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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f\colon X \rightarrow S\) be a flat family of (reduced) varieties of dimension \(d\) with normal crossing singularities. Assume that \(X\) and \(S\) are smooth, \(\Delta \subset S\) and \(Y = f^{-1}(\Delta) \subset X\) are divisors with normal crossings such that \(f\colon X\setminus Y \rightarrow S\setminus\Delta\) and \(f\colon (X, \log Y) \rightarrow (S, \log\Delta)\) are smooth and log smooth morphisms, respectively. Let \({\mathcal L}\) be a line bundle on \(X.\) The author defines projective heat operators which are logarithmic analogues of those from [\textit{B. van Geemen, A. J. de Jong}, J. Am. Math. Soc. 11, No. 1, 189--228 (1998; Zbl 0920.32017)] and describes sufficient conditions for the existence of a projective logarithmic heat operator on \({\mathcal L}\) over \(S\) similarly [loc. cit.] which gives a logarithmic projective connection on \(f_\ast{\mathcal L}.\) This definition is based essentially on the notion of the sheaf of logarithmic differential operators on log schemes introduced by the author. It should be remarked that in a more general context (without assumption of normal crossings) sheaves of logarithmic differential operators were considered and studied in detail in earlier works [\textit{F. J. Calderon-Moreno}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, No. 5, 701--714 (1999; Zbl 0955.14013)]. In conclusion the author proves that a family of generalized Jacobians (moduli spaces of torsion-free sheaves of rank one over nodal curves) satisfies the above mentioned sufficient conditions; in particular, this implies the existence of a projective logarithmic heat operator in this case. flat connections; heat operators; logarithmic differential operators; logarithmic schemes; logarithmic heat equation; moduli spaces; theta divisors; generalized Jacobians Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Heat equation, Algebraic moduli problems, moduli of vector bundles Logarithmic heat projective operators | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 [Georgian Math. J. 22, 563--571 (2015; Zbl 1339.18006)], the author proves that the 2-categorical version of the Seifert-van Kampen theorem holds for the fundamental groupoid of a topological space. As a continuation, the author studies the étale version in the paper under review. More precisely, for a Noetherian scheme \(X\), the assignment \(\Pi_1\) on the site of finite étale covering of \(X\) sends \(Y\) to its étale fundamental groupoid \(\Pi_1(Y)\). The main result is that \(\Pi_1\) is a costack over the finite étale site. Moreover, this costack is the associated costack of the constant trivial pseduofunctor. fundamental groupoid; stack; costack; Seifert-van Kampen; étale covering; associated stack Homotopy theory and fundamental groups in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies The étale fundamental groupoid as a 2-terminal costack | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book is based upon the topic of the Kuwait Foundation Lectures that the author gave at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge in 2005. The goal of this book is to make accessible to graduate students and researchers some central results of transcendental number theory and to apply them to the study of periods and special values of certain classical functions. The book uses results on transcendence and linear independence properties of algebraic \(1\)-forms on commutative algebraic groups.
In Chapter 1, group varieties are introduced and the statement of Analytic Subgroup Theorem due to G. Wüstholz in the early 1980s is given. Then, full details of the consequences of Analytic Subgroup Theorem is given for transcendence and linear independence of periods of algebraic \(1\)-forms on abelian varieties. Chapter 2 features the Schneider-Lang Theorem for functions of one complex variable, which can be used to deduce many transcendence results known before the work of A. Baker. Many corollaries of the Schneider-Lang Theorem are discussed for elliptic and related functions. In Chapter 3, a result of Schneider from 1937 is given, that gives a criterion for complex multiplication on elliptic curves, and its generalization to abelian varieties, due to the author, jointly with H. Shiga and J. Wolfart in the mid-1990s.
In Chapter 4, the components of vectors in polarized lattices inside \(\mathbb C^n\) are introduced as a paring between singular homology and complex de Rham cohomology. Algebraic \(1\)-forms on Riemann surfaces and Jacobians are also discussed. In Chapter 5, the Euler integral representations of classical hypergeometric functions are applied to view them as periods on families of algebraic curves. As the consequences of Analytic Subgroup Theorem from Chapter 1, results on the transcendence of special values of classical hypergeometric functions in one and several variables are given. In Chapter 6 and Chapter 7, most of the transcendence results are due to the papers by the author and joint work with M. D. Tretkoff. Examples related to the level \(2\) Hodge structures of \(K3\) surfaces are given. Focusing on a largely open generalization to polarized Hodge structures of higher level, some affirmative results for certain families of Calabi-Yau manifolds of arbitrary high dimension, appearing in Borcea-Voisin towers, are obtained.
In addition, except for the last chapter, all chapters include exercises suitable for graduate students. hypergeometric functions; periods; elliptic functions Research exposition (monographs, survey articles) pertaining to number theory, Transcendence theory of elliptic and abelian functions, Transcendence theory of other special functions, Complex multiplication and moduli of abelian varieties, Calabