); case (VP) corresponds to valuations composed with projections. anabelian geometry; valuations; section conjecture Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Galois theory, Arithmetic theory of algebraic function fields, Rationality questions in algebraic geometry, Higher symbols, Milnor \(K\)-theory Homomorphisms of multiplicative groups of fields preserving algebraic dependence | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 me cite from the well-written (!) publisher's description: ``In 1948 André Weil published the proof of the Riemann hypothesis for function fields in one variable over a finite ground field. This was a landmark in both number theory and algebraic geometry. Applications included hitherto unattainable bounds for exponential sums, in particular Kloosterman sums. Weil built on innovative work in the 1920's and 1930's of Emil Artin and Helmut Hasse, among others. Later, Grothendieck and Deligne employed profound innovations in algebraic geometry to carry Weil's work much further.
It came as a surprise to the number theory community when Sergei Stepanov gave elementary proofs of many of Weil's most significant results in a series of papers published between 1969 and 1974. Stepanov's method drew inspiration from the work of Axel Thue (1909) in Diophantine approximation. Portraits of Thue, Artin, Hasse and Weil feature on the front cover of this book.
Proofs of Weil's result in full generality, based on Stepanov's ideas, were given independently by Wolfgang Schmidt and Enrico Bombieri in 1973.
The present book contains accounts of both methods. Schmidt's method, which is more elementary, is discussed in Chapters 1 - 6, along with many related matters. This part of the book is, essentially, the content of the first edition, published by Springer in 1976 (see the review in Zbl 0329.12001). The remaining chapters, 7 through 9, cover Bombieri's proof, with necessary material on `Valuations and Places' and `The Riemann-Roch theorem'; developed in Chapters 7 and 8.
All chapters are based on the author's lectures at the University of Colorado. However, the second part existed only in a somewhat rough xeroxed form from 1975 until the present. For the second edition, the whole text has been reset in an attractive typeface, and some inaccuracies have been corrected.
Graduate students will find Wolfgang Schmidt's book a valuable resource. The necessary tools are developed without the need for substantial prerequisites. The style is leisurely, with many well-chosen examples, and proofs are given in full detail''. Schmidt W.: Equations Over Finite Fields: An Elementary Approach, 2nd edn. Kendrick Press, Heber City (2004). Curves over finite and local fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Exponential sums, Other character sums and Gauss sums, Varieties over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic theory of polynomial rings over finite fields, Higher degree equations; Fermat's equation, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Equations over finite fields. An elementary approach | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 proper and integral curve over an algebraically closed field \(k\). The moduli space of line bundles of degree zero on \(X\) is then, in a natural way, a group scheme over \(k\), called the generalized Jacobian \(P(X)\) of \(X\).
If \(X\) is a smooth curve, then \(P(X)\) of \(X\) is just the usual Jacobian of \(X\), i.e., an abelian variety. However, if \(X\) is singular, then \(P(X)\) is only a non-proper group scheme, and the question of whether there is a natural ``compactification'' of \(P(X)\), that is a proper scheme \(\overline P(X)\) containing \(P(X)\) as an open subset and on which \(P(X)\) acts as a group scheme, arises quite inevitably. In fact, such a (so-called) compactified Jacobian can be constructed, namely by considering the functor of families of torsion-free sheaves of rank one on \(X\). This idea of construction goes back to \textit{A. Grothendieck} [cf. ``Fondements de la géométrie algébrique,'' Extraits du Séminaire Bourbaki 1957-1962 (Paris 1962; Zbl 0239.14002)]. Explicit constructions have been carried out, in the sequel, by A. Mayer and D. Mumford (1963), C. D'Souza (1973), T. Oda and C. S. Seshadri (1974), A. Altman, A. Iarrobino and S. Kleiman (1976), C. J. Rego (1980), and others. The fundamental paper of \textit{A. B. Altman} and \textit{S. L. Kleiman} ``Compactifying the Picard scheme'' [Adv. Math. 35, 50-112 (1980; Zbl 0427.14015)] gives the perhaps most general and complete account, with generalizations to higher dimensional varieties \(X\). Finally, the various attempts of constructing compactified Jacobians of curves have led to the now well-established theory of moduli spaces of semistable vector bundles over a curve.
The paper under review deals with the projectivity of the compactified Jacobian \(\overline P (X)\). Using the relative approach by A. Altman and S. Kleiman, the author constructs a Cartier divisor \(\Theta\) on \(\overline P^{g - 1} (X)\), the reduced \((g - 1)\)-component of the compactified Picard scheme \(\overline {\text{Pic}} (X/k)\) with respect to the arithmetic genus \(g\) of \(X\), and proves that \(\Theta\) is ample. -- The proof is given in such a way that it can be generalized to the relative case, i.e., to morphisms \(f : X \to S\) of finite type, flat and projective, whose fibers are integral curves of arithmetic genus \(g\). In this situation, the author's construction leads to a sheaf \({\mathcal D}\) on \(\overline {\text{Pic}}^{g - 1} (X/S)\) which is \(S\)-ample. ampleness; polarization; generalized Jacobian; projectivity of the compactified Jacobian; compactified Picard scheme 15. Soucaris, A.: The ampleness of the theta divisor on the compactified Jacobian of a proper and integral curve, Compositio Math., 93 (1994), 231--242 Jacobians, Prym varieties, Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles The ampleness of the theta divisor on the compactified Jacobian of proper and integral curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper studies the analogue of the Grothendieck-Ogg-Shafarevich formula for curves over local fields. This formula was originally proved for curves over algebraically closed fields [\textit{A. Grothendieck}, SGA 5, Lect. Notes Math. 589, Exposé No. X, 372-406 (1977; Zbl 0356.14005)].
The main theorem in this paper is the following.
Let \(R\) be a complete discrete valuation ring with algebraically closed residue field \(k\); let \(S=\text{Spec}R\); let \(\eta\) be the generic point of \(S\); let \(X\) be a connected normal scheme, proper and flat over \(S\), of relative dimension one, with smooth generic fiber; let \(U\) be an open dense subscheme of \(X\) contained in \(X_\eta\); let \((F_i)_{i\in I}\) be the one-dimensional irreducible components of \(F:=X\setminus U\); let \(\mathbb F\) be a finite field with \(\text{char} \mathbb F\neq\text{char} k\); and let \(\mathcal F\) be a locally constant constructible étale sheaf on \(U\) of \(\mathbb F\)-vector spaces. Assume that the residue field of any point in \(X_\eta\setminus U\) is separable over \(\eta\), and that \(\mathcal F\) has no fierce ramification (see the paper for the definition). Then the main theorem asserts that, for each \(i\in I\), there is an open dense subscheme \(F_i^\circ\) of \(F_i\) such that:
(i) \(F\) is regular along \(F^\circ:=\bigcup_{i\in I}F_i^\circ\);
(ii) if \(x\in F_i^\circ\) for some \(i\in I\), then \(\text{sw}_x^{V/U}(\mathcal F)=\text{sw}_i(\mathcal F)\); and
\[
\begin{multlined}\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathcal F)) =\\ =\text{totdim}_{\mathbb F}(R\Gamma_c(U_{\bar\eta},\mathbb F)) \text{rk}_{\mathbb F}(\mathcal F) +\sum_{i\in I}\chi_c(F_i^\circ)\text{sw}_i(\mathcal F) +\sum_{x\in F\setminus F^\circ}\text{sw}_x^{V/U}(\mathcal F).\end{multlined}\tag{iii}
\]
Here \(\chi_c\) is the Euler-Poincaré characteristic if \(F_i^\circ\) is vertical, or minus the discriminant over \(S\) if it is horizontal; \(\text{totdim}_{\mathbb F}\) is the sum of the dimension over \(\mathbb F\) and the dimension over \(\mathbb F\) of the Swan conductor; and \(R\Gamma_c\) refers to the alternating sum of the groups \(H^i_c\). See the paper for the definition of \(\text{sw}_i(\mathcal F)\).
The first part of the paper is devoted to defining the Swan conductor \(\text{sw}_x^{V/U}(\mathcal F)\) in codimension 2, and the second part gives the proof of the above theorem. A key tool in that proof is the Lefschetz fixed point formula for arithmetic surfaces proved by the author [\textit{A. Abbes}, Compos. Math. 122, 23-111 (2000; see the preceding review Zbl 0986.14014)].
The paper also formulates a conjecture that, for all closed points \(x\in X\), \(\text{sw}_x^{V/U}(\mathcal F)\) is (a) an integer, and (b) independent of the choice of \(V\). Part (a) of this conjecture extends a conjecture of Serre on the existence of Artin representations. Grothendieck-Ogg-Shafarevich formula; Swan conductor; Lefschetz fixed point formula; Artin representation A. Abbès, The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces, Journal of Algebraic Geometry, 9 (2000), 529-576. Arithmetic varieties and schemes; Arakelov theory; heights, Topological properties in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is devoted to the study of the algebraic (or étale) fundamental group of a scheme. This notion was introduced by A. Grothendieck in the 1960s. Later on, in the 1980s, M. Nori has introduced the concept of the fundamental group scheme of a reduced and connected scheme \(X\) over a field \(k\). Both constructions coincide when \(\text{char}(k)=0\), whereas they definitely differ in characteristic \(p>0\). However, Nori has conjectured that two particular, fundamental properties of Grothendieck's étale fundamental group \(\pi_1(X,x)\) have some analogues in the theory of fundamental group schemes \(\pi(X,x)\), which read as follows:
(1) \(\pi(X_k^\times Y,(x,y))= \pi(X,x)_k^\times \pi(Y,y)\) as group schemes;
(2) If \(X\) is a complete, reduced and connected scheme over an algebraically closed field \(k\), and if \(k'\) is an algebraically closed field extension of \(k\), then the canonical homomorphism of group schemes
\[
\pi(X_{k'},x) \to\pi(X_k,x)\times_k \text{Spec}(k')
\]
is an isomorphism.
Now, in the article under review, the authors show that conjecture (1) is indeed true, whilst conjecture (2) is false (in characteristic \(p>0)\). As for the proofs, the authors use a fine analysis of the Tannaka category of stable vector bundles on the related schemes. characteristic \(p\); fundamental group schemes; Tannaka category Mehta V.B., Subramanian S.: On the fundamental group scheme. Invent. Math. 148, 143--150 (2002) Homotopy theory and fundamental groups in algebraic geometry, Group schemes, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Finite ground fields in algebraic geometry On the 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 Let \(\ell\) be a prime number and \(X=X(\ell)\) the modular curve with full level structure \(\ell\). For a subgroup \(H\subset G=\operatorname{GL}_2(\mathbb{F}_\ell)\) containing -1, let us denote by \(X_H\) the corresponding modular curve \(X/H\) given by the action of \(G\) on \(X(\ell)\), and by \(J_H\) its Jacobian variety. For \(C_{ns}\) a non-split Cartan subgroup of \(G\), and its normalizer \(N_{ns}\) the first author proved in [J. Algebra 231, No. 1, 414--448 (2000; Zbl 0990.14012)] that \(J_{C_{ns}}\) and \(J_{N_{ns}}\) are \(\mathbb{Q}\)-isogenous to certain quotients of the modular curve \(X_0(\ell^2)\). Merel conjectured that the isogenies between these varieties can be described by the correspondence induced by the quotient maps from \(X(\ell)\). This was proved in [the first author, Proc. Lond. Math. Soc. (3) 77, No. 1, 1--38 (1998; Zbl 0903.11019)] using the representation theory of \(G\) and identities in finite double coset algebras.
This paper presents an alternative proof of Merel's conjecture, based on arguments by Birch and Zagier. The authors consider a finite field analogue of the complex upper half-plane and the natural action of \(G\) on it by Möbius transformations. This provides a practical description of the quotient of \(G\) by the normalizer of a Cartan subgroup of \(G\) (either split or non-split), where the conjecture is established in quite an elementary way. The proof is extended naturally to quotients of \(G\) by the Cartan groups themselves. modular curves; elliptic curves Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties An explicit correspondence of modular 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 \(f: X\to S\) be a smooth family of curves over a smooth base scheme \(S\) defined over a field of characteristic zero, and let \((E,\nabla_{X/S})\) be a regular relative connection with respect to a vector bundle \(E\) over \(X\). In the paper under review, the authors show the existence of functorial classes \(c_2((E, \nabla_{X/S}))\) and \(c_1((E, \nabla_{X/S}))^2\) in the relative cohomology of the \(d\log\)-complex associated with the family \(f:X\to S\) which appear as a more algebraic version of a related analytic approach due to A. Beilinson. Moreover, the authors explain how this important example transpires from a much more general theory of relative differential characters, the foundations of which were elaborated in a previous paper of \textit{H. Esnault} [in: Regulators in analysis, geometry and number theory, Prog. Math. 171, 89--115 (1996)]. This general framework can be applied to higher-dimensional morphisms and higher functorial classes as well. Furthermore, the authors also give a second construction of their above-mentioned functorial classes, this time using the so-called Weil-algebra homomorphism as constructed in an unpublished work of A. Beilinson and D. Kazhdan. The relevant material from this approach is given in full detail, and it is pointed out how this second approach could be used to generalize the whole theory to the wider class of \(G\)-bundles (instead of just vector bundles \(E\)). Finally, a thorough comparison of the authors' approaches with Beilinson's ideas (private communication to H. Esnault) is delivered at the end of the paper. relative cohomology; Deligne cohomology; sheaves of differentials; Weil algebra; gerbes; characteristic classes Bloch, S. and Esnault, H., Relative algebraic differential characters, (None) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Families, moduli of curves (algebraic), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Relative algebraic differential characters | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(J\) be the Jacobian variety of a curve \(C\) over an algebraically closed field. For every integer \(n>0\), the Weil pairing on the \(n\)-torsion of \(J\) is a perfect pairing
\[
{\mathbf e}_n : J[n]\times J[n] \to\mathbf{G}_m
\]
that takes values in the multiplicative group. The author provides a short proof of a formula for the Weil pairing in terms of tame symbols. In particular, if \((f_x)_{x\in C}\) and \((g_x)_{x\in C}\) are idèles corresponding to \(n\)-torsion bundles \(L\) and \(M\), chosen so that \((f_x^n)\) and \((g_x^n)\) are principal, then the value of the Weil pairing \(e_n([L],[M])\) is
\[
\prod_{x\in C} (f_x,g_x)_x^n,
\]
where \((f_x,g_x)_x\) denotes the tame symbol at \(x\).
This result has been known to experts for a long time, but the first published proof for curves of arbitrary genus appeared in a paper by the reviewer [Math. Ann. 305, No. 2, 387--392 (1996; Zbl 0854.11031)]. This earlier proof involves a reduction to the case of curves over finite fields, and uses class field theory for curves. The author's proof avoids this arithmetic reduction, and involves comparing various fibers of two different pullbacks to \(J\times J\) of the Poincaré bundle. Weil pairing; tame symbol; idèle; Poincaré bundle Algebraic theory of abelian varieties, Abelian varieties of dimension \(> 1\) Weil pairing and tame symbols | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 state that ``the purpose of this note is to prove the following `relative' version of Gersten's conjecture:
Theorem. Let \(X=Spec(R)\) be an affine scheme, flat and of finite type over a discrete valuation ring \(\Lambda\): suppose that there is a finite set of points \(S\subset X\) such that X is smooth over Spec(\(\Lambda\)) at the points of S. Let us write \(A={\mathcal O}_{X,S}\). If \(Y=Spec(R/tR)\to X\) is a principal effective relative (i.e. flat over \(\Lambda\)) divisor, then the exact functors \(M^{(i)}(Y)\to M^{(i)}(Spec(A))\) and \(MF^{(i)}(Y)\to MF^{(i)}(Spec(A))\) [...] both induce the zero map on K-theory.''
In the statement of this result, \(M^{(i)}(Y)\) denotes the category of coherent sheaves of modules over Y supported in codimension i, and \(MF^{(i)}(Y)\) denotes the category of coherent sheaves of modules over Y that are flat over \(\Lambda\). Several applications of this result are given.
\{Reviewers remark: Corollary 5 is very similar to proposition 3.2 in a paper by \textit{L. Claborn} and the reviewer, Ill. J. Math. 12, 228-253 (1968; Zbl 0159.049), where other special cases of this theorem are demonstrated.\} Gersten's conjecture; zero map on K-theory; coherent sheaves \beginbarticle \bauthor\binitsH. \bsnmGillet and \bauthor\binitsM. \bsnmLevine, \batitleThe relative form of Gersten's conjecture over a discrete valuation ring: The smooth case, \bjtitleJ. Pure Appl. Algebra \bvolume46 (\byear1987), no. \bissue1, page 59-\blpage71. \endbarticle \endbibitem Applications of methods of algebraic \(K\)-theory in algebraic geometry, Valuation rings, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((g,f) : ({\mathbb C}^2,0) \rightarrow ({\mathbb C}^2,0)\) be a map defined by two germs of analytic functions \(f\) and \(g\) with no common branches.
The author defines the set of Jacobian quotients of the map \((g,f)\) which contains all the exponents of leading terms of Puiseux expansions associated with branches of the discriminant curve of the map. Based on the Waldhausen theory of differentiable \(3\)-manifolds, she proves that these rational numbers are topological invariants of the map and can be computed in terms of linking numbers of algebraic knots associated with Seifert fibres of the minimal Waldhausen decomposition of \(S_\epsilon^3\) for the link \(K_{fg},\) where \(S_\epsilon^3\) is the sphere centered at the origin of \({\mathbb C}^2\) and \(K_{fg} = (fg)^{-1}(0) \cap S_\epsilon^3.\)
In the case when \(g\) is a linear form transverse to \(f\), the sets of Jacobian quotients of \((g,f)\) and of the polar quotients of \(f\) coincide. Thus, the author confirms the earlier results from \textit{Le Dung Trang}, \textit{F. Michel} and \textit{C. Weber} [Compos. Math. 72, No. 1, 87-113 (1989; Zbl 0705.32021), Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, No. 2, 141-169 (1991; Zbl 0748.32018)], and some others.
It should be also remarked that a method of computing the Jacobian quotients in terms of contact exponents in the minimal resolution of \((fg)^{-1}(0)\) is described in [\textit{H. Maugendre}, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7, No. 3, 497-525 (1998; Zbl 0936.32012)]. plane curves; polar quotients; Jacobian locus; discriminant curve; Waldhausen decomposition; Seifert fibres; linking numbers; Milnor isotopy Maugendre, H.: Discriminant of a germ \({\Phi}\):(C2,0)\$to(C2,0)\( and Seifert fibred manifolds. J. London math. Soc. (2) 59, 207-226 (1999)\) Topological invariants on manifolds, Knots and links in the 3-sphere, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Jacobian problem Discriminant of a germ \(\Phi: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) and Seifert fibred manifolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, the objects of consideration include a regular semilocal Noetherian scheme \(W\), a reductive group scheme \(G\) over \(W\) and a principal \(G\)-bundle over \(\mathbb{P}^1_W\). The main theorem of the paper states that if the restriction of such a \(G\)-bundle to each closed fiber is trivial, then the original bundle is an inverse image of some principal \(G\)-bundle on \(W\). For the case when the scheme \(W\) is equicharacteristic, this theorem was proved in a paper by \textit{I. Panin} et al. [Compos. Math. 151, No. 3, 535--567 (2015; Zbl 1317.14102)] on the Grothendieck-Serre conjecture. That equicharacteristic case of the theorem was used in a paper by \textit{R. Fedorov} and \textit{I. Panin} [Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)], and in another paper by \textit{I. Panin} [``Proof of Grothendieck-Serre conjecture on principal \(G\)-bundles over regular local rings containing a finite field'', Preprint, \url{arXiv:1406.0247}], to prove the Grothendieck-Serre conjecture itself in the equicharacteristic case. The main theorem of the present paper may be useful for proving the general case of the Grothendieck-Serre conjecture. reductive group schemes; principal bundles; Grothendieck-Serre conjecture; mixed characteristic; quotient sheaves Group schemes, Étale and other Grothendieck topologies and (co)homologies A step towards the mixed-characteristic Grothendieck-Serre 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 authors generalize the results of \textit{G. van der Geer} and \textit{M. van der Vlugt} [J. Number Theory 59, 20--36 (1996; Zbl 0872.94052), J. Comb. Theory, Ser. A 70, 337--348 (1995; Zbl 0824.94025)] from curves over fields of characteristic 2 to arbitrary characteristic.
They show how to construct curves with many points attaining the Hasse-Weil bound. For the first construction they use Artin-Schreier curves of the form \(Y^q-Y=XR(X)\), where \(R\) satisfies certain conditions. These curves are no longer hyperelliptic as in the cited articles but share several properties. The \(C_{ab}\) curves (proposed by the second author in his PhD thesis, see also \textit{S. Arita} [Discrete Appl. Math. 130, 13--31 (2003; Zbl 1037.14010)]) have only 1 point at infinity, and for these special curves the number of points over \(F_{q^m}\) can be stated explicitly. The authors show under which conditions on \(q,m\) and \(R\) these curves attain the Hasse-Weil bound. The theorems prove the existence of maximal curves under some conditions and are even constructive, leading to the equation of the curve.
Given a set of such Artin-Schreier curves one can take their fibre product, which can lead to maximal curves as well. Finally, they give the construction of the affine equation of the curve to allow applications in coding theory. curves with many points; number of rational points; quadratic forms; fibre products; Artin-Schreier curves; trace codes; arbitrary characteristic; \(C_{ab}\) curves Curves over finite and local fields, Rational points, Algebraic functions and function fields in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Quadratic forms, fibre products and some plane curves with many points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves that any integral functor \(\Phi_{\mathcal E}\colon D(C)\to D(C')\) between the bounded derived categories of coherent sheaves of two smooth projective curves \(C\) and \(C'\) induces an affine map between the rational Picard groups and then induces a morphism \(\phi\colon J(C)\to J(C')\) between the Jacobian varieties. The main result is that the map \(\phi\) is an isomorphism of abelian varieties which preserves the principal polarizations if and only if \(\Phi_{\mathcal E}\) is an equivalence of categories. This is a combination of the Torelli theorem and the fact that a curve is characterized by its derived category of coherent sheaves. Fourier-Mukai; Jacobians; principal polarizations Lekili, Y., Polishchuk, A.: A Modular Compactification of \({\mathcal{M}}_{1,n}\) from \(\text{A}_{\infty }\)-Structures ArXiv e-prints (2014) Jacobians, Prym varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Fourier-Mukai transforms of curves and principal polarizations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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{J.-P.~Serre} proved in [Comment. Math. Helv. 59, 651--676 (1984; Zbl 0565.12014)] a formula that related the Hasse-Witt invariant of the trace form of an étale algebra over a field of characteristic not \(2\) to the second Stiefel-Whitney class of the permutation representation of the Galois group which corresponds to that algebra. This formula has been generalized by \textit{H. Esnault, B. Kahn} and \textit{E. Viehweg} [J. Reine Angew. Math. 441, 145--188 (1993; Zbl 0772.57028)] who considered symmetric bundles obtained from certain tame finite flat coverings of Dedekind schemes with odd ramification everywhere. This latter result has been generalized further by the present authors in [J. Théor. Nombres Bordx. 12, 597--660 (2000; Zbl 1120.11300)] where a formula was obtained under certain regularity conditions but without further restrictions on the dimensions of the schemes. In [J. Reine Angew. Math. 360, 84--123 (1985; Zbl 0556.12005)], \textit{A. Fröhlich} retrieved Serre's result as a special case of a more general difference formula between the Hasse-Witt invariants of a quadratic form and the twist of this form by an orthogonal representation, involving first and second Stiefel-Whitney classes and spinor norms. The main purpose of the present paper is to obtain a theorem ``à la Fröhlich'' in the geometric setting considered in the authors' earlier paper.
From the authors' abstract: We establish comparison results between the Hasse-Witt invariants \(w_t(E)\) of a symmetric bundle \(E\) over a scheme and the invariants of one of its twists \(E_{\alpha}\). For general twists we describe the difference between \(w_t(E)\) and \(w_t(E_{\alpha})\) up to terms of degree \(3\). Next we consider a special kind of twist, which has been studied by A.~Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. A simple important example of this twisting procedure is the bilinear trace form of an étale algebra, which is obtained by twisting the standard/sum-of-squares form by the orthogonal representation attached to the algebra. We show how to explicitly describe the twist for representations arising from very general tame actions. This involves the `square root of the inverse different' which Serre, Esnault, Kahn, Viehweg and ourselves had studied before. For torsors we show that, in our geometric set-up, Jardine's generalisation of Fröhlich's formula holds. Namely let \((X, G)\) be a torsor with quotient \(Y\), let \(E\) be a symmetric bundle over \(Y\), let \(\rho: G \to\mathbf{O}(E)\) be an orthogonal representation and let \(E_{\rho,X}\) be the corresponding twist of \(E\). Then we verify up to degree 3 that the formula \(w_t(E_{\rho,X}) \text{sp}_t(\rho)=w_t(E)w_t(\rho)\) holds. Here \(\text{sp}_t(\rho)\) and \(w_t(\rho)\) are respectively the spinor invariant and the Stiefel-Whitney class of \(\rho\). The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification, which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by Serre for curves. The comparison result in the tamely ramified case proceeds by reduction to the case of a torsor. The reduction is carried out by means of a partial normalisation procedure, which we had introduced in a previous paper. An important lemma of Esnault, Kahn and Viehweg allows us to express the differe nce between the invariants of bundles before and after the normalisation procedure in terms of Chern classes of certain sub-bundles. As noted elsewhere, this result can be best understood in the context of symmetric complexes and their invariants. Our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours. Hasse-Witt invariant; Stiefel-Whitney class; étale cohomology; vector bundle; symmetric bundle; orthogonal representation \(K\)-theory of quadratic and Hermitian forms, Galois cohomology of linear algebraic groups, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Étale and other Grothendieck topologies and (co)homologies Twists of symmetric 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 \textit{P. Deligne} and \textit{D. Mumford} [Publ.\ Math.\ I.H.E.S.\ 36, 75--109 (1969; Zbl 0181.48803)] proved the stable reduction theorem: a smooth projective curve over a local field acquires stable reduction over a finite separable extension of the ground field. This was strengthened by \textit{A. J.\ de Jong} [Ann.\ Inst.\ Fourier 47, 599--621 (1997; Zbl 0868.14012)] to: a proper curve over an integral quasi-compact excellent scheme admits a semi-stable modification over an alteration of the base scheme.
The author improves this further (by, for example, eliminating the hypothesis of properness) as follows: let \(S\) be a scheme. A multi-pointed \(S\)-curve \((C,D)\) consists of flat finitely presented morphisms from \(C,D\) to \(S\) of pure relative dimensions \(1\) and \(0\), respectively. There are notions of morphisms, modifications and semi-stability for such multipointed curves. Then: if \((C,D)\) has semi-stable generic fiber, there exists a generically etale alteration of the base over which there exists a stable modification which is an isomorphism over the generic fiber. The modification of \(C\) is projective over \(C\). If \(S\) is normal, then this modification is minimal, unique up to unique isomorphism, and an isomorphism over the semi-stable locus. The proofs are independent of the results of Deligne, Mumford and de Jong. stable reduction theorem; modification; alteration Temkin, M.: A new proof of the stable reduction theorem. Ph.D. thesis, Weizmann Institute 2006 (partly) published as: Stable modification of relative curves. J. Alg. Geom. 19, 603--677 (2010) Families, moduli of curves (algebraic), Fine and coarse moduli spaces Stable modification of relative curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this lecture of the proceedings the authors prove Grothendieck's theorem on the specialization of the fundamental group for proper smooth curves over a complete discrete valuation ring in the proper case, i.e. for the algebraic fundamental group [see \textit{A. Grothendieck}, Sémin. de géométrie algébrique, Bois-Marie 1960-1961 (SGA1), Revêtements étales et groupe fondamental, Lect. Notes Math. 224 (1971; Zbl 0234.14002)], as well as the specialization theorem for the tame algebraic fundamental group in the case of an open subset \(U\) of \(X\) whose complement is a finite étale divisor of \(X\) [see \textit{A. Grothendieck} and \textit{J. P. Murre}, ``The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme'', Lect. Notes Math. 208 (1971; Zbl 0216.33001)]. tame fundamental group F. Orgogozo , I. Vidal , Le théorèm de spécialisation du groupe fondamental. Courbes semi-stables et groupe fondamental en géometrie algébrique (J.-B. Bost, F. Loeser, and M. Raynaud, eds.), Prog. in Math. , vol. 187 , Birkhäuser , 2000 , 169 - 184 . MR 1768100 | Zbl 0978.14033 Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Schemes and morphisms The specialization theorem of the fundamental group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) denote a nonsingular curve of genus two with a rational Weierstrass point defined over a field \(k\). \textit{D.~Grant} [J. Reine Angew. Math. 411, 96--121 (1990; Zbl 0702.14025)] defined a projective embedding of the Jacobian of \(X\) into \({\mathbb P}^8_k\) using the hyperelliptic \(\sigma\)-function and some of the partial derivatives of \(\log \sigma\), if the characteristic of \(k\) is not 2, and he determined the group law. Using these same hyperelliptic functions and the function \(\phi_n(u)=\sigma(nu)/\sigma(u)^{n^2}\), the author finds explicit formulas for multiplication by \(n\) in such a Jacobian, if the characteristic of \(k\) is not 2 or 3. \textit{D.~G.~Cantor} [J. Reine Angew. Math. 447, 91--145 (1994; Zbl 0788.14026)] also found formulas for multiplication by \(n\) in such a Jacobian by different methods. hyperelliptic curve; hyperelliptic function; Jacobian; division polynomial Kanayama N.: Division polynomials and multiplication formulae of Jacobian varieties of dimension 2. Math. Proc. Camb. Philos. Soc. 139, 399--409 (2005) Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Special algebraic curves and curves of low genus Division polynomials and multiplication formulae of Jacobian varieties of dimension 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors consider the moduli functor of curves with an action of a finite group and the moduli functor of Galois covers and show that both can be represented by closed subschemes of the Sato Grassmannian.
More precisely, consider first curves with an action of a finite group \(G\), rigidified by the choice of a distinguished orbit, a formal trivialisation along this orbit and a group monomorphism of \(G\) into the group \(\text{QP}_r^m\) of \(r\times r\) matrices of quasipermutations with coefficients in the group of \(m\)th roots of unity, where \(r\) is the number of distinct points in the orbit and \(m={1\over r}|G|\). The authors prove that the corresponding moduli functor \({\mathcal M}^{\infty}_G(r)\) is represented by a closed subscheme of the Sato Grassmannian \(\text{Gr}(V)\), where \(V:=k((z_1)\times\ldots\times k((z_r))\), and give a characterisation of this subscheme. This generalises a previous result of the authors and \textit{J.~M.~Muñoz Porras} [Math. Ann. 327, No. 4, 606--639 (2003; Zbl 1056.14039)].
For the case of Galois covers, the moduli space is a subspace of the Hurwitz space previously studied by \textit{J.~M.~Muñoz Porras} and \textit{F. Plaza Martín} [Equations of Hurwitz schemes in the infinite Grassmannian, \texttt{math/0207091}]. The authors characterise its points as the points of the Hurwitz space whose stabilisers under the action of \(\text{QP}_r^m\) have order \(rm\). The representability of the corresponding functor by a subscheme of \(\text{Gr}(V)\) follows from this. The authors also construct the moduli spaces for Galois covers with fixed group and with fixed curve.
The final section is concerned with explicit equations for the moduli spaces described above. curves with automorphisms; Galois covers; Hurwitz spaces; infinite Grassmannians Automorphisms of curves, Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Curves with a group action and Galois covers via infinite Grassmannians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S\) be a curve over an algebraically closed field \(k\) of characteristic \(p\geq 0\). To any family of representations \(\rho=(\rho_{\ell}: \pi_{1}(S) \to \text{GL}_{n}(\mathbb{F}_{\ell}))\) indexed by primes \(\ell \gg 0\) one can associate abstract modular curves \(S_{\rho,1}(\ell)\) and \(S_{\rho}(\ell)\) which, in this setting, are the modular analogues of the classical modular curves \(Y_{1}(\ell)\) and \(Y(\ell)\). The main result of this paper is that, under some technical assumptions, the gonality of \(S_{\rho}(\ell)\) goes to \(+\infty\) with \(\ell\). These technical assumptions are satisfied by \(\mathbb{F}_{\ell}\)-linear representations arising from the action of \(\pi_{1}(S)\) on the étale cohomology groups with coefficients in \(\mathbb{F}_{\ell}\) of the geometric generic fiber of a smooth proper scheme over \(S\). From this, we deduce a new and purely algebraic proof of the fact that the gonality of \(Y_{1}(\ell)\), for \(p\nmid\ell(\ell^{2} - 1)\), goes to \(+\infty\) with \(\ell\). Cadoret, A.; Stix, J., \textit{note on the gonality of abstract modular curves}, The arithmetic of fundamental groups - PIA 2010, 89-106, (2012), Springer Special divisors on curves (gonality, Brill-Noether theory) Note on the gonality of abstract modular 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 presents the general yoga of the duality theory in Galois cohomology, étale cohomology and flat cohomology. The first results in this direction were discovered in the late fifties and early sixties by J. Tate in the framework of Galois cohomology of finite modules and abelian varieties over local and global fields. Later on, Artin and Verdier extended some of these results to étale cohomology groups over integers in local and global fields. These results wer subsequently generalized by several people to flat cohomology groups. However, much of this work was never published in details. The main purpose of the book under review is to offer a selfcontained and systematic treatment of these developments. - The book contains three chapters and a number of appendices.
In the first chapter one proves a very general duality theorem in Galois cohomology that applies whenever one has a class formation. This theorem is used then to prove a duality theorem for modules over the Galois group of a local field. Next one shows that these results can be used to prove Tate's duality theorem for abelian varieties over a local field. Then one considers the global fields, proving duality results in this context (including Tate's duality theorem for abelian varieties over global fields). One also shows that the validity of the conjecture of Birch and Swinnerton-Dyer for abelian varieties over a number field depends only on the isogeny class of the variety in question. In the last part of the first chapter one proves duality theorems for tori and one gives some arithmetic applications (to the Hasse principle for finite modules and algebraic groups, to the existence of forms of algebraic groups, to Tamagawa numbers of algebraic tori over global fields, or to the central embedding problem for Galois groups).
Chapter two deals with étale cohomology, where first the duality theorem for \({\mathbb{Z}}\)-constructible sheaves on the spectrum of a Henselian discrete valuation ring with finite residue field is proved. Then one presents a generalization of the duality theorem of Artin and Verdier to \({\mathbb{Z}}\)-constructible sheaves on the spectrum of the ring of integers in a number field, or on curves over finite fields. This result is then extended to some other situations. This chapter ends with duality theorems for abelian schemes, or for schemes of dimension \(\geq 2.\) The necessary prerequisites on étale cohomology needed in this chapter are rather elementary.
The last chapter is concerned with duality theorems for the flat cohomology of finite flat schemes or Néron models of abelian varieties. The results of this chapter are more tentative than the corresponding ones in the first two chapters, and some of them are new. duality theory; Galois cohomology; étale cohomology; flat cohomology; abelian varieties; global fields; Tamagawa numbers; algebraic tori; Néron models . Milne, J.S. , '' Arithmetic Duality Theorems '', Academic Press, Orlando, 1986. Local ground fields in algebraic geometry, Global ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Galois cohomology, Topological properties in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Arithmetic duality theorems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the germs of curves embedded in a rational surface singularity \((S,P)\) from the point of view of proximity. We generalize the geometric theory of Enriques, obtaining necessary and sufficient conditions to impose on a finite set of points infinitely near \(P\) with assigned orders to get effective Weil (resp. Cartier) divisors going through these points with these orders (where, since a Weil divisor \(C\) on \((S,P)\) is \(\mathbb{Q}\)- Cartier, the orders of \(C\) are defined to be the rational numbers determined by the valuations given by the exceptional curves in the respective point blowing ups). We apply this result to give a formula to compute the minimal numbers of generators of any \({\mathfrak m}\)-primary complete ideal \(I\) of \({\mathcal O}_{S,P}\), generalizing the formula given by Hoskin and Deligne in the nonsingular case. We also give an algorithm to describe a minimal system of generators of \(I\).
The germs of reduced curves in \((S,P)\) are classical up to a notion of equisingularity which generalizes the equisingularity of germs of plane curves. The equisingularity class of such a germ of curve \(C\) in \((S,P)\) consists of the weighted dual graph of the minimal embedded desingularization of \(C\) in \((S,P)\), together with some weighted arrows corresponding to the branches of \(C\). We describe an algorithmic procedure giving the sequence of multiplicities of the successive strict transforms of each branch of \(C\) from the equisingularity class of \(C\). curves embedded in a rational surface singularity; proximity; divisors; equisingularity Reguera, A. J.: Courbes et proximité sur LES singularités rationnelles de surface. CR acad. Sci. Paris sér. I math. 319, 383-386 (1994) Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Curves in algebraic geometry, Local deformation theory, Artin approximation, etc. Curves and proximity in rational surface singularities. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J\) be the generalized Jacobian of an integral projective curve \(C\) over an algebraically closed field, and let \(\overline J\) its compactification parametrizing torsion-free sheaves of rank 1 and degree 0 on \(C\).
Suppose that \(C\) has at worst double points as singularities and is of positive genus, for a fixed line bundle \({\mathcal L}\) of degree 1, the Abel map \(A_{{\mathcal L}}: C\to\overline J\) is a closed embedding.
In a previous paper jointly with \textit{M. Cagné} [J. Lond. Math. Soc., II. Ser. 65, No. 3, 591--610 (2002; Zbl 1060.14045)] the authors proved that the pullback map \(A^*_{{\mathcal L}}: \text{Pic}^0_{\overline J}\to J\) is an isomorphism which is independent of \({\mathcal L}\). The closure of \(\text{Pic}^0_{\overline J}\) in the moduli space of torsion-free sheaves of rank 1 on \(\overline J\) is a natural compactification.
The main result of the paper under review is
Theorem 4.1. Let \(C/S\) be a flat projective family of integral curves with at worst ordinary nodes and cusps, then the extended map \(A^*_{{\mathcal L}}\) is an isomorphism. compactified Jacobian; autoduality; curves with double points M. Melo, A. Rapagnetta and F. Viviani. \textit{Fourier-Mukai and autoduality for compactified Jacobians. I}. At http://arxiv.org/abs/1207.7233, with an appendix by A.C. López-Martín. Picard schemes, higher Jacobians, Jacobians, Prym varieties The compactified Picard scheme of the compactified Jacobian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Since A.~Grothendieck discovered that the foundation of algebraic geometry has to be the language of schemes it became rather hard to non-specialists to follow basic ideas of the subject as well as to get an feeling for recent research in algebraic geometry. This is in particular true for mathematicians working on complex analysis, which had and has great influence of algebraic geometry. The book under review is intended for the working mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. The book is not conceived as a subsitute for an introduction to algebraic geometry as given by \textit{R.~Hartshorne} [``Algebraic geometry'', Graduate Texts in Math. 52 (1977; Zbl 0367.14001)], or by \textit{I. R. Shafarevich} [``Basic algebraic geometry'', Grundlehren 213 (1974); translation from the Russian (1972; Zbl 0258.14001)], for example. It is the authors' goal to inspire the readers to undertake a more serious study of the subject, i.e. to invite them to algebraic geometry.
The book grew out of a course of the first author at the University of Jyväskylä (Finland) for Ph. D. students and interested mathematicians whose research is quite far removed from algebra. The text is divided into eight chapters (affine algebraic varieties, algebraic foundations, projective varieties, quasi-projective varieties, classical constructions, smoothness, birational geometry, and maps to projective space) with an appendix (sheaves and abstract algebraic varieties). Starting with the affine algebraic varieties as the zero loci of sets of polynomials the basic commutative algebra is introduced and projective and quasi-projective varieties are sketched. All the material is nicely presented. Several constructions are illustrated by instructive pictures. The chapter on classical constructions covers Veronese maps, Segre products, Grassmannians, and the Hilbert polynomial. The chapter on smoothness sketches the Jacobian criterion, smoothness in families and the Bertini theorem. The chapter `Birational geometry' explains the problem of resolution of singularities, blowing up along a subvariety, birational equivalence, and the classification problem. Most of the chapters provide some information about current research. The final chapter introduces line bundles, rational maps, and very ample line bundles from a very concrete point of view. The appendix contains a short introduction to some basic notions of modern algebraic geometry.
Because of the nature of this book as an invitation it was necessary to omit several proofs and sacrifice some rigor. For all details there are references and explanations. Only a few prerequisites are presumed beyond a basic course in linear algebra. So the lectures are also intended to an interested undergraduate student. The reviewer believes that the authors convey in their invitation that the main objects in algebraic geometry and the main research questions about them are as interesting and accessible as ever, even for mathematicians far from the subject. affine variety; projective variety Smith, K.-E.; Kahanpää, L.; Kekäläinen, P.; Traves, W., An invitation to algebraic geometry, (2000), Springer-Verlag New York Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Foundations of algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Affine geometry, Real algebraic and real-analytic geometry An invitation 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 The paper under review contains three theorems. The first one is a generalization to the singular case of the Weil bound for the number of rational points of a smooth irreducible curve defined over a finite field. Note that this was also (and independently) obtained by \textit{Y. Aubry} and \textit{M. Perret} in [Manuscr. Math. 88, 467-478 (1995; Zbl 0862.11042)]. This generalization states that the Weil bound remains true if one replaces in the formula the geometric genus of the curve \(C\) by its arithmetic genus (this statement can be slightly sharpened; see the complete statement of the theorem). This is proved by desingularization of the curve: one has only to evaluate the number of points of the smooth model \(\widetilde C\) of \(C\) lying over a given singular point of \(C\), which is the object of lemma 4.1. Note that this lemma is not the best one: the given bound is twice the correct one, which is given in [op. cit.] (in particular, the sharpened version of the theorem is not the best one\dots).
The second theorem states that the arithmetic genus of a curve of degree \(d\) in projective \(n\)-space is less than \({d(d-1) \over 2}\). This is done by induction, asserting that the projection from a generic point on the hyperplane at infinity can only make an increase in the arithmetic genus and leaves the degree unchanged. Note a misprint in the proof of theorem 3.2: one should read ``\(p_a(C) \leq p_a(D)\)''.
Finally, the third theorem gives another upper bound for the arithmetic genus of a curve in projective \(n\)-space in terms of the degrees of the \((n-1)\) first polynomials defining it (note that the curve need not be a complete intersection). One has to prove that the arithmetic genus of the curve is less than the arithmetic genus of any complete intersection containing it. singular curves; Weil bound; number of rational points; smooth irreducible curve; finite field; arithmetic genus Bach, E., Weil bounds for singular curves, Appl. Algebra Eng. Commun. Comput., 7, 289-298, (1996) Curves over finite and local fields, Singularities of curves, local rings Weil bounds for singular 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 A germ of a morphism of a variety in the multiplicative linear group defines locally a divisor. \textit{P. Cartier} remarked in his thesis [Bull. Soc. Math. Fr. 86, 177--251 (1958; Zbl 0091.33501)] that we can as well consider morphisms with values in an arbitrary (say commutative) algebraic group; thus we obtain the notion of a ``divisor of type \(G\)''. In the case of the usual divisors, the problem of constructing the Picard scheme (i.e. putting a geometrical structure on the group of divisor classes in a natural way, that is in a functorial way) has drawn quite a lot of attention; in this thesis the author studies the construction of the Picard variety of type \(G\). One of the main theorems asserts, that in case \(G\) is a linear, commutative group variety, the Picard variety of type \(G\) of a complete variety exists (theorem 2.3 on page 94). The method of proof is inspired by the work of \textit{C. Chevalley} on the Picard variety which was developed around 1958 [Variétés de Picard, Sém. C. Chevalley 3 (1958/59), No. 7, 6 p. (1960; Zbl 0122.38804); Am. J. Math. 82, 435--490 (1960; Zbl 0127.37701)]; thus in this thesis we find the generalisations to the case of divisors of type \(G\) of the following notions: continuity of algebraic families of divisors, and of divisor classes, infinitesimal divisors, descent theorems and rationality questions. The work concludes with an example which shows that the restriction to linear commutative groups is quite natural, namely by showing that for a commutative group variety \(G\) in general the Picard variety of type \(G\) of a variety is not representable.
Although the style of the author is clear, thus facilitating the reading of this thesis, the reader may have some difficulties: there are several misprints (e.g. on pages 100/101 on various places wrong formulas are used), and references are not always clear. It seems that the properties of divisors of type \(G\) can be understood better if not the Zariski topology, but for example the faithfully flat quasi compact topology is used. In this direction \textit{M. Miyanishi} found results analogous to those of the present author [On the cohomologies of commutative affine group schemes. J. Math. Kyoto Univ. 8, 1--39 (1968; Zbl 0181.48801)]. As the construction of the Picard variety (in the ordinary sense) was already simplified by Grothendieck and Murre, it remains unclear why the author uses still the techniques of Chevalley, instead of for example the representability criterion of \textit{J.-P. Murre} [Publ. Math., Inst. Hautes Étud. Sci. 23, 5--43 (1964; Zbl 0142.18402)]. algebraic geometry Algebraic geometry Variété de Picard de type linéaire commutatif | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Deligne and Grothendieck have conjectured the existence of a unique natural transformation \(c_*\) from the constructible function functor to the homology functor such that the value \(c_*(1_X)\) of the characteristic function of a smooth complex algebraic variety \(X\) is the Poincaré dual of the total Chern cohomology class \(c(T_X)\cap [X]\). The conjecture has been affirmatively solved by \textit{R. MacPherson} [Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001)], see also \textit{G. Kennedy} [Commun. Algebra 18, No. 9, 2821-2839 (1990; Zbl 0709.14016)], \textit{M. Kwieciński} [``Sur le transformé de Nash et la construction du graphe de MacPherson'' (Thèse, Université de Provence 1994)] and \textit{C. Sabbah} [Astérisque 130, 161-192 (1985; Zbl 0598.32011)]. The distinguished value of the characteristic function of a variety under this transformation (called the Chern-Schwartz-MacPherson one) is isomorphic to the Schwartz class via the Alexander duality. A bivariant theory (BT) assigns to each morphism of complex algebraic varieties an abelian group. A BT is equipped with three commuting operations: product, proper push-forward and pull-back where product is associative, push-forward and pull-back are functorial, and there is a projection formula. A Grothendieck transformation between two BTs is a collection of homomorphisms preserving the above three operations. The notion of BT has been introduced by \textit{W. Fulton} and \textit{R. MacPherson} [Mem. Am. Math. Soc. 243, 165 p. (1981; Zbl 0467.55005)]. They have conjectured the existence of a Grothendieck transformation from the BT of constructible functions to the bivariant homology theory in the category of complex algebraic varieties which extends the original Chern-Schwartz-MacPherson transformation. The conjecture was solved by \textit{J.-P. Brasselet} [Astérisque 101-102, 7-22 (1981; Zbl 0529.55009)] for a certain reasonable category. The authors show that there exists a unique Grothendieck transformation from the BT of constructible functions to a certain operational bivariant Chow theory. For a constructible function \(\alpha\) on a variety \(X\), bivariant with respect to a morphism \(f:X\rightarrow Y\), they get a collection \(\gamma (\alpha)\) (which is an element of a bivariant Chow group) \((\gamma (\alpha))(g):A(Y')\rightarrow A(X\times _YY')\), one for each morphism \(g:Y'\rightarrow Y\), which are compatible with proper push-forwards and intersection products. Compatibility with flat pull-back is not required. algebraic variety; bivariant theory; intersection product; Chern-Schwartz-MacPherson class; specialization; Grothendieck transformation; characteristic function; Chow group; push-forward; pull-back L Ernström, S Yokura, Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups, Selecta Math. 8 (2002) 1 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of holomorphic vector fields and foliations Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\{\Gamma_ t | t \in \mathbb{P}^ 1\}\) be a linear pencil of projective plane curves of degree \(m\geq 3\) defined over an algebraically closed field of arbitrary characteristic. Assume that this linear pencil satisfies the following conditions:
(A1) Every member \(\Gamma_ t\) is irreducible, and the general member is smooth.
(A2) The \(m^ 2\) base points \(P_ 0,P_ 1,\ldots,P_{m^ 2-1}\) of the pencil are distinct.
Under these conditions, the generic member \(\Gamma\) of the pencil is a smooth plane curve of genus \((m-1)\cdot(m-2)/2\), defined over the rational function field \(K=k(t)\). The base point \(P_ 0\) (for example) defines an embedding of the general member into the Jacobian \(J\) of \(\Gamma\), and the remaining base points \(P_ 1,\ldots,P_{m^ 2-1}\) in \(\Gamma\) may be regarded as \(K\)-rational points in \(J\), i.e., as elements of the group \(J(K)\) of \(K\)-rational points of \(J\).
The theorem of Manin-Shafarevich states that, in the special case of \(m=3\) and under the conditions (A1) and (A2), the eight base points \(P_ 1,\ldots,P_ 8\) are independent and generate a subgroup of index 3 in the Mordell-Weil group of the elliptic curve \(\Gamma\). Recently, the author of the present paper has generalized the notion of Mordell-Weil lattices to the higher-genus case [cf. Proc. Japan Acad., Ser. A 68, 247- 250 (1992)] and, as an application of this concept, he provides a corresponding generalization of the Manin-Shafarevich theorem in this brief note under review. His generalization of the theorem of Manin- Shafarevich to linear pencils of plane curves of degree \(m\geq 3\), satisfying conditions (A1) and (A2), states that the group \(J(K)\) of \(K\)- rational points in the Jacobian \(J\) is a torsion-free abelian group of rank \(m^ 2-1\), and the points \(P_ 1,\ldots,P_{m^ 2-1}\) are independent and generate a lattice of index \(m\) in \(J(K)\).
In fact, this generalized version of the theorem of Manin-Shafarevich is an immediate corollary of a more general result on Mordell-Weil lattices, which is also proved in the present note. More precisely, the author proves the following theorem: The Mordell-Weil lattice \(J(K)\) is an integral unimodular lattice of rank \(m^ 2-1\), positive-definite with respect to the height pairing. It is even if and only if \(m\) is odd. Moreover, the points \(P_ 1, \ldots,P_{m^ 2-1}\) generate a sublattice of index \(m\), and there is a unique point \(Q\in J(K)\) such that \(mQ=P_ 1+\cdots+P_{m^ 2-1}\) and \(J(K)\) is freely generated by \(\{P_ 1, \ldots, P_{m^ 2-1},Q\}\). -- At the end, the author points out that his proof can be generalized to Lefschetz pencils of hyperplane sections of smooth algebraic surfaces of degree \(d\) in a projective space \(\mathbb{P}^ N\) with trivial Picard variety. The proof will be published elsewhere, but the result is illustrated by the instructive example of the Fermat surface of degree 4 in \(\mathbb{P}^ 3\). Jacobian variety; rational points; linear pencil of projective plane curves; Mordell-Weil lattices; Manin-Shafarevich theorem; height pairing; Lefschetz pencils of hyperplane sections Shioda, T.: Generalization of a theorem of Manin-Shafarevich. Proc. Japan acad. 69A, 10-12 (1993) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves, Special surfaces Generalization of a theorem of Manin-Shafarevich | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 singular curve of genus \(g\) over the complex field, with \(g\) nodes as singularities and having a normalization \(\pi:\mathbb{P}^ 1\to C\). It is known [cf. \textit{D. Mumford}, ``Tata lectures on theta. II'', Prog. Math. 43 (1984; Zbl 0549.14014); chapter III b, \S5] that the (generalized) Jacobian \(\text{Jac} C\) is isomorphic to the product of \(g\) copies of the multiplicative group \(\mathbb{C}^*\). This space can be compactified, as analytic space, to \(\text{Jac} C \cong (\mathbb{P}^ 1)^ g/ \sim_ \nu\), where \(\sim_ \nu\) is a suitable equivalence relation determined by a symmetric matrix \(\nu\) such that, for \(1\leq i\), \(j\leq g\), \(\nu_{ii}=0\) and \(\nu_{ij}\) is invertible for \(i\neq j\).
The author proves that, when \(k\) is a ring and \(\nu\) is a symmetric matrix with entries in \(k\), satisfying the above conditions, then the quotient \(A_ \nu\) of \((\mathbb{P}^ 1)^ g \times k\) with respect to the equivalence relation induced by \(\nu\), is a stable quasi-abelian scheme and he explicitly describes that scheme as \(\text{Proj} R\), where \(R\) is a graded ring of totally degenerate theta functions relative to \(\nu\). By means of this description the author proves that the quotient \(A_ \nu\) is a projective scheme and that the canonical map \((\mathbb{P}^ 1)^ g \times k \to A_ \nu\) is finite. generalized Jacobian; totally degenerate theta function; singular curve Azad, H., and J. J. Loeb,Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N. S.3(4) (1992), 365--375. Theta functions and abelian varieties On the algebra of totally degenerate theta 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 [For the entire collection see Zbl 0747.00038.]
The paper proceeds as follows: We first review relevant facts from commutative algebra, étale and crystalline cohomology. These are mostly somehow known, or at least dwell on well known ideas, but they cannot be found in the literature in such a form that we can use them directly. We also use the occasion to generalise many results. Some of the main new features are:
We extend the comparison-theory of \textit{J.-M. Fontaine} and \textit{G. Laffaille} [cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547-608 (1982; Zbl 0579.14037)] to families of \(F\)-crystals.
We construct logarithmic versions of crystalline cohomology, for non proper varieties with a good compactification (hidden behind this there is a whole theory of ``logarithmic commutative algebra'', which however we choose not to develop in detail).
We make precise the relation between étale cohomology and Galois- cohomology.
We allow nontrivial systems of coefficients.
After that we prove the comparison results, first in the absolute case, and then more general for direct images under ``log-smooth'' maps. We conclude by an application to the theory of finite flat group schemes, giving a complete description in terms of ``semilinear algebra''. Finally we show that our method also allows to settle the ``de Rham conjecture''. de Rham conjecture; crystalline cohomology; étale cohomology; Galois- cohomology Faltings, G., Crystalline cohomology and \textit{p}-adic Galois representations, (Algebraic Analysis, Geometry, and Number Theory. Algebraic Analysis, Geometry, and Number Theory, Baltimore, MD, 1988, (1989), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD), 25-80 \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies, de Rham cohomology and algebraic geometry Crystalline cohomology and \(p\)-adic Galois-representations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1949 \textit{A.~Weil} [Bull. Am. Math. Soc. 55, 497--508 (1949; Zbl 0032.39402)] made his famous conjectures concerning the number of solutions of equations over finite fields and their relation to the topological properties of the variety defined by the corresponding equations over \(\mathbb C\). In the sixties, in attempts to solve the problem, M.~Artin and A.~Grothendieck developed an adequate étale cohomology theory, which was later used by \textit{P.~Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 43, 273--307 (1974; Zbl 0287.14001)] in his proof of the Weil conjectures. A Weil cohomology theory is a contravariant functor from the category of irreducible, smooth projective varieties over an algebraically closed field \(k\) of prime characteristic \(p\) to the category of finite-dimensional graded anticommutative algebras over a coefficient field \(F\), satisfying certain sophisticated axioms. M.~Artin and A.~Grothendieck developed a Weil cohomology theory with coefficients in \({\mathbb Q}_l\), the field of \(l\)-adic numbers, for any prime \(l\neq p\). An intermediate step was to develop a Weil cohomology theory with coefficients in the residue rings \({\mathbb Z}/l^n\mathbb Z\), for all \(n>0\). The author combines model-theoretic insight and the existing theory of étale cohomology with torsion coefficients to give a new instance of a Weil cohomology theory with coefficients in a field of characteristic zero, specifically, in a pseudo-finite field (\(=\) infinite model of the first order theory of finite fields). He considers an ultraproduct of Weil cohomologies with coefficients in \({\mathbb F}_l\), where \(l\) runs over an infinite set of primes, and checks the axioms of Weil cohomology for the ultraproduct cohomology, which has its coefficients in the pseudo-finite field of characteristic zero that is the ultraproduct of the fields \({\mathbb F}_l\). A different model-theoretic approach was proposed by \textit{A.~Macintyre} [in: Connections between model theory and algebraic and analytic geometry, Quad. Mat. 6, 179--199 (2000; Zbl 1078.14021)] who showed that the axioms of Weil cohomology are, in a natural sense, first order, and that, in particular, one can form ultraproducts of Weil cohomology theories and these are again models of the axioms. ultraproduct; pseudo-finite field Tomašić, Ivan: A new Weil cohomology theory, Bull. London math. Soc. 36, No. 5, 663-670 (2004) Étale and other Grothendieck topologies and (co)homologies, Ultraproducts and related constructions A new Weil cohomology 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 author constructs the local Jacobian of a relative formal curve and proves a relative duality formula.
Let \({\mathcal F}\) be the \(S\)-group extension of the completion \(\check W\), of the universal \(S\)-Witt vector group \(W\), by the group of units \({\mathcal O}_S [[T]]^*\).
The author proves the following theorem: Let \(S= \text{Spec} (A)\) be a noetherian affine scheme. Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Let \(B\) be an \(A\)-algebra adic isomorphic to \( A[[T]]\), \({\mathcal X} = \text{Spf} (B)\), \({\mathcal U}= \text{Spec} (A[[T]][T^{-1}])\). In the case then for any section \(\sigma \in G({\mathcal U})\) there exists unique homomorphism \( h: {\mathcal F} \to G\) such that \(\sigma = h_{\text{omb}} \circ f\) where \(f: {\mathcal U} \rightarrow {\mathcal F}_{\text{omb}}\) is an Abel-Jacobi morphism of \(({\mathcal X}, {\mathcal F})\), i.e. the arrow \(\mathrm{Hom}_{S\text{-gr}} ({\mathcal F},G) \rightarrow G({\mathcal U})\) is bijective.
The proof of the theorem is reduced to the proof of the theorem 1.4.4 (see below).
This paper is the next in a sequence of papers by the author in which he is involved with the Grothendieck program concerning global and local dualities with continuous coefficients. The relative generalized Jacobian of the smooth curve \(X - D\) and an Abel-Jacobi morphism \(X - D \rightarrow J\) are constructed in the author's papers [C. R. Acad. Sci., Paris, Sér. A 289, 203--206 (1979; Zbl 0447.14005); Prog. Math. 87, 69--109 (1990; Zbl 0752.14023)].
Let \({\mathfrak X}\) be the formal completion of \(X\) along \(D\), \(\text{omb}({\mathfrak X}) = \text{Spec} (\Gamma({\mathfrak X}, {\mathcal O}_{\mathfrak X}) = \text{Spec} (A[[T]])\).
Let \(\rho: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}^{0}, G) \to F(G)\) and \(\rho^{+}: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}, G) \to F^{+} (G)\).
Theorem 1.4.4. The notations are taken above. Let \(A\) be a Noetherian ring and \(S = \text{Spec} (A)\). Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Then \(\rho \) and \(\rho^{+}\) are isomorphisms.
The proof of the theorem is given in sections 2--5.
This interesting article is done (is presented) by the author in the spirit of the algebraic geometry by Grothendieck, Verdier, Artin, Deligne, Saint-Donat [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)], by Grothendieck and Demazure [Schémas en groupes. I: Propriétés générales des schémas en groupes. Exposés I à VIIb. Séminaire de Géométrie Algébrique 1962/64, dirigé par Michel Demazure et Alexander Grothendieck. Revised reprint. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0207.51401)] together with some new developments by \textit{A. Beilinson} [Fields Institute Communications 56, 15--82 (2009; Zbl 1186.14019)], by \textit{K. Rülling} [J. Algebr. Geom. 16, No. 1, 109--169 (2007; Zbl 1122.14006)], by \textit{M. Kapranov} and \textit{É. Vasserot} [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113--133 (2007; Zbl 1129.14022)] and by \textit{A. N. Parshin} [in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 362--392 (2011; Zbl 1266.11118)].
The paper under review closes with some results on the autoduality of \({\mathcal F}\) in the sense of Cartier. local symbol(s); tame symbol; Contou-Carrère symbol; local Abel- Jacobi morphism; universal Witt bivectors; cartier duality; local Jacobian; relative formal curve; Witt residues; local relative class field theory; Rosenlicht Jacobian Contou-Carrère, C., Rend. Semin. Mat. Univ. Padova, 130, 1-106, (2013) Group schemes, Symbols and arithmetic (\(K\)-theoretic aspects) Local Jacobian of a relative formal 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 In recent papers Masser and Zannier have proved various results of ``relative Manin-Mumford'' type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay [\textit{D. Bertrand} et al., Proc. Edinb. Math. Soc., II. Ser. 59, No. 4, 837--875 (2016; Zbl 1408.14139)] settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell's equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves. elliptic families; Manin-Mumford; Pell's equation; abelian varieties Heights, Algebraic functions and function fields in algebraic geometry, Elliptic curves, Group schemes, Classical hypergeometric functions, \({}_2F_1\) Relative Manin-Mumford in additive extensions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0655.00011.]
In Part I [in Algebraic and topological theories, Kinosaki 1984, 305-327 (1986; Zbl 0800.14008)], the author generalized Maruyama's method of elementary transformations of algebraic vector bundles on locally Noetherian schemes. The present article is a direct continuation of Part I, in the following sense. By analyzing the generalized elementary transformations and their basic properties more closely, the author establishes a general geometric method of construction for algebraic vector bundles on arbitrary Noetherian schemes. Moreover, this method of construction allows one to describe the basic properties of the corresponding bundles in a rather explicit way and, in particular, to rediscover some of the known explicit vector bundles on special algebraic varieties.
More precisely, let \(X\) be a Noetherian scheme, \(Z\) a normal Cartier divisor in \(X\), \(\{W_1,\cdots,W_r\}\) a finite set of effective Weil divisors in \(Z\), and \(\mathcal{L}\) a linear system on \(Z\) with \(\dim \mathcal{L}=r-1\). Assume that the Weil divisors \(W_i\), \(1\leq i\leq r\), are mutually rationally equivalent and invertible (in a sense specified in the paper). Moreover, assume that \(\mathcal{L}\) is---in the same specified sense---invertible along \(Z\). Then the author constructs, with respect to these data, rank-\(r\) vector bundles \(\mathcal{E}(Z,W_1,\cdots,W_r)\) and \(\mathcal{E}(Z,\mathcal{L})\) on \(X\), together with global sections \(s_1,\cdots,s_r\) which redefine \(Z\) and \(W_i\) as determinantal schemes. In view of the main theorem of Part I, the author shows that on a nonsingular quasiprojective variety \(X\) defined over an algebraically closed field \(k\), any algebraic vector bundle is obtained, up to tensoring by a line bundle, by this construction procedure.
All this is done in Section 1 of the paper, whilst Section 2 is devoted to the study of the fundamental properties of the bundles \(\mathcal{E}(Z,W_1,\cdots,W_r)\) and \(\mathcal{E}(Z,\mathcal{L})\). Among other things, the author provides a base change theorem for these bundles, isomorphism theorems, explicit computations concerning their Chern classes and, in particular, a stability criterion for the rank-2 bundles \(\mathcal{E}(Z,W_1,W_2)\).
The final section 3 deals with a concrete application of the foregoing results. Namely, the author constructs the well-known Horrocks-Mumford bundle on \({\mathbb{P}}^4_{\mathbb{C}}\) by the method described above, and studies its properties from the viewpoint of elementary transformations and the explicit results obtained in section 2. This yields, for example, another elementary proof of the results on the structure of the cohomology groups \(H^i({\mathbb{P}}^4_{\mathbb{C}},\mathcal{E}(m))\) of the twisted Horrocks-Mumford bundle \(\mathcal{E}(m)\), as they were obtained by \textit{G. Horrocks} and \textit{D. Mumford} [cf. Topology 12, 63-81 (1973; Zbl 0255.14017)] in a different way.
Altogether, the author's work on elementary transformations of algebraic vector bundles is a fundamental contribution to the constructive theory and classification theory of vector bundles in general. Cartier divisor; Weil divisor; determinantal schemes; (twisted) Harrocks-Mumford bundle; constructive theory; classification theory of vector bundles Hideyasu Sumihiro, Elementary transformations of algebraic vector bundles. II, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 713 -- 748. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic cycles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Elementary transformations of algebraic vector bundles. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraically closed field of characteristic \(p>0\)
and let \(S\subset M_g\) be any complete subvariety of positive
dimension of the coarse
moduli space \(M_g\) over \(k\) of proper smooth curves of genus \(g\geq 2\).
In this paper the author shows the existence of some specialization
of points \(x\rightsquigarrow y\) on \(S\) such that the
specialization homomorphism of geometric fundamental groups of the
represented curves \(\pi_1(C_{\bar x})\to \pi_1(C_{\bar y})\)
is not an isomorphism. For the proof, we may assume
\(S\) is a complete curve with generic point \(\eta\).
Then, combining Tamagawa's technique of equi-characteristic
deformation of generalized Prym varieties
[\textit{A. Tamagawa}, J. Algebr. Geom. 13, No.~4, 675--724 (2004; Zbl 1100.14021)] with the isotriviality
of complete families of ordinary abelian varieties due to Raynaud, Szpiro and Moret-Baily
[\textit{L. Moret-Bailly}, Pinceaux de variétés abéliennes. Astérisque, 129
(1985; Zbl 0595.14032)], one finds a closed point \(s_0\) on \(S\) and a prime
\(\ell (\neq p)\gg 0\) such that the Jacobians of certain corresponding \(\ell\)-cyclic étale covers
of \(C_{\bar s_0}\) and of \(C_{\bar \eta}\) have different \(p\)-ranks. anabelian geometry Mohamed Saïdi, On complete families of curves with a given fundamental group in positive characteristic, Manuscripta Math. 118 (2005), no. 4, 425 -- 441. Coverings of curves, fundamental group, Étale and other Grothendieck topologies and (co)homologies, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) On complete families of curves with a given fundamental group in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A theorem generalizing the Lang-Weil estimates for the number of rational points of a variety over a finite field is proved by model-theoretic methods in [\textit{Z. Chatzidakis}, \textit{L. van den Dries} and \textit{A. Macintyre}, J. Reine Angew. Math. 427, 107-135 (1992; Zbl 0759.11045)]. The paper under review gives an algebraic proof of the theorem. An algorithm is constructed to find formulas and constants occurring in the theorem. The authors use Galois stratification as the main tool in their investigations.
For the convenience of the reader the notion of a ring cover and the Artin symbol are defined and their basic properties are stated. A section of a paper is dedicated to some basic definitions and concepts from algebraic geometry. There exist several slightly different definitions of Galois stratification. The authors use the version of \textit{M. D Fried} and \textit{M. Jarden} [Field arithmetic (1986; Zbl 0625.12001)]. Another important tool in the paper is a non-regular analog of the Chebotarev density theorem. Lang-Weil estimates; number of rational points; variety over a finite field; Galois stratification; ring cover; Artin symbol M. D. Fried, D. Haran and M. Jarden, Effective counting of the points of definable sets over finite fields, Israel J. Math. 85(1--3) (1994), 103--133. Curves over finite and local fields, Finite fields (field-theoretic aspects), Finite ground fields in algebraic geometry, Rational points Effective counting of the points of definable sets 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 These notes are based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. The notes provide an introduction to and summary of many of the major ideas and results in the theories of rational and integral points on algebraic varieties.
Beginning with a study of height functions, the book progresses to a proof of the weak Mordell-Weil theorem using the Chevalley-Weil theorem. After a brief discussion of descent arguments the full Mordell-Weil theorem is proved.
Following this is a description of Mordell's conjecture, and a discussion of the major results (Chabauty's theorem, the Manin-Demyanenko theorem, Mumford's theorem) and applications that preceded Faltings' proof. - Integral points and quasi-integral points on curves are studied via Siegel's theorem and Baker's method. The study includes a discussion of effectivity, and applications to the arithmetic of curves.
The problem of lifting rational points under morphisms is introduced with the notion of thin sets and Hilbert's irreducibility theorem. Applications to the construction of Galois extensions with certain prescribed Galois groups, and to the construction of elliptic curves of large rank are given. There is also a discussion of the large sieve, and applications to the study of thin sets.
Finally, an appendix contains some remarks about the class number 1 problem, and connections with elliptic and modular curves. rational points; integral points; height functions; Mordell-Weil theorem; Mordell's conjecture; lifting rational points; Hilbert's irreducibility theorem; class number one problem Serre, J.P.; ; Lectures on the Mordell-Weil Theorem: Braunschweig, Germany 1989; . Rational points, Elliptic curves, Elliptic curves over global fields, Heights, Arithmetic ground fields for curves, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to number theory, Global ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Arithmetic ground fields for abelian varieties, Units and factorization Lectures on the Mordell-Weil theorem. Transl. and ed. by Martin Brown from notes by Michel Waldschmidt | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a reductive group scheme over a regular local ring \(R\). Grothendieck-Serre conjecture predicts that any principal \(G\)-bundle \(\mathcal{P}\) over \(R\) is trivial if \(P\) is generically trivial. This conjecture has been proved earlier by Fedorov-Panin when \(R\) contains a field.
This paper concerns the mixed characteristic case, i.e. when \(R\) does not contain a field. By some reductions, this case can be reduced to the situation when \(R\) is the local ring of a closed point \(x\) in an integral scheme \(X\) projective over \(\mathbb{Z}_{(p)}\), where \(\mathbb{Z}_{(p)}\) which is the localization of \(\mathbb{Z}\) at the prime \(p\). The main result of this paper is that, under the following assumptions: 1) the fiber \(X_p\) is generically reduced; 2) the set of points where \(X\) is not regular intersects \(X_p\) in a subset codimension at least two in \(X_p\); 3) the group scheme \(G\) is split, the Grothendieck-Serre conjecture holds.
This paper also proves another theorem that is used in proving the main theorem. Let \(R\) be a Noetherian local ring and let \(H\) be a split reductive group scheme over \(R\). The theorem asserts that If a principal \(H\)-bundle \(\mathcal{P}\) over \(\mathbb{A}^1_R\) is trivial over the complement of a closed subscheme that is finite over \(\mathrm{Spec} R\), then \(\mathcal{P}\) is trivial over \(\mathbb{A}^1_R\). reductive group scheme; principal bundle; Grothendieck-Serre conjecture Group schemes, Arithmetic ground fields (finite, local, global) and families or fibrations, Homogeneous spaces and generalizations, Linear algebraic groups over adèles and other rings and schemes, Exceptional groups, Quadratic forms over local rings and fields On the Grothendieck-Serre conjecture on principal bundles in mixed 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 According to the author, this booklet is a progress report purporting ``to establish the correspondence between curves and function fields in one variable'' and then to give a proof of the celebrated theorem of A. Weil, the ``Riemann Hypothesis'' for a smooth projective curve defined over a finite field. Apart from some elementary prerequisites from commutative algebra, the Riemann-Roch theorem for curves, and the theorem on the existence of a smooth model of a curve, which are stated without proof, the author's exposition is self-contained. Weil's ``Riemann Hypothesis'' is proved by a version of Stepanov's method going back to \textit{E. Bombieri} [Proc. Symp. Pure Math. 28, De Kalb 1974, 269-274 (1976; Zbl 0351.14009)]. The reader interested in this history of and the literature on this subject may consult, for instance, the monograph by \textit{S. A. Stepanov} [Arithmetic of algebraic curves, Monographs in Contemporary Mathematics (1994; Zbl 0862.11036)]. exponential sums; Weil's Riemann hypothesis; zeta functions; curves; function fields in one variable Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Curves, function fields and the Riemann hypothesis | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Galois stratification for formulas in the language of rings, introduced by \textit{M.~Fried} and \textit{G.~Sacerdote} [``Solving diophantine problems over all residue class fields of a number field and all finite fields'', Ann. Math. (2) 104, 203--233 (1976; Zbl 0376.02042)], provides an explicit description of definable sets over finite fields by giving a quantifier elimination procedure for so-called Galois formulas associated with Galois covers of varieties. This strengthened results by \textit{J.~Ax} [``The elementary theory of finite fields'', Ann. Math. (2) 88, 239--271 (1968; Zbl 0195.05701)] on the theory of finite fields. \textit{E.~Hrushovski} [``The elementary theory of the Frobenius automorphisms'', \url{arXiv:math/0406514}] generalized several aspects of Ax' work by developing a theory of difference schemes and studying the elementary theory of Frobenius difference fields, i.e.~algebraically closed fields of positive characteristic together with a power of the Frobenius endomorphism.
The present paper develops a theory of Galois stratification for formulas in the language of difference rings. For this, the notions of twisted Galois covers of difference schemes and twisted Galois stratifications are introduced, and a quantifier elimination is proven. This leads to an algebraic description of definable sets over Frobenius difference fields.
After publication, a significantly extended version under the same title appeared as [\url{arXiv:1112.0802}]. difference ring; difference scheme; Galois stratification; Frobenius automorphism Tomašić, Ivan: Twisted Galois stratification, (2012) Difference algebra, Model-theoretic algebra, Automorphisms and endomorphisms of algebraic structures, Coverings in algebraic geometry, Finite ground fields in algebraic geometry Twisted Galois stratification | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0717.00009.]
\textit{P. Deligne} proved the Weil conjectures in the context of \(\ell\)- adic étale cohomology [the signum \([De]\) refers to his paper `La conjecture de Weil. II', Publ. Math., Inst. Hautes Étud. Sci. 52, 137- 252 (1980; Zbl 0456.14014) which, however, mistakingly fails in the list of references]. The present paper contains analogous results with \(\mathbb{Q}_ p\)-crystalline cohomology, for instance how the weights of a mixed convergent \(F\)-isocrystal (analogue of \(\mathbb{Q}_ \ell\)-adic étale sheaf) carry over to its cohomology. In order to treat open varieties, one has to consider differentials with logarithmic poles at infinity, etc., and to this end logarithmic analogues of notions of commutative algebra are given, from the notion of a log-étale map (reminding the very beginning of SGA) to that of logarithmic crystalline cohomology. Results include the crystalline Lefschetz trace formula and unipotency of crystalline monodromy. open varieties; logarithmic poles; log-étale map; logarithmic crystalline cohomology; Weil conjectures Gerd Faltings, \?-isocrystals on open varieties: results and conjectures, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 219 -- 248. \(p\)-adic cohomology, crystalline cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) F-isocrystals on open varieties. Results and conjectures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone-von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is \(\text{Spec}\overline{\mathbb F}_p\)), we obtain the new notion of mod \(p\) Weil representations of \(p\)-adic metaplectic groups on \(\overline{\mathbb F}_p\)-vector spaces. The mod \(p\) Weil representations admit an alternative construction starting from a \(p\)-divisible group with a symplectic pairing. We have been motivated by a few possible applications, including a conjectural mod \(p\) theta correspondence for \(p\)-adic reductive groups and a geometric approach to the (classical) theta correspondence. abelian varieties; Weil representations; Heisenberg groups Shin, S.W.: Abelian varieties and Weil representations. Algebra Number Theory 6(8), 1719--1772 (2012) Theta series; Weil representation; theta correspondences, Abelian varieties of dimension \(> 1\), Theta functions and abelian varieties Abelian varieties and Weil representations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A. Grothendieck constructed the Hilbert scheme \(\text{Hilb}^n_X\) of \(n\) points on \(X\), for any quasi-projective scheme \(X\) on a noetherian base scheme \(S\). In the paper under review, the authors are interested in showing the existence of the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\), where \(P\) is a (non necessarily closed) point on such a scheme \(X\). A natural candidate would be \(\bigcap_{P\in U_\alpha}\text{Hilb}^n_{U_\alpha}\), where \(U_\alpha\) varies in the set of open subsets of \(X\) containing \(P\). But in general an infinite intersection of open subschemes of a scheme is not a scheme. It is a scheme if one takes only locally principal open subschemes.
The authors introduce and study the notion of generalized fraction rings and localized subschemes, of which \(\text{Spec}({\mathcal O}_{X,P})\) is a particular case. They prove that, if \(X\) is a scheme such that \(\text{Hilb}^n_X\) exists, then the functor of points of a localized scheme \({\mathcal S}^{-1}X\) is representable. As a particular case, they get the following result:
If \(X\rightarrow S\) is a projective morphism of Noetherian schemes and \(P\) is a point in \(X\), then the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\) exists and coincides with the intersection of the Hilbert schemes of \(n\) points of the open subschemes of \(X\) containing \(P\). localized schemes; determinants; fraction rings Parametrization (Chow and Hilbert schemes) Infinite intersections of open subschemes and the Hilbert scheme of points. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a field admitting a discrete valuation ring \({\mathcal O}_K\) with algebraically closed residue field of positive characteristic and spectrum \(S\). Let \(X_K\) be a torsor of an elliptic curve over \(K\) and \(X\) a proper minimal regular model of \(X_K\) over \(S\). The relative Picard functor \(\text{Pic}^0_{X/S}\) is in general not representable. However, \textit{M. Raynaud} showed [Publ. Math., Inst. Hautes Étud. Sci. 38, 27--76 (1970; Zbl 0207.51602)] that there exists an epimorphism of fppf-sheaves \(q: \text{Pic}^0_{X/S}\to J\), where \(J\) denotes the identity component of the Néron model of \(\text{Pic}^0_{X_K/K}\) over \(S\). It is the aim of this paper to study the morphism \(q\).
The main result is the determination of the kernel of \(q\) and to show that, using Greenberg realization functors, that \(q\) is pro-algebraic in nature, hence giving an exact sequence of pro-algebraic groups. The morphism \(q\) can be thought of as an analogue of the norm map in local class field theory studied by \textit{J.-P. Serre} [Bull. Soc. Math. Fr. 89, 105--154 (1961; Zbl 0166.31103)]. Moreover, the results of the paper are interpreted in relation to the duality theory for torsors under abelian varieties due to Shafarevich. elliptic fibrations; models of curves; Shafarevich pairing; abelian varieties; Picard functor; pro-algebraic groups Bertapelle, A.; Tong, J., On torsors under elliptic curves and serre's pro-algebraic structures, Math. Z., 277, 91-147, (2014) Arithmetic ground fields for abelian varieties, Picard schemes, higher Jacobians On torsors under elliptic curves and Serre's pro-algebraic 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 We show that assuming the standard conjectures, for any smooth projective variety \(X\) of dimension \(n\) over an algebraically closed field, there is a constant \(c>0\) such that for any positive rational number \(r\) and any polarized endomorphism \(f\) of \(X\), we have
\[
\Vert G_r \circ f \Vert \le c \deg (G_r \circ f),
\]
where \(G_r\) is a correspondence of \(X\) so that for each \(0\le i\le 2n\), its pullback action on the \(i\)-th Weil cohomology group is the multiplication-by-\(r^i\) map. This inequality is known to imply the generalized Weil Riemann hypothesis and is a special case of a more general conjecture by the authors' work [``A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity'', Preprint, \url{arXiv:2102.04405}]. polarized endomorphism; positive characteristic; standard conjectures; correspondence; Weil's Riemann hypothesis; algebraic cycle Positive characteristic ground fields in algebraic geometry, Algebraic cycles, Étale and other Grothendieck topologies and (co)homologies, Dynamical systems over finite ground fields, Arithmetic ground fields for abelian varieties An inequality on polarized endomorphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0547.00007.]
Let \(M_ g\) (resp. \(A_ g)\) be the moduli space of complete smooth curves of genus g (resp. principally polarized abelian varieties of dimension g). Here we work only over the field of complex numbers. There is a canonical holomorphic map \(i: M_ g\to A_ g\) by associating a curve C with its Jacobian variety J(C), which is known to be injective (Torelli theorem). There arises naturally a problem to characterize the image of i, which is now called ''the Schottky problem'' (in a wide sense). Here the author gives a compact but very well organized review on recent development on this subject. According to \textit{D. Mumford}'s earlier work ''Curves and their Jacobians'' (1975; Zbl 0316.14010) the author classifies 4 approaches and gives recent results in each direction: [1] algebraic equations (to find the relations of theta constants characterizing the closure of Im i in \(A_ g)\); [2] d trisecants (to use special properties of the Kummer surface of the Jacobian); [3] geometry of the moduli space (to use a special embedding \(A_ g(2,4)\) in \({\mathbb{P}}^ N\) by theta constants); [4] rings of differential operators (to characterize the image as the set of points where a certain theta function with the corresponding modulus satisfies the so-called K-P equation (abbr. to Kadomtsev-Petviashvili-Novikov's conjecture). The author himself gives a main contribution to the case [3] with van Geemen. The most definite result is Shiota's solution for the Novikov's conjecture by using the theory of K-P hierarchy due to M. Mulase. Now almost all papers in the references have been published: e.g. \textit{B. van Geemen} [Invent. Math. 78, 329-349 (1984; Zbl 0568.14015)]; \textit{T. Shiota} [Invent. Math. 83, 333-382 (1986)] and \textit{G. E. Welters} [Ann. Math., II. Ser. 120, 497- 504 (1984; Zbl 0574.14027)]. moduli space of complete smooth curves; principally polarized abelian varieties; Torelli theorem; Schottky problem; Kummer surface of the Jacobian; theta constants; K-P equation Jacobians, Prym varieties, Families, moduli of curves (analytic), Theta functions and abelian varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Partial differential equations of mathematical physics and other areas of application, Families, moduli of curves (algebraic) The Schottky 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 In part I of this paper [Invent. Math. 114, No. 3, 565-623 (1993; Zbl 0815.14014)], \textit{M. S. Narasimhan} and the author had proved that the space of generalized theta functions on the moduli space of semi-stable rank-2 vector bundles of degree \(d\) over a complex curve of genus \(g\geq 4\), with precisely one node, is (non-canonically) isomorphic to the corresponding space of generalized theta functions for the desingularization of the base curve. Moreover, it was shown, and used for the proof of the isomorphism theorem, that (again for \(g\geq 4\)) the first cohomology group of the theta bundle on the moduli space \(U_X (d)\) of semi-stable rank-2 bundles of degree \(d\), over the nodal base curve \(X\), is trivial. Finally, the authors had announced that the restricting condition \(\geq 4\) for the genus of \(X\) can actually be dropped and, as to that fact, referred to a forthcoming paper.
The present article, the second in a series of planned papers, provided the promised proof of the announced more general result. The proof of the vanishing theorem: ``\(H^1 (U_X (d), \Theta)= (0)\) for a given nodal curve of arithmetic genus \(g\geq 1\) and with precisely one node'', together with the derivation of a concrete Verlinde formula for the dimension of the space \(H^0 (U_X (d), \Theta)\) of generalized theta functions on \(U_X (d)\), is based on two new ingredients. One of them is a rather general result from geometric invariant theory and invariant cohomology, which is certainly of independent interest. More precisely, this auxiliary result states that, for a given projective variety acted on by a reductive algebraic group \(G\), the invariant cohomology (with coefficients in an ample \(G\)-linearized line bundle \(L\)) coincides with that of the open set of semi-stable points (with respect to \(G\)). The second basic ingredient is a statement about the effect of certain ``Hecke transformations'' on quasi-parabolic rank-2 sheaves over genus-zero curves with one node.
The proof of the Verlinde formula for \(\dim_C H^0 (U_X (d), \Theta)\) then uses the vanishing theorem, \(H^1 (U_X (d), \Theta)= (0)\), and an induction argument with respect to the genus \(g\). As to the starting point \((g=0)\), the corresponding Verlinde formula was established by \textit{K. Gawȩdzki} and \textit{A. Kupiainen} in 1991 [cf.: Commun. Math. Phys. 135, No. 3, 531-546 (1991; Zbl 0722.53084)]. generalized theta functions; vanishing theorem; Verlinde formula Ramadas T.R.: Factorization of generalized theta functions. II. The Verlinde formula. Topology 35, 641--654 (1996) Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Theta functions and abelian varieties Factorisation of generalised theta functions. II: The Verlinde formula | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review aims to study a new Grothendieck topology on the category of quasi-compact quasi-separated schemes, the so-called arc-topology. It is a slight refinement of the v-topology and h-topology for more general non-noetherian schemes. Covers in the arc-topology are tested via rank \(\leq 1 \) valuation rings. Examples of arc sheaves include étale cohomology with torsion coefficients and perfect complexes on perfect \(\mathbb{F}_p\)-schemes. Using the arc-desent, Bhatt (the first author) and Scholze established the étale comparison theorem for prismatic cohomology.
As applications, the authors reproved two classical results in étale cohomology: (i) the Gabber-Huber affine analog of proper base change theorem; (ii) the Fujiwara-Gabber theorem for punctured henselian pairs. They also gave an application to Artin-Grothendieck's vanishing theorem in rigid analytic geometry, which strengthens the results of \textit{D. Hansen} [Compos. Math. 156, No. 2, 299--324 (2020; Zbl 1441.14085)]. Artin-Grothendieck vanishing theorem; étale cohomology; excision; Grothendieck topologies; proper base change; rigid analytic geometry; valuation rings Étale and other Grothendieck topologies and (co)homologies, Sheaves in algebraic geometry, Rigid analytic geometry The \(\operatorname{arc}\)-topology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The book under review is the first part of the author's two-volume text ``Lectures on Algebraic Geometry''. Grown out of some series of lectures given by the author at the University of Bonn, Germany, these two volumes are to provide a thorough introduction to the methods of modern abstract algebraic geometry in their scheme-theoretic and cohomological setting. As the author points out in the preface to the current book, his original goal was to write a profound monograph on the cohomology theory of arithmetic groups, his preponderant topic of interest and research, which, however, necessarily had to be based on large parts of the extensive modern framework of homological algebra, sheaf cohomology, algebraic geometry, and arithmetic geometry. Thus, in order to develop this indispensable framework of its own interest and importance independently, thereby providing a comprehensive and appropriated introduction to modern algebraic geometry in itself, the author has decided to publish these fundamental parts in the first two volumes of his planned overall treatise, which intendedly will culminate in a forthcoming third volume on the ultimate, principal topic: the cohomology of arithmetic groups and their allied quotient spaces in algebraic number theory and arithmetic algebraic geometry.
The present first volume is predominantly devoted to those concepts, methods, and techniques which are absolutely basic for both formulating and treating algebraic geometry in its scheme-theoretic and cohomological language à la Grothendieck and his successors. Actually, two thirds of this volume deal with the fundamental prerequisites from general category theory, abstract homological algebra, sheaf theory, and cohomology theory of sheaves, nevertheless so with a prevailing view toward their (later) applications to algebraic geometry, algebraic topology, and complex-analytic geometry. The first steps into actual algebraic geometry are undertaken in the last third of the book, where mainly the analytic theory of compact Riemann surfaces and of abelian varieties serves as an entrance to what will be the subjects of the subsequent second volume.
As to the more precise contents, the current book comprises five chapters, each of which is subdivided into numerous sections and subsections dedicated to a large variety of specific topics.
Chapter 1 introduces the relevant basics of category theory in a brief, concise and pleasantly informal style, including products, projective and direct limits, the Yoneda lemma, and representable functors. Examples from algebraic number theory are given to illustrate the abstract concepts.
Chapter 2 provides an introduction to the methods of homological algebra. With a view to the anticipated third volume on the cohomology of arithmetic groups, the discussion is mainly based on the relevant example of group cohomology, with particular emphasis on derived functors, their associated long exact sequences, and the crucial functors Ext and Tor.
Chapter 3 briefly develops the basic notions concerning presheaves and sheaves. Locally ringed spaces, manifolds in their sheaf-theoretic setting, the sheafication of a presheaf, direct images and pull-backs of sheaves, the adjunction formula, and further operations on sheaves are the main topics touched upon in this third introductory chapter.
Chapter 4 is much more elaborated and advanced, with its main theme being the cohomology of sheaves. Apart from the basic definitions and functorial properties of several sheaf cohomology theories, the author also offers a rather wide panoramic view of their various aspects and applications.
Along the way, the reader gets acquainted with fibre bundles, vector bundles, principal \(G\)-bundles, spectral sequences, derived categories, cup products, cohomology of differentiable manifolds, Poincaré duality, the isomorphism theorems of de Rham and Dolbeault, Chern classes, Kähler manifolds, some basic Hodge theory, and other related topics. Numerous instructive examples from various areas enhance this fine and multifarious discussion of sheaf cohomology in very effective a manner.
Chapter 5 gives then first applications of the entire foregoing sheaf cohomological framework to the classical subject of compact Riemann surfaces and complex abelian varieties, that is, the author prepares the reader for an introduction to modern algebraic geometry (via schemes) by means of explaining a very important, classical, prototypical and utmost fascinating branch of it in the first place, and that in modern, non-classical language. The starting point of the author's approach is the cohomological version of the Riemann-Roch theorem for compact Riemann surfaces, from which their algebraic nature is then concluded. This leads to a discussion of smooth projective curves of low genus, including elliptic curves, as well as to a first treatment of Jacobians, Abel's theorem, and the Riemann period relations. Whereas all this will be deepened, from a more advanced viewpoint, in the subsequent second volume of the treatise, the author proceeds here with complex tori, their line bundles, polarized complex abelian varieties, and an outlook toward their algebraic theory. Along this path, the allied concept of theta functions as well as an outlook to moduli spaces of polarized abelian varieties serves as both an illustration of the enormous usefulness of the methods developed so far and a first encounter with the theory of modular forms, the latter of which will play a significant role in the arithmetic context of the announced third volume.
After this first introduction to algebro-geometric objects, namely algebraic curves and abelian varieties, the forthcoming second volume ``Lectures on Algebraic Geometry II'' will focus on the basic concepts of the theory of algebraic schemes, their underlying (local) commutative algebra, the fundamental properties of projective schemes, the algebraic theory of curves and their associated abelian varieties, Picard schemes for curves and Jacobians, and on an outlook to further developments regarding étale cohomology and other related subjects.
No doubt, the great lucidity of exposition, the masterly style of writing, the broad spectrum of topics touched upon, and the purposeful, very special disposition of the subject matter make this text, together with its expected companion book(s), a very particular and outstanding enrichment of the existing textbook literature in algebraic geometry and its intimately related areas. textbook (algebraic geometry); textbook (category theory, homological algebra); sheaves; sheaf cohomology; Riemann surfaces; complex tori; abelian varieties Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Riemann surfaces; Weierstrass points; gap sequences, Analytic theory of abelian varieties; abelian integrals and differentials Lectures on algebraic geometry I. Sheaves, cohomology of sheaves, and applications to Riemann surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves that the germ of a reduced complex analytic curve \((C,x)\) through a quotient singularity \((X,x)\) of dimension two is biholomorphic to the germ \((Y,D,y)\) of a projective curve on a projective surface. Along the way he gives an explicit description of the divisor class group of a quotient singularity.
A detailed, self-contained proof is given; the key point is that equivariant functions on \((\mathbb{C}^ 2,0)\) with isolated singularity are finitely determined. The proper context, especially for generalisations, is the theory of ``geometrically defined subgroups of \({\mathcal A}\) and \({\mathcal K}\)'' [cf. \textit{J. Damon}, Proc. Symp. Pure Math. 40, Part 1, 233- 254 (1983; Zbl 0519.58014)]. divisor class group; quotient singularity R. Blache, A GAGA type theorem on germs of analytic curves through germs of quotient surface singularities. Arch. Math.62, 308--314 (1994). Complex surface and hypersurface singularities, Singularities in algebraic geometry A GAGA type theorem on germs of analytic curves through germs of quotient surface singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X \subseteq \mathbb{P}^ N\) be a nondegenerate, absolutely irreducible algebraic curve over \(\mathbb{F}_ q\). In a previous paper, \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc., III. Ser. 52, 1- 19 (1986; Zbl 0593.14020)] introduced the notion of ``\(q\)-Frobenius order-sequence of \(X\),'' and gave a geometric proof of Weil's basic theorem on rational points of curves over \(\mathbb{F}_ q\). The order sequence of \(X\) is the set of possible intersection multiplicities of \(X\) with the hyperplanes of \(\mathbb{P}^ N\) at general points. It is basic that the \(q\)-Frobenius sequence differs from the order sequence by the deletion of one order. The paper being reviewed studies this phenomenon and establishes some geometric results, e.g. on the inseparability degree of Gauss maps. rational points of curves over finite fields; Frobenius sequence; Weil number Arithmetic ground fields for curves, Finite ground fields in algebraic geometry, Rational points, Curves over finite and local fields, Special algebraic curves and curves of low genus Frobenius order-sequences 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 authors prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture in the situation where the ambient space is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is finitely generated. As a consequence a proof of the Mordell-Lang conjecture that does not depend on model theory methods but rather on algebraic-geometric methods is provided. In sectin 2 of the paper, the authors developed the necessary machinery to work with relative jet schemes via the reduction of scalars functor (represented by \(W\)). With the use of jet spaces, some natural ``critical schemes'' are constructed in section 3 to ``catch rational points''. Using Galois-theory methods, the proof of proposition 4.1 in section 4 allows to show the sparsity of points over finite fields that are liftable to highly \(p\)-divisible unramified points. Exceptional sets were also introduced by \textit{A. Buium} [Ann. Math. (2) 136, No. 3, 557--567 (1992; Zbl 0817.14021)], where the structure of Exc is studied via differential equations. However, the use of Galois theory to study the critical sets is essentially new to the approach of the authors in this paper. The fundamental result relevant to the logical connection between the Manin-Mumford and Mordell-Lang conjectures is the following.
Theorem. Let \(K_0\) be the function field of a smooth curve over \(\bar{F}_p\). Let \(A\) be an abelian variety over \(K_0\), and let \(X \hookrightarrow A\) be a closed integral scheme. Also let \(\Gamma \subset A(K_0)\) be a finitely generated subgroup. Suppose that for any field extension \(L_0|K_0\) and every \(Q \in A(L_0)\), the set \(X_{L_0}^{+,Q} \cap \mathrm{Tor}(A(L_0))\) is not Zariski dense in \(X_{L_0}^{+,Q}\), then \(X \cap \Gamma\) is not Zariski dense in \(X\).
In the statement of the theorem, \(X_{L_0}^{+,Q}\) is denoting the set theoretical image of \(X_{L_0}\) under translation by \(Q\) in \(A_{L_0}\).
In section 2B of the article, the authors present the theory of jet schemes of smooth commutative groups, with the definition of the maps
\[
\lambda_n^W : W(U) \rightarrow J^n(W/U)(U),
\]
while working with \(U\)-schemes and associated jets \(J^n(W/U)\). Hints for a discussion of cases when the Manin-Mumford could be considered special case of Mordell-Lang or Mordell-Lang conjecture reduce to Manin-Mumford are included in remark 4.9 and remark (important) in the introduction. positive characteristic; abelian variety; jet schemes; Galois theory; restriction of scalars functor; lifts of points; \(p\)-divisible points; Manin-Mumford conjecture; Mordell-Lang conjecture Rössler, D, On the Manin-Mumford and Mordell-lang conjectures in positive characteristic, Algebra Number Theory, 7, 2039-2057, (2013) Rational points, Subvarieties of abelian varieties, Positive characteristic ground fields in algebraic geometry, Abelian varieties and schemes, Abelian varieties of dimension \(> 1\) On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a finite group and \(S\) be a smooth complex projective \(K3\) surface with a \(G\)-action. It is known that the number of rational curves in an integral linear system on \(S\) can be calculated using the relative compactified Jacobian. In this paper, this strategy is used in the setting of \(G\)-representations.
In the first important result, the author considers the Hodge-Deligne polynomial of \(S^{[n]}\), but with coefficients being the \(G\)-representations \([H^{p,q}(S^{[n]},\mathbb C)]\) instead of the usual integers \(h^{p,q}(S^{[n]},\mathbb C)\). That is, the coefficients are in the ring of \(G\)-representations \(R_{\mathbb C}(G)\). The result is an explicit description of the polynomial in terms of \([H^{p,q}(S,\mathbb C)]\).
Let \(\mathcal{C}_n\) be the tautological family of curves over any \(n\)-dimensional integral \(G\)-stable linear system. The birationality of \(S^{[n]}\) with the relative compactified Jacobian \(\overline{J^n(\mathcal{C}_n)}\), implies that their Hodge-Deligne polynomials in \(R_{\mathbb C}(G)[u,v]\) are the same.
This last result is used to compute the topological Euler characteristic and count the number of rational curves in the linear system. The drawback of using the topological Euler characteristic in \(R_{\mathbb C}(G)[u,v]\) instead of the classical one is that the latter may be zero, even if the former is not. So that, this method may count also non-rational curves. The author solves this problem by proving that a necessary condition for a \(G\)-orbit of curves with nodal singularities to contribute is that a certain quotient of the normalized curve is rational.
The last part of the paper is dedicated to some applications of the theory in some explicit example. curve counting; \(K3\) surfaces; group representations; Hilbert schemes of points; compactified Jacobians Moduli, classification: analytic theory; relations with modular forms, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects), Automorphisms of surfaces and higher-dimensional varieties Counting rational curves on \(K3\) surfaces with finite group 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 A theorem of Riemann says that the theta divisor of the Jacobian of a smooth projective curve of genus \(g\) is given by the image of the \((g-1)\)th symmetric power of the Abel-Jacobi map.
The present paper shows the following converse: Let \((A,\theta)\) be an indecomposable principally polarized abelian variety of dimension \(g\geq 2\). If there is a curve \(C\) and a \((g-2)\)-dimensional subvariety \(Y\) such that \(\theta= C+Y\), then \(C\) is smooth and \((A,\theta)\) is the Jacobian of \(C\).
The paper also proves a generalization of this, characterizing Jacobians by the existence of subvarieties of smaller dimension with a curve summand whose twisted ideal sheaf is a generic vanishing sheaf in the sense of Pareschi-Popa. The proof of the first theorem uses cohomological and geometric techniques which originated in the work of Ran and Welters as well as the theorem of \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 10, No. 1, 243--258 (1997; Zbl 0901.14028)] on the singularities of the theta divisor. The proof is then reduced to Matsusaka's characterization of Jacobians by the fact that the cohomology class \({1\over(g-1)}[\theta]^{g-1}\) can be represented by a curve. There are several interesting consequences determining all theta divisors that can be dominated by products of curves. theta divisor; Schottky problem Schreieder, Stefan, Theta divisors with curve summands and the Schottky problem, Math. Ann., 365, 3-4, 1017-1039, (2016) Theta functions and curves; Schottky problem, Subvarieties of abelian varieties, Rational and birational maps, Jacobians, Prym varieties, Theta functions and abelian varieties Theta divisors with curve summands and the Schottky 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 \(C\) be a smooth curve with canonical divisor \(K\) and let \(D\) be an effective divisor of degree \(d\) on \(C.\) Let \({\mathcal M}_C(2,D)\) be the moduli space of semistable rank \(2\) vector bundles over \(C,\) with determinant \({\mathcal O}_C(D).\) We can associate, using Serre's duality, hyperplanes in the projective space \(|K+D|\) with naturally defined equivalence classes of extensions of the form \(0\to {\mathcal O}_C \to E \to {\mathcal O}_C(D)\to 0.\) This gives rise to the so called Serre's correspondence, a rational (in general) map \(\psi: |K+D|^* \to {\mathcal M}_C(2,D).\)
\textit{M. Thaddeus} [Invent. Math. 117, No. 2, 317-353 (1994; Zbl 0882.14003)] gave a description of the Serre correspondence for large \(d\) in terms of a sequence of simple birational transformations between suitably constructed moduli spaces of stable pairs, and a final contraction. -- The paper under review gives first a thorough introduction to the theory of stable pairs and triples and a detailed explanation of Thaddeus' interpretation of Serre's correspondence. The author then reinterprets the results in the light of the minimal model program. The chain of maps described above is now looked upon as a sequence of log flips. Notice that the varieties involved in the sequence are not just threefolds. This interpretation, the author suggests, might help in understanding the situation in the cases of small \(d,\) where Thaddeus' construction does not apply. The paper is a very readable and well written effort at combining, as the author himself points out, three different worlds in algebraic geometry : curves in projective spaces, moduli of vector bundles, pairs and triples on curves, and the minimal model program. The introduction of the paper contains interesting questions on possible developments of the issues at hand. Serre's correspondence; stable pairs; stable triples; minimal model program; log flips; birational maps; moduli space over smooth curve Bertram, A.: Stable pairs and log flips, Proc. sympos. Pure math. 62, 185-201 (1997) Vector bundles on curves and their moduli, Divisors, linear systems, invertible sheaves, Algebraic moduli problems, moduli of vector bundles, Minimal model program (Mori theory, extremal rays), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Stable pairs and log flips | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 prove a generalisation of the Brill-Noether theorem for the variety of special divisors \(W^r_d(C)\) on a general curve \(C\) of prescribed gonality. Our main theorem gives a closed formula for the dimension of \(W^r_d(C)\). We build on previous work of \textit{N. Pflueger} [Adv. Math. 312, 46--63 (2017; Zbl 1366.14031)], who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne [\textit{D. Cartwright} et al., Can. Math. Bull. 58, No. 2, 250--262 (2015; Zbl 1327.14266)]. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of \textit{D. E. Speyer} [Algebra Number Theory 8, No. 4, 963--998 (2014; Zbl 1301.14035)] on genus \(1\) curves to arbitrary genus.''
More precisely, the main result is the following:
Theorem. Let \(C\) be a general curve of genus \(g\) and gonality \(k\ge 2\) over the complex numbers. Assume that the quantity \(g-d+r\) is positive. Then \[\dim W^r_d(C)=\max_{\ell\in\{0,\dots,r'\}} g-(r-\ell+1)(g-d+r-\ell)-\ell k\] where \(r'=\min\{r, g-d + r-1\}\). Special divisors on curves (gonality, Brill-Noether theory), Combinatorial aspects of tropical varieties Brill-Noether theory for curves of a fixed gonality | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 monograph may be regarded as a natural generalization of the thirty years old article ``Diophantine equations with special references to elliptic curves'', J. Lond. Math. Soc. 41, 193--291 (1966; Zbl 0138.27002)] by \textit{J. W. S. Cassels}, to genus 2 curves and their Jacobians. Arithmetic (i.e., number theoretic properties) of genus 2 curves over fields (e.g., \(\mathbb Q\), number fields, or local fields) are explored, generalizing the methods and techniques, such as group laws, 2-descents, formal groups, torsion points, heights, a weak Mordell-Weil theorem, etc., developed for elliptic curves to genus 2 curves and their Jacobians (or rather the associated Kummer surfaces).
It might be said that the principal goal of the monograph is to compute the Mordell-Weil groups of genus 2 curves defined over \(\mathbb{Q}\). Indeed, the Mordell-Weil groups (at least the rank) are determined explicitly for a large number of genus 2 curves.
Let \(k\) be a perfect field of characteristic \(\neq 2\). A curve of genus 2 defined over \(k\) is shown to be birationally equivalent to a curve in a canonical form \({\mathcal C} :Y^2= F(X)\) where \(F(X)=f_0+f_1 X+f_2X^2+f_3X^3 + f_4X^4+f_5X^5 + f_6X^6\in k[X]\) has no multiple factors. The Jacobian variety \(J({\mathcal C})\) of \({\mathcal C}\) is constructed, as a surface in 15-dimensional projective space \(\mathbb P^{15}\) by computing explicitly the 16 basis elements defining \(J({\mathcal C})\). The group law on \(J({\mathcal C})\) is described generically. The Jacobian \(J({\mathcal C})\) is unfortunately too large for any practical and computational purposes. To remedy this difficulty, the associated Kummer surfaces \({\mathcal K}= {\mathcal K} ({\mathcal C})\) and their Jacobians are investigated in detail. The Kummer surfaces are quartic surfaces in \(\mathbb P^3\) determined by four of the 16 basis elements, and they contain much of the information on the Jacobians. Unfortunately, the group structures on the Jacobian \(J({\mathcal C})\) will be lost in passing from the Jacobians to the Kummer surfaces, though addition on 2-division points still remain to be significant on the Kummer surfaces.
The classical result that every abelian variety of dimension 2 (over an algebraically closed field) is isogenous to the Jacobian of a genus 2 curve is generalized to abstract Kummer surfaces defined over \(\mathbb Q\).
The computation of the Mordell-Weil group \({\mathfrak G}\) of \(J({\mathcal C})\) is one of the principal results of the monograph, when the field \(k\) is a number field. Analogous to genus 1 case, the weak Mordell-Weil theorem asserting the finiteness of the group \({\mathfrak G}/2 {\mathfrak G}\) is proved. This is done by constructing a group homomorphism with kernel \(2{\mathfrak G}\) from \({\mathfrak G}\) to an easily treated group. For this the Kummer surfaces as well as the dual Kummer surfaces are used. Then the Mordell-Weil theorem that \({\mathfrak G}\) is finitely generated is established, invoking the use of a height function on the jacobian (as in the genus 1 case).
The torsion group of \(J({\mathcal C})\) is discussed imposing two concrete problems: given a genus 2 curve, (1) how can one find the rational torsion group in \(J({\mathcal C})\)? and (2) given an integer \(N\), how does one try to find a curve whose Jacobian has a rational point of order \(N\)? In an attempt to answer these questions, the group law, the formal group, and isogenies are explicitly described. Then a crude algorithm which might lead to a complete solution to the question (1) is presented. For (2), a method for finding large torsion elements over \(\mathbb{Q}\) are obtained, and computation of torsion elements of orders \(\leq 29\) are carried out for a number of genus 2 curves. The actual computation of the Mordell-Weil group is carried out by performing complete 2-descents on several genus 2 curves. Most of the examples discussed here have Mordell-Weil rank 0 or 1, with small torsion subgroups. For instance, \({\mathcal C}: Y^2=X(X-1)\) \((X-2)\) \((X-5)\) \((X-6)\) has the Mordell-Weil rank 1 with \({\mathfrak G}/2 {\mathfrak G} =\langle {\mathfrak G}_{\text{tors}}, \{(3,6), \infty\} \rangle\) and \({\mathfrak G}_{\text{tors}}\) consisting only of the 2-torsion group of order 16. Curves with large Mordell-Weil rank are briefly discussed.
Other results are concerned with finding all the rational points on genus 2 curves, the number of which is known to be finite by a theorem of Faltings. Here the authors make use of Chabauty's theorem to find rational points on a genus 2 curve. One of worked out examples is the curve \({\mathcal C}: Y^2=(X^2-2X-2)\) \((-X^2+1)\) \((2X)\), all whose rational points are given by \({\mathcal C} (\mathbb Q)=\{(0,0), \infty, (\pm 1,0)\), \((-1/2, \pm 3/4)\}\). -- The endomorphism ring of the Jacobian which is strictly larger than \(\mathbb Z\) is described for some genus 2 curves. These give rise to examples of genus 2 curves with complex, or real multiplication.
The monograph consists of 18 chapters, some theoretical, and others computational. For instance, the chapters 8, 10, 11, 12, 13 are of computational nature. The contents are: Chapter 1: Curves of genus 2; Chapter 2: Constructions of the Jacobian; Chapter 3: The Kummer surface; Chapter 4: The dual of the Kummer surface; Chapter 5: Weddel's surface; Chapter 6: \({\mathfrak G}/2 {\mathfrak G}\); Chapter 7: The Jacobian over local fields -- formal groups; Chapter 8: Torsion; Chapter 9: The isogeny -- theory; Chapter 10: The isogeny -- applications; Chapter 11: Computing the Mordell-Weil group, Chapter 12: Heights, Chapter 13: Rational points -- Chabauty's theorem: Chapter 14: Reducible Jacobians; Chapter 15: The endomorphism rings; Chapter 16: The desingularized Kummer surface; Chapter 17: A neoclassical approach; and Chapter 18: Zukunftsmusik. In the appendix, MAPLE programs and other files available by anonymous ftp are listed.
There are numerous worked out examples of genus 2 curves, demonstrating the power of computer algebra as a research tool in diophantine equations, especially, for genus 2 curves. Mordell-Weil groups; genus 2 curves; Jacobian variety; Kummer surfaces; torsion group; rational points; computer algebra; Diophantine equations Cassels, J. W. S.; Flynn, E. V., Prolegomena to a Middlebrow Arithmetic of Curves of Genus \(2\), London Mathematical Society Lecture Note Series 230, xiv+219 pp., (1996), Cambridge University Press, Cambridge Special algebraic curves and curves of low genus, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, 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, Arithmetic varieties and schemes; Arakelov theory; heights, Computational aspects of algebraic curves, Computer solution of Diophantine equations Prolegomena to a middlebrow arithmetic of curves of genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a curve over the finite field \(\mathbb{F}_ q\) \((q=p^ m)\) with \(\infty\in X\) a fixed closed point; we set \(A\) to be the affine ring of \(X-\infty\). In his fundamental paper ``Elliptic modules'' [Math. USSR, Sb. 23, 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)] \textit{V. G. Drinfeld} introduced elegant analogs (now called ``Drinfeld modules'') of elliptic curves and abelian varieties for \(A\). Although various instances of this theory go back to L. Carlitz in the 1930's, Drinfeld's seminal paper marked the beginning of the modern theory of function fields over finite fields. In particular, Drinfeld was able to give a moduli theoretic construction of the maximal abelian extension of \(k\) (= fraction field of \(A\)) which is split totally at \(\infty\), as well as to give a Jacquet-Langlands style two-dimensional reciprocity law where the infinite component is a Steinberg representation. The two-dimensional reciprocity law is established by decomposing the étale cohomology of the compactified rank 2 moduli scheme. The impediment to implementing this procedure for arbitrary \(d>2\) is the difficulty of obtaining a good compactification in general.
The operator \(\tau: x\mapsto x^ q\) satisfies many analogies with the classical differentiation operator \(D:= d/dx\) and these analogies go surprisingly deep. A prime example of this was given in 1976 when Drinfeld found an interpretation of a Drinfeld module \(\varphi\) in terms of a locally free sheaf \({\mathcal F}\) on \(X\) (of the same rank as \(\varphi\)) with maps relating \({\mathcal F}\) and its twist by the Frobenius map; this sheaf-theoretic interpretation being analogous to results of I. Krichever on differential operators.
An excellent reference for all this is the paper by \textit{D. Mumford} [An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related nonlinear equations, [Proc. int. symp. on algebraic geometry, Kyoto, 115- 153 (1977; Zbl 0423.14007)]. Such locally free sheaves are examples of ``shtuka'', ``\(F\)-sheaves'', or ``elliptic sheaves'' -- this last being the notation used in the paper being reviewed. More generally, shtuka appear when the axioms in the
\[
\text{elliptic modules} \longleftrightarrow \text{elliptic sheaves}
\]
dictionary are weakened a bit.
The paper under review contains the basic (and profound) work on ``\({\mathcal D}\)-elliptic sheaves'', where \({\mathcal D}\) is a central simple algebra over \(k\). These can be viewed, at least to first order, as ``elliptic sheaves with complex multiplication by \({\mathcal D}\)'' (and, among other axioms the dimension of \({\mathcal D}\) must also be the rank of the elliptic sheaf). A notion of ``level structure'' can be given generalizing that for Drinfeld modules. The point being that such \({\mathcal D}\)-elliptic sheaves have good, smooth, moduli spaces, and, when \({\mathcal D}\) is a division algebra, this moduli is projective (think of the theory of Shimura curves). Thus, and importantly, the problems of non-compact moduli spaces are avoided. In particular, using the cohomology of such moduli spaces the authors prove a reciprocity law generalizing the one given in Drinfeld's original paper. As a consequence, the authors find enough representations to establish the basic local Langlands conjecture for \(\text{GL}_ d\) (\(d\) arbitrary) of a local field of equal characteristic; this completes a program that was first begun by P. Deligne in the 1970's using Drinfeld's original work for \(d=2\). The proofs of these reciprocity laws are highly nontrivial and use many of the main techniques available in harmonic analysis such as the Selberg trace formula and the Grothendieck-Lefschetz fixed point formula. The authors also discuss applications of their results to the Tate conjectures for their varieties. Drinfeld modules; elliptic sheaves; elliptic modules; level structure; central simple algebra; moduli spaces; reciprocity law; local Langlands conjecture; Selberg trace formula; Grothendieck-Lefschetz fixed point formula; Tate conjectures Laumon, G.; Rapoport, M.; Stuhler, U., \(\mathcal{D}\)-elliptic sheaves and the Langlands correspondence, Invent. Math., 113, 217-338, (1993) Drinfel'd modules; higher-dimensional motives, etc., Algebraic functions and function fields in algebraic geometry, Langlands-Weil conjectures, nonabelian class field theory, Finite ground fields in algebraic geometry, Spectral theory; trace formulas (e.g., that of Selberg), Arithmetic theory of algebraic function fields, Formal groups, \(p\)-divisible groups \({\mathcal D}\)-elliptic sheaves and the Langlands correspondence | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 smooth irreducible curve, \(\mathcal{A} \rightarrow S\) an abelian scheme of relative dimension \(g \geq 2\), and \(C\) an irreducible curve in \(\mathcal{A}\) not containing in a proper subgroup scheme of \(\mathcal{A}\), even after a finite base extension.
Suppose that \(S\), \(\mathcal{A}\) and \(C\) are defined over \(k\).
Then, in this paper, it is proved that the intersection of \(C\) with the union of all flat subgroup schemes of \(\mathcal{A}\) of codimension at least 2 is a finite set. Furthermore, if the intersection of \(C\) with the union of all flat subgroup schemes of \(\mathcal{A}\) of codimension at least \(m\), where \(1 \leq m \leq g\), is infinite, it is proved, using the above result, that there exists a finite cover \(S^{\prime} \rightarrow S\) such that \(C\times_S S^{\prime}\) is contained in a flat subgroup scheme of \(\mathcal{A}\times_S S^{\prime}\) of codimension at least \(m-1\).
This result has an interesting application in the study of solvability of almost Pell-equations in polynomials. More precisely, let \(S\) be a smooth irreducible curve defined over a number field \(k\), \(k(S)\) its function field and \(\overline{k(S)}\) the algebraic closure of \(k(S)\). Further, let \(D,F\in k(S)[X]\setminus \{0\}\) with \(D\) squarefree of even degree \(\geq 6\). Suppose that the jacobian of the hyperelliptic curve defined by \(Y^2 = D(X)\) contains no one-dimensional abelian subvariety over \(\overline{k(S)}\). Then, if the equation \(A^2-DB^2 = F\) does not have a non-trivial solution in \(\overline{k(S)}[X]\), it is proved that there are at most finitely many \(s_0\in S(\mathbb{C})\) such that the specialized equation \(A^2-D_{s_0}B^2 = F_{s_0}\) has a solution \(A,B \in \mathbb{C}[X]\) with \(B \neq 0\). abelian variety; polynomial Pell equation; abelian scheme; flat subgroup scheme; elliptic surface Algebraic theory of abelian varieties, Elliptic curves over global fields, Global ground fields in algebraic geometry, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Heights, Model theory (number-theoretic aspects) Unlikely intersections in families of abelian varieties and the polynomial Pell equation | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of the article is an existence theorem on geometric quotients for proper actions of algebraic group schemes on algebraic spaces. Let \(G\) be an affine algebraic group scheme of finite type over some excellent base scheme \(S\). Suppose that \(G\) acts morphically on the algebraic space \(X\) of finite type over \(S\). The notion of a geometric quotient for the action of \(G\) on \(X\) is defined by the author basically in analogy to D. Mumford's concept for the case of schemes. Whereas in the category of schemes a geometric quotient is necessarily categorical and hence unique, these two properties are no longer satisfied in the category of algebraic spaces, as the author shows in an explicit example. Now assume that the action of \(G\) on \(X\) is proper.
The main result of the article ensures that there exists a geometric quotient \(q : X \to Y\) for the action of \(G\) on \(X\) if one of the following conditions is valid:
(a) \(G\) is reductive over \(S\),
(b) \(S\) is the spectrum of a field of positive characteristic.
Moreover, then the algebraic space \(Y\) over \(S\) is separated and \(q\) is in fact categorical.
The author obtains also important properties of quotient morphisms. In fact, he works under slightly weaker assumptions and considers the notion of an approximate quotient for the action of \(G\) on \(X\). For \(G\) universally open over \(S\) it is shown that such a quotient \(p: X \to Y\) is an affine morphism. If furthermore \(G\) is flat over \(S\), then for every coherent \(G\)-sheaf \(F\) on \(X\) the sheaf \((p_*F)^G\) of invariants on \(Y\) is coherent. The main result yields the existence of certain moduli spaces that was known before only in characteristic zero. In the meantime \textit{S. Keel} and \textit{S. Mori} [Ann. Math., II. Ser. 145, No. 1, 193-213 (1997; see the following review)] presented a proof for the existence of geometric quotients by proper actions of flat algebraic group schemes with finite stabilizer on algebraic spaces in a more general framework. algebraic group actions; geometric quotients; algebraic spaces \textsc{J. Kollár}, \textit{Quotient spaces modulo algebraic groups}, Ann. of Math. (2) 145 (1997), no. 1, 33--79. DOI 10.2307/2951823; zbl 0881.14017; MR1432036; arxiv alg-geom/9503007 Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Geometric invariant theory Quotient spaces modulo algebraic groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian problem asks whether a polynomial endomorphism of complex affine \(n\)-space with non-vanishing Jacobian determinant is an isomorphism. Such a morphism is étale and surjective modulo codimension two. The generalized Jacobian problem asks whether an étale morphism from a simply connected complex variety of dimension \(n\) to complex \(n\)- space which is surjective modulo codimension two is an isomorphism. If the generalized problem had an affirmative answer, so would the original one. In this paper, the author constructs a counterexample to the generalized problem. His method is to show the equivalence between the generalized problem and a condition on fundamental groups of complex affine plane curve complements, namely whether such can have proper subgroups of finite index generated by geometric generators. He shows that if \(\overline D\) is a degree 4 projective plane curve with three cuspidal singularities and \(L\) is a projective line transversally intersecting \(\overline D\) in four points then the fundamental group of the plane curve \(D-L\) has a geometrically generated proper finite index subgroup.
He also considers the related local question, namely whether the fundamental group of the complement of a germ of an analytic curve in a two dimensional complex ball can have a proper finite index subgroup generated by geometric generators, and shows that this is equivalent to the question of whether such a ball can be the image by the germ of an étale surjective holomorphic map of degree more than one from a simply connected analytic surface. He shows that this second question has a negative answer, which thus settles the first negatively in the analytic case also.
The paper also establishes two additional equivalent formulations of the Jacobian problem: first, that injective Lie endomorphisms of the set of derivations of the polynomial ring are automorphisms, and the second in terms of properties of the differential equations associated to sets of certain derivations of complex \(n+1\) space. generalized Jacobian problem; étale morphism; germ of an analytic curve Kulikov, V.S.: Generalized and local Jacobian problems, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. \textbf{56}(5), 1086-1103 (1992); translation in Russian Acad. Sci. Izv. Math. \textbf{41}(2), 351-365 (1993) Automorphisms of curves, Polynomial rings and ideals; rings of integer-valued polynomials, Germs of analytic sets, local parametrization Generalized and local Jacobian 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 book under review is the first monograph, in the entire literature of algebraic geometry, that is exclusively devoted to the subject of algebraic stacks. The authors' aim is to provide a systematic, comprehensive, rigorous and general exposition of the theory of algebraic stacks which, over the past thirty years, has become a powerful and indispensible tool for constructing and investigating moduli schemes in algebraic geometry. The concept of algebraic stack is still far from being generally familiar to, or even used by the majority of active researchers in the classification theory of algebro-geometric objects, and many features of algebraic stacks are only used intuitively, superficially, reluctantly, in an awkward ad-hoc manner, or sometimes even in a sloppy folklore style.
On the other hand, algebraic stacks arise quite naturally from A. Grothendieck's functorial approach to algebraic geometry, and they proved their ubiquity in many concrete situations, long before they were formally introduced and recognized as objects that are just as important as schemes and sheaves themselves. According to Grothendieck's re-foundation of algebraic geometry, the category of schemes can be interpreted in two ways, namely
(1) as full subcategory of the category of ringed spaces (i.e., from the geometric viewpoint), or
(2) as full subcategory of the category of covariant functors from the category of rings to the category of sets (i.e., from the functorial viewpoint).
The second point of view is particularly useful in those situations where schemes with certain universal properties are to be established (e.g., Hilbert schemes, Picard varieties, moduli schemes, etc.). Based on Grothendieck's ideas and techniques developed along these lines, which had led him to introduce objects such as étale topologies, sheaves of categories, sites, and topoi, Mumford (1963), Deligne-Mumford (1969), and M. Artin (1974) extended the concept of a sheaf of categories to the one of an ``algebraic stack'' and used it in the moduli theory of algebraic curves and singularities. -- More precisely, Deligne and Mumford used their algebraic stack of stable curves to construct a compactification of the moduli space of smooth curves of given genus \(g\), and M. Artin applied his version of an algebraic stack to create construction techniques for algebraic spaces and versal deformations of singularities.
The past twenty-five years have seen various applications of these approaches to moduli problems via algebraic stacks, and also a few attempts to develop a general theory of stacks and their intersection theory, but up to now, no systematic, comprehensive, or at least compiling treatise on that subject had emerged.
The authors of the book under review have filled this painful gap in a thorough, masterly and rewarding manner. They focus on precisely that approach to a theory of algebraic stacks, which has been initiated by Deligne-Mumford and M. Artin, and they present it in its most general, deep-going and methodically perfect form. -- The text consists of nineteen chapters, one appendix, an up-to-date bibliography, and a carefully arranged terminological index.
Chapter 1 resumes briefly the basic facts from the theory of algebraic spaces. As for more details, the reader is referred to the original works of M. Artin, D. Knutson, and B. Moishezon. Chapter 2 provides the generalities on the 2-category of groupoids over a given base scheme \(S\). -- This is used, in chapter 3, to introduce the category of \(S\)-stacks, \(S\) being a fixed base scheme, the notion of an 1-morphism between representable stacks, and the first properties of the so-called ``gerbes'' (i.e., stack-like fibre spaces). -- Chapter 4 discusses diverse properties of algebraic stacks (à la Deligne-Mumford-Artin) and their 1-morphisms. -- Chapter 5 turns to the topological aspects of algebraic stacks and their 1-morphisms, culminating in a stack-theoretic generalization of Chevalley's theorem on constructible sets. -- Chapter 6 deals with the local structure of algebraic stacks and provides various results on the existence of schemes over stacks.
Valuation criteria, specialisation properties, and special morphisms between algebraic stacks are the subject of chapter 7, while chapter 8 gives characterizations for algebraic spaces and for Deligne-Mumford stacks. Chapter 9 adds some remarks on the most important Grothendieck topologies, which are used, in chapter 10, to discuss M. Artin's algebraicity criterion for general \(S\)-stacks. Chapter 11 gives the definitions and properties of algebraic points, residual sheaves, residual gerbes, and gerbes on algebraic stacks. Then, in chapter 12, the authors begin the study of general sheaves over the so-called smooth-étale site of an algebraic stack, and chapter 13 extends this study to quasi-coherent module sheaves on algebraic stacks. This includes such functorial properties of quasi-coherent modules, which lead to a general variant of the theory of faithfully flat descent for them.
Chapter 14 deals with local constructions over algebraic stacks such as generalized vector bundles and generalized projective bundles, on the one hand, and with the concepts of Picard stacks and the Deligne correspondence between Picard stacks, on the other hand. Chapter 15 is devoted to the study of functorial properties of coherent sheaves over locally noetherian algebraic stacks. The totality of the preceding concepts, methods and results is utilized in chapters 16 to 18, where the deep results in the theory of stacks are proved. Chapter 16 describes a (partial) generalization of ``Zariski's main theorem'' to 1-morphisms of algebraic stacks, a variant of ``Chow's lemma'' for Deligne-Mumford stacks of finite type, and some applications to algebraic spaces. Chapter 17 turns to a stack-theoretic generalization of the cotangent complex of a morphism. This extends L. Illusie's construction for morphisms of ringed topoi to the 2-category of algebraic stacks and represents the probably deepest result and, without any doubt, the hardest part of the entire book. -- Chapter 18 gives an account on constructible sheaves over the smooth-étale site of an algebraic stack, mainly with a view towards their functorial and cohomological properties. The material presented here generalizes, to a major extent, related results obtained by P. Deligne in 1977 [cf. \textit{P. Deligne}, ``Cohomologie étale'', in: SGA \(4{1\over 2}\), Lect. Notes in Math. 569 (1977; Zbl 0349.14008-14013)]. The concluding chapter 19 is devoted to a very brief survey on some more recent results in the theory of algebraic stacks. As for details and complete proofs, the authors refer to the corresponding original papers listed in the bibliography. However, their brief comments are very inspiring, and illustrate the steadily growing role of algebraic stacks in algebraic geometry, particularly in view of their applications to the intersection theory of cycles in moduli spaces. -- Finally, in a short appendix, the authors have compiled (with full proofs) some complementary results on algebraic spaces, which were used in the course of the text.
Altogether, this comprehensive treatise on algebraic stacks is not at all easy to read. It is written in the abstract style that the matter dictates, and the reader is required to be reasonably familiar with Grothendieck's EGA [``Élements de géométrie algébrique''] as well as with most of the volumes of ``Seminaire de géométrie algébrique du Bois Marie'' (SGA 1-7). The style is concise, but extremely elegant and efficient, and the reader, who is willing to work hard while studying the material, will profit a great deal from this effort. Also, the authors have provided the mathematical community with a standard source book on algebraic stacks, which is unique, so far, and is therefore unchallenged and indispensible. gerbes; algebraic stacks; moduli theory of algebraic curves and singularities Laumon, Gérard; Moret-Bailly, Laurent, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). A Series of Modern Surveys in Mathematics, vol. 39, (2000), Springer-Verlag: Springer-Verlag Berlin Formal methods and deformations in algebraic geometry, Fine and coarse moduli spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Algebraic stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For part I see Trans. Am. Math. Soc. 301, 163-188 (1987; Zbl 0616.14014).]
A \textit{Grothendieck-Ogg-Shafarevich} (GOS) formula expresses the Euler-Poincaré characteristic of a constructible sheaf of \(\mathbb{F}_l\)-modules on a proper smooth curve over an algebraically closed field of characteristic \(p > 0\), where \(p \neq l\), as a sum of a global term and local terms [\textit{M. Raynaud}, in Sém. Bourbaki 17 (1964/65), Exp. No. 286 (1966; Zbl 0204.54301)]. A previously known result removes the restriction on the characteristic for \(p\)-torsion sheaves trivialized by \(p\)-extensions. A new proof of this known result was given by the author in part I of this paper (loc. cit.). In the same paper a \(p\)-torsion analogue of the GOS formula was conjectured and proved for constructible sheaves of \(\mathbb{F}_p\)-modules in the case that the generic stalk has rank \(p\). In the paper under review this conjecture is reduced to another conjecture on group schemes which is then proved to be valid in a number of additional cases. This leads to some new results on surfaces and the \(\mathbb{F}_q\)-vector schemes of Raynaud, where \(q = p^r\), \(r \geq 1\). A generalization of the GOS formula to higher dimensions is stated as a conjecture. characteristic \(p\); Grothendieck-Ogg-Shafarevich formula; Euler-Poincaré characteristic of a constructible sheaf; étale cohomology Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The étale cohomology of \(p\)-torsion sheaves. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors chronicle the main results on plane algebroid curves over fields \(k\) of arbitrary characteristic \(p \geq 0\). The work described covers many decades, so this book will be valuable to the reader wanting to learn about the subject without having to collect the many papers and books it is based on. Below we summarize the material of the six chapters.
Chapter 1: Weierstrass theorem and applications. The authors give background results on power series rings \(R = k[[x_1, \dots, x_n]]\) in \(n\) variables, based on [\textit{N. Bourbaki}, Éléments de mathématique. Paris: Hermann \& Cie (1962; Zbl 0142.00102); \textit{O. Zariski} and \textit{P. Samuel}, Commutative algebra. Vol. II. Princeton, N.J.-Toronto-London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801); \textit{H. Grauert} and \textit{R. Remmert}, Analytische Stellenalgebren. Unter Mitarbeit von O. Riemenschneider. (Analytic place algebras). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0231.32001); \textit{T. de Jong} and \textit{G. Pfister}, Local analytic geometry. Basic theory and applications. Braunschweig: Vieweg (2000; Zbl 0959.32011)]. The topics include results on summable families of power series, the Weierstrass division and preparation theorems, factorization results (including the fact that \(R\) is a UFD), the implicit and inverse function theorems, and Hensel's lemma. The last section gives some specialized results in the case of \(n=2\) variables.
Chapter 2: Plane algebraic curves (basic concepts). The main topic of the book begins here, \textit{plane algebroid curves} given by a power series \(f(x,y) \in k[[x,y]]\) with \(f(0,0)=0\). If \(f(x,y) = f_1^{m_1} \dots f_r^{m_r}\) with \(f_i\) irreducible and pairwise coprime, then the curves \(\{f_i = 0\}\) are the irreducible components with multiplicities \(m_i\). One result is the local normalization theorem giving a resolution of singularity by a finite sequence of quadratic transformations due to \textit{M. Noether} [Rend. Circ. Mat. Palermo 4, 89--108 (1890; JFM 22.0712.02)] and \textit{O. Zariski} [Am. J. Math. 87, 507--536 (1965; Zbl 0132.41601)]. Intersection multiplicities are defined in terms of valuations \(v_f\) associated to branches \(f\) and the semigroup associated to a branch \(f\) is the set \(\Gamma (f) \subset \mathbb N\) of all \(v_f (h)\) taken over all power series \(h\) with \(h \not \in (f)\). Two branches \(f,g\) are equisingularity if \(\Gamma (f) = \Gamma (g)\), the same as (a)-equivalence in the sense of Zariski ([\textit{Acevedo}, , Fort Wayne, IN: Purdue University (PhD Thesis) (1967)] and \textit{R. Waldi} [Commun. Algebra 28, No. 9, 4389--4401 (2000; Zbl 0969.14019)]. Log-distance and Newton diagrams are introduced.
Chapter 3: Plane algebroid branches. After giving generalities on (virtual) conductors and minimal systems of generators for numerical semigroups, the authors study the structure of the semigroup \(\Gamma (f)\) associated to a branch, following \textit{A. Seidenberg} [Trans. Am. Math. Soc. 57, 387--425 (1945; Zbl 0060.07101)]. Existence of the \textit{key polynomials} is proved and the Zariski characteristic sequence of a branch is introduced. Tools used are the strong triangle inequality and log distance between branches developed by \textit{E. García Barroso} and \textit{A. Płoski} [Rev. Mat. Complut. 28, No. 1, 227--252 (2015; Zbl 1308.32032); Colloq. Math. 156, No. 2, 243--254 (2019; Zbl 1434.32042)]. Following \textit{O. Zariski} [in: C.I.M.E. 3. Ciclo Varenna 1969, Quest. algebr. Varieties, 261--343 (1970; Zbl 0204.54503)], the key polynomials are also constructed from the Puiseux sequence. The chapter closes with the characterization of semigroups associated with branches due to \textit{H. Bresinsky} [Proc. Am. Math. Soc. 32, 381--384 (1972; Zbl 0218.14003)] and \textit{G. Angermüller} [Math. Z. 153, 267--282 (1977; Zbl 0331.14015)] and the intersection formula.
Chapter 4: Equisingularity invariants. Following the book of \textit{A. Campillo} [Algebroid curves in positive characteristic. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0451.14010)], the authors study the classical invariants of plane curve singularities: the degree of the conductor \(c\), the delta invariant \(\delta\), and the Milnor number \(\mu\) (in arbitrary characteristic it is defined by \(\mu = c-r+1\), where \(r\) is the number of irreducible components). They relate the semigroup of a plane branch to the multiplicity sequence and prove that two branches are equisingular if and only if they have the same multiplicity sequence. The chapter closes with the \textit{Hamburger-Noether expansions} as a method to study algebroid curves in arbitrary characteristic, which were introduced by \textit{G. Ancochea} [Courbes algébriques sur corps fermés de caractéristique quelconque. Acta Salamantic., Ci., Sec. Mat. 1 (1946; Zbl 0063.00081)] to avoid using the characteristic zero Puiseux series.
Chapter 5: Simple curve singularities. In characteristic zero, \textit{V. I. Arnol'd} classified simple singularities as the famous A-D-E-singularities [Russ. Math. Surv. 29, No. 2, 10--50 (1974; Zbl 0304.57018)]. \textit{G. M. Greuel} and \textit{H. Kröning} [Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)] modified the characterization of simple singularities to positive characteristic, using the definition of \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565--573 (1985; Zbl 0553.14012)]. Here the authors give a new proof using the theorem of finite determinacy and the consequence that an isolated plane curve singularity given by \(f=0\) is determined by its \(2 \tau\)-jet in the sense of contact equivalence, where \(\tau\) is the Tjurina number of \(f\). The answer depends on whether char \(k \geq 7\) or char \(k \leq 5\).
Chapter 6: Computational aspects. Using standard bases and Sagbi bases, the authors show how computer algebra systems can be used to compute singularity invariants. They illustrate with examples using SINGULAR. plane algebroid curves; equisingularity; numerical semigroups; branches; simple curve singularities Research exposition (monographs, survey articles) pertaining to algebraic geometry, Plane and space curves, Arithmetic ground fields for curves Plane algebroid curves in 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 \(S\) be a scheme, and consider \(S\)-schemes \(Z\) such that \(g : Z \to S\) is proper and equidimensional of relative dimension \(d\); under additional hypotheses one can identify the relative dualizing sheaf \(g^ ! {\mathcal O}_ S\) with the sheaf \(\omega_ Z\) of regular \(d\)-forms [cf. \textit{E. Kunz} and \textit{R. Waldi}, ``Regular differential forms'', Contemp. Math. 79 (1988; Zbl 0658.13019)]. Now let \(f:X \to Y\) be a suitable restricted morphism of such schemes; by a result of \textit{S. L. Kleiman} [Compos. Math. 41, 39-60 (1980; Zbl 0423.32006)] there is a canonical map \(\eta : f^* \omega_ Y \otimes \omega_ f \to \omega_ X = f^ ! \omega_ Y\) where \(\omega_ f\) is the canonical dualizing sheaf for \(f\). On the other hand, \textit{R. Hübl} [Manuscr. Math. 65, No. 2, 213-224 (1989; Zbl 0704.13004)] gives a rather explicit description by methods of commutative algebra of a morphism \(\varphi : f^* \omega_ Y \otimes \omega_ f \to \omega_ X\).
The bulk of the present paper consists of showing that \(\eta = \varphi\). To prove this result the author makes use of the residue formalism developed by \textit{R. Hübl} and \textit{E. Kunz} [J. Reine Angew. Math. 410, 53-83 (1990; Zbl 0712.14006) and ibid. 84-108 (1990; Zbl 0709.14014)]. This paper contains many results on differential forms, residues, generalizations of the residue theorem of \textit{R. Hübl} and \textit{P. Sastry} [Am. J. Math. 115, No. 4, 749-787 (1993; Zbl 0796.14012)] and related topics and should be a must for anybody interested in these questions. With respect to residues of differential forms, one may also consult the Habilitationsschrift of \textit{R. Hübl} [``Residues of differential forms, de Rham cohomology and Chern classes'' (Regensburg 1994)]. sheaf of regular \(d\)-forms; canonical dualizing sheaf; residue formalism; differential forms J. Lipman and P. Sastry,Regular differentials and equidimensional scheme-maps, Journal of Algebraic Geometry1 (1992), 101--130. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Modules of differentials Regular differentials and equidimensional scheme-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 The setting of the article under review is twofold: (i) a smooth algebraic stack \(X\) with a smooth divisor \(D \subset X\), (ii) a degeneration \(\pi : W \to B\) over a curve with special fibre given by the union of two smooth algebraic stacks meeting transversally along a smooth divisor.
In the setting (i) the authors construct the relative configuration space \(X_D^{[n]}\), which is a natural compactification of the cartesian product \((X \setminus D)^n\). In the setting (ii) they construct the stack \(W^{[n]}_\pi\), which compactifies the space of \(n\) ordered points on the smooth fibres of \(\pi\). Their methods rely on the universal approach to expanded pairs and expanded degenerations, developed in [\textit{D. Abramovich} et al., Commun. Algebra 41, No. 6, 2346--2386 (2013; Zbl 1326.14020)]. If the spaces in (i) are schemes, an alternative construction of \(X_D^{[n]}\) already appeared in [\textit{B. Kim} and \textit{F. Sato}, Sel. Math., New Ser. 15, No. 3, 435--443 (2009; Zbl 1177.14029)].
In the second part of the paper, the authors apply their constructions to prove the properness of the stack \(K_{\mathrm{pd}}\) of predeformable expanded stable maps in the sense of \textit{J. Li} [J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540)], in both the settings (i) and (ii). When the spaces in (i) and (ii) are schemes, the properness of \(K_{\mathrm{pd}}\) was already proved in [loc. cit.] with different methods. This kind of properness result is the first ingredient in the statement and the proof of degeneration formulas in Gromov-Witten theory. moduli spaces; Gromov-Witten invariants; algebraic stacks; configuration spaces Stacks and moduli problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Configurations of points on degenerate varieties and properness of moduli spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve. Quot schemes; Grothendieck ring of varieties (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli On the motive of Quot schemes of zero-dimensional quotients on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C_ g\) be the coarse moduli space of genus g of pointed curves over the complex numbers and \(W^ n_ g=\{(X,p)\in C_ g\); is an n- Weierstrass point of \(X\}\). - The main result is the following theorem:
(i) \(W^ n_ g\) is irreducible for \(g\geq 4\), \(n\geq 1\) and for \(g=3\), \(n\neq 2.\)
(ii) \(W^ 2_ 3=W^ 1_ 3+S\) where S is an irreducible divisor.
Among the many interesting results of this paper I should mention: (a) A new proof of the following theorem of Lax: If \(g\geq 3\) then a general smooth curve of genus g has only ordinary n-Weierstrass points. (b) An n- Weierstrass point p is said to be dimensionally proper if the locus of n- Weierstrass points near p in the versal deformation family has codimension equal to the weight at p. The authors prove the following existence theorem for n-Weierstrass points: Let \(\beta =(\beta_ 0,...,\beta_ r)\) where \(r=(2n-1)(g-1)-1\) with \(0\leq \beta_ i\leq \beta_{i+1}\); let \(w(\beta):=\sum^{r}_{i=0}\beta_ i \). For \(n\geq 3\) and \(g\geq 2\) (or for \(n=2\) and \(g\geq 3)\) there exists a dimensionally proper n-Weierstrass point on some smooth curve of genus \(g\) having \(\beta\) as its ramification sequence, if \(w(\beta)\leq g-1\) (or respectively \(w(\beta)\leq g-2)\). (c) The authors compute the linear equivalence class of the closure of \(W^ n_ g\) in the moduli space of stable pointed curves. coarse moduli space; n-Weierstrass point Cukiermann F., Fong L.-Y. (1991). On higher Weierstrass points. Duke Math. J. 62: 179--203 Riemann surfaces; Weierstrass points; gap sequences, Fine and coarse moduli spaces On higher Weierstrass points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result (Theorem 1.1) of this very informative and thorough paper is the following. Let \(S\) be a Dedekind scheme with function field \(K\), and let \(X_K\) be a proper smooth connected curve of strictly positive genus over \(K\). Let \(X\) be a minimal proper regular model of \(X_K\) over \(S\), and let \(X_{\text{sm}}\) be the smooth locus of \(X\). Then \(X_{\text{sm}}\) is a Néron model of \(X_K\). For more general curves, the authors also study under which condition Néron models or Néron lft-models exist.
The idea of the proof is as follows. First a study is made of \(S\)-morphisms \(f : Y \to X\) with \(Y\) smooth and \(X\) a normal relative curve with smooth generic fiber. It is shown (Proposition 3.6) that if a fiber \(Y_s\) over a closed point \(s \in S\) is irreducible, then \(f (Y_s)\) is either a point or \(X\) is smooth at all points of this image. This implies that if \(X\) is an integral relative curve with smooth generic fiber, \(Y\) is smooth and irreducible, and \(f : Y \to X\) is an \(S\)-morphism with dominant generic fiber \(f_K : Y_K \to X_K\), then \(f\) factors through the minimal desingularization of \(X\).
The authors then use this property and the closed immersion \(f : X_{K, \text{sm}} \to \text{Pic}_{X_K | K}^1\) to reduce the problem to the known case of Jacobians, thereby proving the main theorem. Note, however, that \(X_{\text{sm}}\) may not be faithfully flat over \(S\), nor need it embed into the Néron model of \(J_K\). Still, a finite morphism \(Y_K \to X_K\) from a proper smooth connected curve \(Y_K\) always extends to a morphism \(Y_{\text{sm}} \to X_{\text{sm}}\). For general smooth proper algebraic varieties \(X_K\) that admit a proper regular model \(X\) without rational curves in the fibers over elements of \(S\), it is also shown that \(X_{\text{sm}}\) is a Néron model of \(X_K\).
After studying the local case and the special case of conics, the authors show the following results (among others) for more general curves in the case where \(S\) is excellent:{\parindent=0.7cm\begin{itemize}\item[--] Let \(X_K\) be a proper regular (but not necessarily smooth) curve of strictly positive arithmetic genus over \(K\). Then the smooth locus of \(X_K\) admits a Néron model, which is constructed exactly as before. \item[--] (Theorem 7.10) Let \(U_K\) be an affine smooth connected curve over \(K\). Then \(U_K\) admits a Néron lft-model over \(S\) if \(U_K \ncong \mathbb{A}_L^1\) for any finite extension \(L\) of \(K\). \item[--] (Proposition 7.11) Let \(U_K\) be an affine smooth geometrically connected curve over \(K\), not isomorphic to \(\mathbb{A}_K^1\), and with regular compactification \(C_K\). Let \(\Delta_K = C_K \setminus U_K\). Let \(C\) be a relatively minimal regular model of \(C_K\) over \(S\), and let \(\Delta\) be the reduced Zariski closure of \(\Delta_K\) in \(C\). Then the Néron lft-model above is in fact a Néron model if and only if \(\Delta_K (K_s^{sh}) = \emptyset\) for all closed points \(s \in S\) and moreover \(\Delta_s \cap C_{\text{sh}, s} (k (s)^{\text{sep}}) = \emptyset\) for almost all \(s \in S\). In particular, the Néron lft-model is not a Néron model if \(K\) is of characteristic \(0\), \(S\) is infinite, and \(\Delta_K (K^{\text{sep}}) \neq \emptyset\).
\end{itemize}} Néron model; curve; good reduction Liu, Q; Tong, J, Néron models of algebraic curves, Trans. Amer. Math. Soc., 368, 7019-7043, (2016) Arithmetic ground fields for curves, Local ground fields in algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights, Varieties over global fields Néron models 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 We generalize a formula of \textit{S.-W. Zhang} [Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031), Theorem 3.4.2], which describes local arithmetic intersection numbers of three Cartier divisors with support in the special fiber on a self-product of a semi-stable arithmetic surface using elementary analysis. By an approximation argument, Zhang extends his formula to a formula for local arithmetic intersection numbers of three adelic metrized line bundles on the self-product of a curve with trivial underlying line bundle. Using the results on intersection theory from [the author, Abh. Math. Semin. Univ. Hamb. 86, No. 1, 97--132 (2016; Zbl 1391.14047)] we generalize these results to \(d\)-fold self-products for arbitrary \(d\). For the approximations to converge, we have to assume that \(d\) satisfies the vanishing condition [the author, loc. cit., 4.7], which is true at least for \(d\in\{2,3,4,5\}^3\). Local ground fields in algebraic geometry, Elliptic curves over global fields, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rigid analytic geometry An analytic description of local intersection numbers at non-Archimedian places for products of semi-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 The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic \({\mathcal D}\)-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)]. As another application, we compare the push-forward as arithmetic \({\mathcal D}\)-modules and the rigid cohomologies taking Frobenius into account. These theorems will be used to prove ``\(p\)-adic Weil II'' and a product formula for \(p\)-adic epsilon factors. \(p\)-adic cohomology; arithmetic \(\mathcal D\)-module; rigid cohomology Abe, Tomoyuki, Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \(\mathcal{D}\)-modules, Rend. Semin. Mat. Univ. Padova, 131, 89-149, (2014) \(p\)-adic cohomology, crystalline cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, \(p\)-adic differential equations Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \(\mathcal D\)-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 This paper is a report on the current status of attempts to generalize, to the Prym setting, two famous approaches to the Torelli problem for Jacobians: the ``base locus of quadric tangent cones'' method of \textit{A. Andreotti} and \textit{A. Mayer} [Ann. Sc. Norm. Supér. Pisa, Sci. Fis. Mat. (3) 21, 189--238 (1967; Zbl 0222.14024)], \textit{D. Mumford} [Curves and their Jacobians, Univ. Mich. Press (1975; Zbl 0316.14010)], and \textit{M. Green} [Invent. Math. 75, 85--104 (1984; Zbl 0542.14018)] and the ``branch divisor of the Gauss map'' method of \textit{A. Andreotti} [Am. J. Math. 80, 801--828 (1958; Zbl 0084.17304)], both of which refer to construction on the theta divisor. We will also describe for Prym varieties the ``infinitesimal variation of Hodge structures'' (IVHS) approach to Torelli problems of \textit{J. A. Carlson} and \textit{P. A. Griffiths} [Journées de geometrie algébrique, Angers/Fr. 1979, 51--76 (1980; Zbl 0479.14007)] and an analog of a result of \textit{G. Kempf} [Math. Centrum, Amsterdam, Afd. zuivere Wisk. ZW6/71 (1971; Zbl 0223.14018), p. 16; and Ann. Math. (2) 110, 243--273 (1979; Zbl 0452.14011), Cor. 4.4, p. 253] linking the Torelli problem to properties of ``Picard sheaves'', i.e. (higher) direct images of Poincaré bundles on Abelian varieties.
Although the article's goal is primarily expository, much of the material discussed is very recent, some arguments (such as the IVHS argument for degree one Torelli for Pryms) seem not to have occurred in print before and some results (such as the density of double points in the stable singular locus of Prym theta divisors and the requisite Riemann singularities theorem for double points which are both stable and exception are new. base locus of quadrics; Prym varieties; Jacobian Gauss divisors; Prym Gauss divisors; birational Donagi conjecture; double covers R. Smith and R. Varley, ''The Prym Torelli problem: An update and a reformulation as a question in birational geometry,'' in Symposium in Honor of C. H. Clemens, Bertram, A., Carlson, J. A., and Kley, H., Eds., Providence, RI: Amer. Math. Soc., 2002, pp. 235-264. Torelli problem, Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians The Prym Torelli problem: An update and a reformulation as a question in birational 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 paper is part of a very interesting new project of the two authors of developing nonstandard methods for algebraic geometry. The present article is a remarkable application of these techniques to the theory of étale cohomology. A drawback of étale cohomology, an important tool in arithmetic geometry, is the fact that it has good properties only with torsion or even finite coefficients, although one would like to obtain cohomology groups that are vector spaces over a field of characteristic zero. There are different ways to overcome this problem, one is the well-known \(\ell\)-adic cohomology, another more general and more sophisticated one is Jannsen's and Dwyer-Friedlander's continuous cohomology which agrees with the latter one for smooth projective schemes over a field.
The paper under review provides an astonishing new approach. Using infinite prime numbers in nonstandard integers one can construct fields of characteristic zero which behave in many respects like finite fields. This yields an étale cohomology theory with characteristic zero coefficients but the good behavior of finite coefficients. In order to achieve this goal, the authors study an enlarged category of nonstandard étale sheaves and the associated sheaf cohomology theory, which is in fact a Weil cohomology theory. In particular, the definition of nonstandard étale cohomology is given as a derived functor which is also one of the advantages of \textit{U. Jannsen}'s construction of continuous cohomology for schemes in [Math. Ann. 280, No.2, 207--245 (1988; Zbl 0649.14011)]. Moreover, there is also a generalized Hochschild-Serre spectral sequence for nonstandard étale cohomology. In the final section nonstandard étale and \(\ell\)-adic cohomology are compared and shown to agree for smooth projective varieties over algebraically closed fields for infinitely many primes \(\ell\). nonstandard methods; étale cohomology; \(\ell\)-adic sheaves; \(\ell\)-adic cohomology Étale and other Grothendieck topologies and (co)homologies, Nonstandard models in mathematics Nonstandard étale cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper contains two fundamental contributions.
{1. Dualizability.} The author studies dualizability for commutative group stacks.
{2. Albanese morphism.} The author defines and generalizes scheme theoretic Albanese torsors and morphisms to stacks.
The author then uses these fundamental contributions to describe universal torsors of Colliot-Thélène and Sansu in terms of the Albanese torsor. The author also uses these fundamental results to prove that Grothendieck's section conjecture holds for some root stacks over Severi-Brauer varieties.
Let us now describe the two main contributions of the paper.
{1. Dualizability.}
Let \(G\) be a commutative group stack \(G\) over a base scheme \(S\). One associated to \(G\) a commutative group stack \(D(G)\) defined as \(\mathscr{H}om (G , B \mathbb{G}_m) \). There is a canonical map \(e_G : G \to D (D(G))\). Naturally, one says that \(G\) is dualizable if \(e_G\) is an isomorphism. After recalling that Deligne \(1\)-motives and Beilinson \(1\)-motives are dualizable, the author attacks the question of finding sufficient conditions for \(G\) to be dualizable. The author has the following Conjecture.
{Conjecture.} Let \(S\) be a scheme with \(2 \in \mathcal{O} _S ^{\times}\). Let \(G\) be a proper, flat, and finitelly generated algebraic commutative group stack over \(S\), with finite and flat inertia stack. Then:
(1) \(D(G)\) is algebraic, proper, flat and finitely presented, with finite diagonal.
(2) \(G\) is dualizable.
The author has fundamental results in this direction. Under the assumptions of the conjecture, Brochard proves that \(D(G)\) is algebraic and finitely presented with quasi-finite diagonal and proves that (2) is a consequence of (1). Moreover the author proves the conjecture with the additional assumption that \(H^0(G)\) is cohomologically flat. The author gives other results with the same flavour in the paper. The author notes that the assumption that \(2 \in \mathcal{O} _S ^{\times}\) might be superfluous.
{2. Albanese morphism.}
Let \(X\) be an algebraic stack over a base \(S\) and denote by \(f: X \to S\) its structural morphism. Assume that \(\mathcal{O} _X \to f_* \mathcal{O} _X\) is universally an isomorphism and that \(f\) locally has sections in the \(fppf\) topology. The author defines the Albanese stack \(A^0_{\tau}(X)\) as the dual \(D(\mathrm{Pic}^{\tau} _{X/S})\) of the torsion component of the the Picard stack. Then Brochard defines an \(A_{\tau}^0(X)\)-torsor \(A^1_{\tau}(X)\) and a morphism \(a_X : X \to A_{\tau}^1(X)\) called the Albanese morphism. After introducing a ''duabelian condition'', the author proves the following Theorem.
{Theorem.} Assume that \(\mathrm{Pic}^{\tau} _{X/S}\) is a duabelian group. Then the Albanese morphism
\[
a_X : X \to A^1 _{\tau} (X)
\]
is initial among maps to torsors under abelian stacks, in the following sense. For any triple \((B, T, b)\) where \(B\) is an abelian stack, \(T\) is a \(B\)-torsor and \(b : X \to T\) is a morphism of algebraic stacks, there is a triple \((c_0, c_1, \gamma)\) where \(c_0 : A^0 _{\tau} (X) \to B\) is a homomorphism of commutative group stacks, \(c_1 : A^1 _{\tau} (X) \to T \) is a \(c_0\)-equivariant morphism, and \(\gamma\) is a 2-isomorphism \(c_1 \circ a_X \Rightarrow b.\) Such a triple \((c_0, c_1, \gamma)\) is unique up to a unique isomorphism. commutative groups stacks; dualizability; Picard stacks; Albanese morphism Generalizations (algebraic spaces, stacks), Picard schemes, higher Jacobians Duality for commutative group stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X'\rightarrow X\) be an étale cover of degree 2 where \(X\) and \(X'\) are geometrically connected smooth complete curves over a finite field of characteristic \(> 2\) and \(F\subset F'\) function fields. Let \(G = \text{PGL}_2\). Consider an everywhere unramified cuspidal automorphic representation \(\pi\) of \(G(F_{\mathbb A})\) and its base change \(\pi_{F'}\) to \(F'\). The main result of the article is a geometric expression for the central derivatives of the \(L\)-function of \(\pi_{F'}\). The result is obtained by the comparison of two relative trace formulas. One of these is an adaptation of Jacquet's relative trace formula, the other one is of geometric nature.
The authors define for each even number \(r\) a cycle (called Heegner-Drinfeld cycle) on a moduli stack of shtukas for \(G\) with \(r\)-modifications. The intersection number of this cycle with its transform by a Hecke function is defined. It is studied in two ways. Using intersection theory it is written as a trace. The decomposition of the cohomology of the stack under the Hecke action gives a decomposition of the Heegner-Drinfeld cycle and the spectral decomposition of the intersection number.
In order to compare the two relative trace formulas, the orbital integrals of the analytic formula are interpreted as traces of Frobenius. The orbital sides being identified (up to a factor \({(\text{log} q)}^r\)), the spectral sides are also equal. That gives the expression of the \(r\)-th derivative of the \(L\)-function as a self-intersection number of a component of the Heegner-Drinfeld cycle. \(L\)-functions; Drinfeld Shtukas; Gross-Zagier formula; Waldspurger formula Yun, Zhiwei; Zhang, Wei, Shtukas and the {T}aylor expansion of {\(L\)}-functions, Ann. of Math. (2). Annals of Mathematics. Second Series, 186, 767-911, (2017) Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Modular and Shimura varieties, Representation-theoretic methods; automorphic representations over local and global fields, Vector bundles on curves and their moduli Shtukas and the Taylor expansion of \(L\)-functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this expository paper is to present a catalogue of locally free replacements of the sheaves of principal parts for (families of) Gorenstein curves. In the smooth category, locally free sheaves of principal parts are better known as jet bundles, understood as those locally free sheaves whose transition functions reflect the transformation rules of the partial derivatives of a local section under a change of local coordinates. Being a natural globalisation of the fundamental notion of Taylor expansion of a function in a neighborhood of a point, jet bundles are ubiquitous in mathematics. They proved powerful tools for the study of deformation theories within a wide variety of mathematical situations and have a number of purely algebraic incarnations: besides the aforementioned principal parts of a quasi-coherent sheaf we should mention, for instance, the theory of arc spaces on algebraic varieties, introduced by Nash to deal with resolutions of singular loci of singular varieties.
The issue we want to cope with in this survey is that sheaves of principal parts of vector bundles defined on a singular variety \(X\) are not locally free. Roughly speaking, the reason is that the analytic construction carried out in the smooth category, based on gluing local expressions of sections together with their partial derivatives, up to a given order, is no longer available. Indeed, around singular points there are no local parameters with respect to which one can take derivatives. This is yet another way of saying that the sheaf \(Omega^1_X\) of sections of the cotangent bundle is not locally free at the singular points.
If \(C\) is a projective reduced singular curve, it is desirable, in many interesting situations, to dispose of a notion of global derivative of a regular section. If the singularities of \(C\) are mild, that is, if they are Gorenstein, locally free substitutes of the classical principal parts can be constructed by exploiting a natural derivation \(O_C \to omega_C\), taking values in the dualising sheaf, which by the Gorenstein condition is an invertible sheaf. This allows one to mimic the usual procedure adopted in the smooth category. Related constructions have recently been reconsidered by \textit{A. Patel} and \textit{A. Swaminathan} [``Inflectionary invariants for isolated complete intersection curve singularities'', \url{arXiv:1705.08761}], under the name of sheaves of invincible parts, motivated by the classical problem of counting hyperflexes in one-parameter families of plane curves. Besides, locally free jets on Gorenstein curves have been investigated by a number of authors, starting about twenty years ago. The reader can consult other references for several applications. Arcs and motivic integration Jet bundles on Gorenstein curves and applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By classical work of Serre, Tate and Hartshorne, Grothendieck's trace map for a smooth, projective curve over a finite field can be expressed as a sum of residues over all closed points of the curve. Subsequently, this result was generalized in several ways, up to algebraic varieties of any dimension, using higher-dimensional adèles. In all these generalizations one only deals with varieties over a field.
The paper gives the first extension non-varieties, namely to arithmetic surfaces. Let \({\mathcal O}_k\) be a Dedekind domain of characteristic zero and with finite residue fields, and let \(K\) be its field of fractions. The extension given in Theorem 3.1 of the paper applies to a normal scheme, proper and flat over \(S=\text{Spec}{\mathcal O}_K\), whose generic fibre is a smooth, geometrically connected curve.
This extension requires three main steps: i) the definition of suitable local residue maps, either on spaces of differential forms or on local cohomology groups; ii) using the local residue maps, the definition of the dualizing sheaf; iii) patching together the local residue maps to define Grothendieck's trace map on the cohomology of the dualizing sheaf.
The first two steps are essentially already contained in a previous paper of the same author [New York J. Math. 16, 575--627 (2010; Zbl 1258.14031)], while the third step is achieved in the present paper.
The paper ends with applications to adelic duality for the arithmetic surface. reciprocity laws; higher adèles; arithmetic surfaces; Grothendieck duality; residues Morrow, M.: Grothendieck's trace map for arithmetic surfaces via residues and higher adèles, Algebra number theory 6, No. 7, 1503-1536 (2012) Arithmetic ground fields for curves, Local cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Grothendieck's trace map for arithmetic surfaces via residues and higher adèles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(g>0\) be an integer and \(M_g\) (resp. \(A_g\)) be the coarse moduli space of smooth, irreducible and projective curves of genus \(g\) (resp. principally polarized abelian varieties of dimension \(g\)) over the complex field. These two important moduli spaces are related through the \textit{Torelli map} \(j\) which associates to the isomorphism class of a curve in \(M_g\), the isomorphism class of its Jacobian with its canonical polarization in \(A_g\). \textit{Thomae-like formulae} (called here Thomae-Weber formulae) can be seen as an explicit description of \(j\). Indeed, as \textit{D. Mumford} showed in [Invent. Math. 1, 287--354 (1966; Zbl 0219.14024)], a principally polarized abelian variety can be written down as intersection of explicit quadrics in a projective space and the coefficients of these quadrics are determined by a certain projective vector of constants called \textit{Thetanullwerte} (or Thetanulls or theta constants). Thomae-like formulae express these constants (or quotients raised to a certain power) in terms of the geometry of the curve.
In the case of a hyperelliptic curve, \textit{J. Thomae} himself [J. Reine Angew. Math. 71, 201--222 (1870; JFM 02.0244.01)] found such a formula.
For non-hyperelliptic curves, several other authors have worked precise formulae in specific cases. \textit{H. Weber} for instance gives such a formula in [Preisschrift. Berlin. G. Reimer. \(4^{\circ}\) (1876; JFM 08.0293.01)] for non-hyperelliptic curves of genus \(3\) seen as smooth plane quartics in terms of the coefficients or their bitangents. His method, revisited in [\textit{E. Nart} and \textit{C. Ritzenthaler}, Contemp. Math. 686, 137--155 (2017; Zbl 1374.14026)] is actually sufficiently flexible to work out Thomae-like formulae for any non-hyperelliptic curve, not as a closed formula but algorithmically. This requires a good understanding of the combinatoric of the generalization of the bitangents (called multitangents) to the canonical embedding of the curve. The explicit labeling deduced from the combinatoric is worked out thanks to a tricky algorithm presented in this paper.
The hardest problem is to actually find a case where this is computable as, for a generic curve over \(\mathbb{Q}\), the multitangents are defined over a too large Galois extension to be handled exactly. For genus 4, the author works with examples coming from del Pezzo surfaces of degree 1 for which she manages to compute the equations of the tritangents and to control their combinatoric. theta constants; non-hyperelliptic curve; del Pezzo surface; algorithm; genus 4; Torelli map Theta functions and curves; Schottky problem, Quadratic forms over general fields Thomae-Weber formula: algebraic computations of theta constants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 survey article, the author provides an overview of one of the central topics in the study of complex projective algebraic curves: Brill-Noether theory. Roughly speaking, Brill-Noether theory describes the geometry of special divisors and linear systems on generic curves from various points of view. Starting from the pioneering classical work of \textit{A. Brill} and \textit{M. Noether} [Gött. Nachr. 1873. 116--132 (1873; JFM 05.0348.03)] more than 135 years ago, this topic has undergone a tremendous development in the last 30 years, especially in the context of moduli spaces of curves and their geometrically defined subschemes. Whereas much of the foundational material concerning modern Brill-Noether theory can be found in the standard text ``Geometry of algebraic curves'' of \textit{E. Arbarello, M.~Cornalba, P.~A. Griffiths}, and \textit{J. Harris} [Grundlehren der mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)], the author focuses here on discussing the following more general aspects:
1. What do we mean by Brill-Noether theory, and which role does it play in our understanding of algebraic curves?
2. Classical Brill-Noether theory (up to the 1980s) and its main results.
3. Recent results and directions for further study I: The geometry of general curves in projective space. -- This section is devoted to possible variants of the so-called ``Maximal Rank Conjecture'' for general projective curves and related conjectural statements of cohomological (or syzygy-theoretic) nature.
4. Recent-results and directions for further study II: Linear systems on special curves. -- In this concluding section, the author addresses the following general question: To what extent do the basic theorems of classical Brill-Noether theory continue to hold for curves that are general in low-dimensional subvarieties of the corresponding moduli space? Together with some tentative dimension counts for certain components of related Hilbert schemes, both a precise conjecture and a concrete strategic research problem are given.
Overall, this survey article provides a very enlightening and inspiring introduction to some important aspects of Brill-Noether theory in algebraic geometry, thereby linking its fascinating genesis and its contemporary significance in a masterly manner. The author, one of the leading experts in the field, anew gives proof of his vivid style of mathematical writing and expository mastery. survey article (algebraic geometry); algebraic curves; Brill-Noether theory; moduli spaces of curves; Hilbert schemes; special divisors; linear systems Special divisors on curves (gonality, Brill-Noether theory), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Divisors, linear systems, invertible sheaves, Parametrization (Chow and Hilbert schemes), Picard groups, Syzygies, resolutions, complexes and commutative rings Brill-Noether theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper contains a lecture on the stable reduction theorem for curves as proved by \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)]. After an introduction to normal, regular and semistable models of curves, the Picard functor of a singular curve is explained and it is used to link semistable reduction of the curve with that of its Jacobian. This is the path to the proof of Deligne and Mumford. Finally, this theorem is applied to show that moduli spaces of stable curves are proper. algebraic curve; regular model; stable reduction Curves over finite and local fields, Families, moduli of curves (algebraic) Models 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 concepts of étale topology and étale cohomology were initially defined by A. Grothendieck in the early 1960's and, in the sequel, developed as a complete methodical framework in algebraic geometry by him, M. Artin, P. Deligne, and others. The original purpose for introducing this étale theory was to rigorously elaborate A. Weil's discovery (1949) that the complex topology of an algebraic variety over \(\mathbb{Z}\) seemingly determines the number of rational points with respect to the finite field obtained by reduction (of the defining equations) modulo a prime number. In this regard, étale cohomology theory proved itself to be very useful and effective. Moreover, it has become one of the most powerful methods in algebraic geometry and number theory, which has led to fundamentally new insights, solutions of long-standing problems, and various applications within these areas.
The present booklet provides a first introduction into the basic concepts and method of étale cohomology. It grew out of a lecture course held by the author at the University of Göttingen in 1975/1976, and the German edition of the text appeared in 1979 as mimeographed lecture notes [cf. ``Einführung in die étale Komologie'', Regensb. Trichter 17 (1979; Zbl 0473.14010)]. The text under review is an English translation of the completely unchanged German original.
Quite doubtless, it is a highly welcome circumstance that the author has made his well-tried lecture notes once more available and, this time, even accessible to a wider international public. In fact, the text has maintained, over the past twenty years, its unique introductory character, and it still represents an excellent guide through the basics of étale cohomology. Even now, these notes provide a masterly arranged preparatory material for those, who want to learn étale cohomology and its various applications more thoroughly, in particular by studying the very advanced standard books ``SGA 4''=Séminaire de géométrie algébrique 1963-64, Lect. Notes Math. 269, 270 and 305 (1972; Zbl 0234.00007), (1972; Zbl 0237.00012) and (1973; Zbl 0245.00002); ``SGA \(4{1 \over 2}\)'' [cf. Lect. Notes Math. 569 (1977; Zbl 0345.00010)], and \textit{J. S. Milne}'s ``Étale cohomology'' [cf. Princeton Math. Ser. 33 (1980; Zbl 0433.14012)]. étale topology; étale cohomology G. Tamme, Introduction to étale cohomology, Universitext, Springer, Berlin 1994. Étale and other Grothendieck topologies and (co)homologies, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Introduction to étale cohomology. Translated by Manfred Kolster | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Hilbert scheme and the Quot scheme, both constructed by Grothendieck, are fundamental objects in algebraic geometry: the first one parametrizes the subschemes of a projective space with fixed Hilbert polynomial, the second one the quotients with fixed Hilbert polynomial of a fixed coherent sheaf on a projective space.
Here the author considers an affine variety \(X\), with an action of a reductive group \(G\), and a coherent sheaf \(\mathcal M\) on \(X\), that is supposed to be \(G\)-linearized. First of all he proves the existence of a quasi-projective scheme, the invariant Quot scheme, parametrizing the quotients \(\mathcal L\) of \(\mathcal M\), such that the space of global sections \(H^0(X,\mathcal L)\) is a direct sum of simple \(G\)-modules with fixed finite multiplicities: the datum of these multiplicities is here the analogous of the Hilbert polynomial. The invariant Quot scheme is a natural generalization of the invariant Hilbert scheme recently introduced by \textit{V. Alexeev} and \textit{M. Brion} [J. Algebr. Geom. 14, 83--117 (2005; Zbl 1081.14005)]. The construction relies on the multigraded Quot scheme of Haiman and Sturmfels, corresponding to the case in which \(G\) is a torus [cf. \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)].
In the second part of the article, the author focuses on a special example, the cone \(X\) of primitive vectors of a simple \(G\)-module, with a free sheaf \(\mathcal M\) generated by another simple \(G\)-module. He proves that, in this case, the invariant Quot scheme has only one point, and that it is reduced unless \(X\) is the cone of the primitive vectors of a quadratic vector space \(V\) of odd dimension \(2n+1\) and \(G= \text{Spin}(2n+1)\times H\), for a connected reductive group \(H\). The Quot scheme of this example is isomorphic to \(\text{Spec}({\mathbb C}[t]/\langle t^2\rangle)\). Hilbert scheme; Quot scheme; reductive group; primitive vector Jansou, S., Le schéma quot invariant, J. Algebra, 306, 2, 461-493, (2006) Parametrization (Chow and Hilbert schemes), Geometric invariant theory, Group actions on varieties or schemes (quotients) The invariant Quot 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 Let \(\mathcal O\) denote the structure sheaf of a smooth, projective curve \(X\) over a finite field \(k\), and \(\text{Cl}(\mathcal O[G])\) the reduced Grothendieck group of \(\mathcal O[G]\)-vector bundles, where \(G\) is a finite abelian group. Any tamely ramified Galois covering \(f:Y \to X\) of smooth projective curves over \(k\) with Galois group \(G\) (``tame \(G\)-cover of \(X\)'') and any \(G\)-stable line bundle \(\mathcal A\) on \(Y\) yields a ``realizable'' class \((f_* \mathcal A) \in \text{Cl}(\mathcal O[G])\). The main aim of this paper is to give an explicit description of the set of classes arising from the structure sheaves of all tame (resp. étale) \(G\)-covers of \(X\), and of that subgroup of \(\text{Cl}(\mathcal O[G])\), which is generated by all realizable classes. Considering the natural Euler characteristic classes of \(\mathcal O[G]\)-vector bundles as introduced by \textit{T. Chinburg} [in: Sémin. Théor. Nombres Bordx., Sér. II 4, No. 1, 1-18 (1992; Zbl 0768.14008)], the authors apply their results to describe the realizable classes in \(\text{Cl} (k[G])\) as well.
To attack this problem, it suffices to consider the cases that \(G\) is a cyclic \(p\)-group, or \(p \nmid \#G\), resp., where \(p=\text{char}(k)\). In the former case, the proof uses Witt vectors and the cohomology groups of \textit{J.-P. Serre} [in: Sympos. int. Topol. Algebr. 24-53 (1958; Zbl 0098.13103)] to describe the realizable classes (theorem 2.5). In the latter one, the Hom-description of class groups -- originating from \textit{A. Fröhlich} [``Galois module structure of algebraic integers'' (1983; Zbl 0501.12012)] and adapted for function fields by \textit{R. J. Chapman} [in: The arithmetic of function fields, Proc. Workshop Ohio State Univ., Columbus 1991, Ohio State Univ. Math. Res. Inst. Publ. 2, 403-411 (1992; Zbl 0801.11047)] is employed to explicitly gain control of \(\text{Cl}(\mathcal O[G]) \simeq \text{Pic}(\mathcal O[G])\).
For the description of the realizable classes (theorem 2.8, in the paper referred to as theorem 2.9), the authors adapt the main ideas of \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259-306 (1987; Zbl 0619.12008)], who used resolvents and Stickelberger maps to answer the corresponding question for abelian extensions of number fields, and of the second author [\textit{D. Burns}, Math. Proc. Camb. Philos. Soc. 118, No. 3, 383-392 (1995; Zbl 0863.11077)]. reduced Grothendieck group; equivariant line bundles on curves; curve over a finite field; resolvent; Stickelberger map; Euler-Poincaré characteristic map; Picard group; tamely ramified Galois covering; class groups 10.1353/ajm.1998.0045 Vector bundles on curves and their moduli, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Group actions on varieties or schemes (quotients) On the Galois structure of equivariant line bundles on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper generalizes several known results for smooth curves and their Jacobians to the following setting: a curve \(X\), which is reduced, projective and with locally planar singularities. The Jacobian of the smooth curve is replaced by two objects: on the one hand, \(J(X)\), the generalized Jacobian, parametrizing line bundles on \(X\) that have degree zero on each irreducible component; on the other hand, a fine compactified Jacobian \(\overline J_X(\underline q)\), which depends on a stability condition \(\underline q\). The latter was constructed in this setting by \textit{E. Esteves} [Trans. Am. Math. Soc. 353, No. 8, 3045--3095 (2001; Zbl 0974.14009)], and the authors studied these compactifications already in [Trans. Am. Math. Soc. 369, No. 8, 5341--5402 (2017; Zbl 1364.14007)], the results of which serve as a basis for the current paper.
The main goal is to show that the Fourier Mukai transform of \(\overline J_X(\underline q)\) is an autoequivalence of its bounded derived category, generalizing the celebrated result for smooth curves and their Jacobians established by \textit{S. Mukai} [Nagoya Math. J. 81, 153--175 (1981; Zbl 0417.14036)]. In the current paper, the authors show that the Fourier Mukai transform corresponding to a Poincaré line bundle on \(J(X) \times \overline J_X(\underline q)\) is fully faithful (Theorem A). This is then used in the subsequent paper by the same authors [Geom. Topol. 23, No. 5, 2335--2395 (2019; Zbl 1430.14009)] to establish the above described autoequivalence (in fact, more generally, the derived equivalence of any two fine compactified Jacobians).
The authors give several additional applications of their result in the current paper: they calculate cohomology of line bundles on \(\overline J_X(\underline q)\) (Corollary B); show that \(\mathrm{Pic}^0(\overline J_X(\underline q)) \simeq J(X)\) as algebraic groups (Theorem C); and they show that line bundles on \(\overline J_X(\underline q)\) are algebraically equivalent to \(0\) if and only if they are numerically equivalent to \(0\) (Theorem D). Each of these again extend important results from the smooth case to that of reduced curves with locally planar singularities. compactified Jacobians; Fourier-Mukai transforms; locally planar curve singularities Singularities of curves, local rings, Jacobians, Prym varieties, Vector bundles on curves and their moduli, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Fourier-Mukai and autoduality for compactified Jacobians. 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 ``We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curve E over \({\mathbb{F}}_ q\), given by a Weierstrass equation, one can compute the number of \({\mathbb{F}}_ q\)- rational points of E in time O((log q)\({}^ 9)\). Our result is the following. Let A be an Abelian variety over \({\mathbb{F}}_ q\). Then one can compute the characteristic polynomial of the Frobenius endomorphism of A in time O((log q)\({}^{\Delta})\), where \(\Delta\) and the implied constant depend only on the dimension of the embedding space of A, the number of equations defining A and the addition law, and their degrees. The method, generalizing that of Schoof, is to use the machinery developed by Weil to prove the Riemann hypothesis for Abelian varieties. By means of this theory, the calculation is reduced to ideal-theoretic computations in a ring of polynomials in several variables over \({\mathbb{F}}_ q\). As applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over \({\mathbb{Q}}\) in time polynomial in log p: we show also that, for a fixed prime \(\ell\), we can compute the \(\ell th\) roots of unity mod p, when they exist, in polynomial time in log p. This generalizes Schoof's application of his algorithm to find square roots of a fixed integer x mod p.'' Abelian varieties over finite fields; algorithm of Schoof; rational points; Frobenius endomorphism; roots of unity mod p; polynomial time J. Pila, ''Frobenius maps of abelian varieties and finding roots of unity in finite fields,'' Math. Comp., vol. 55, iss. 192, pp. 745-763, 1990. Number-theoretic algorithms; complexity, Finite ground fields in algebraic geometry, Analysis of algorithms and problem complexity, Complex multiplication and moduli of abelian varieties, Varieties over finite and local fields, Complex multiplication and abelian varieties Frobenius maps of abelian varieties and finding roots of unity in 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 Let \(F\) be a finite extension of \(\mathbb Q_p\), and let \(\Sigma_N\) be an inverse system of rigid analytic étale Galois coverings of the nonarchimedean upper half space \(\Omega \subset \mathbb P^{n-1}_F\) indexed by a positive integer \(N\). These coverings are equivariant for the action of \(GL(n,F)\) and for the group \(B^\times\) of units in a division algebra \(B\) over \(F\) with invariant \(1/n\). It was conjectured by \textit{H. Carayol} [Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. II, Perspect. Math. 11, 15-40 (1990; Zbl 0704.11049)] that the direct limit of the cohomology of the \(\Sigma_N\) decomposed as the sum, over square-integrable representations \(\pi\) of \(GL(n,F)\), of \(\pi \otimes JL(\pi) \otimes \widetilde{\sigma} (\pi)\), where \(JL(\pi)\) is the representation of \(B^\times\) associated to \(\pi\) by the generalized Jacquet-Langlands correspondence, and where \(\widetilde{\sigma} (\pi)\) is the representation of the Weil group of \(F\) associated to \(\pi\) by the Langlands correspondence, dualized and slightly twisted. Under certain assumptions, Carayol proved his conjecture for \(n=2\) and sketched a program to generalize his methods to higher dimensions. In this paper, the author carries out most of Carayol's program for the part of the cohomology corresponding to supercuspidal representations of \(GL(n,F)\). The results of this paper include the realization of the generalized Jacquet-Langlands correspondence, and a partial description of the associated Weil group representations. supercuspidal representations; automorphic forms; Jacquet-Langlands correspondence; Galois representations; cohomology of Drinfel'd upper half spaces; étale Galois coverings; Weil group representations Michael Harris, Supercuspidal representations in the cohomology of Drinfel\(^{\prime}\)d upper half spaces; elaboration of Carayol's program, Invent. Math. 129 (1997), no. 1, 75 -- 119. Representation-theoretic methods; automorphic representations over local and global fields, Étale and other Grothendieck topologies and (co)homologies Supercuspidal representations in the cohomology of Drinfel'd upper half spaces; elaboration of Carayol's program | 0 |
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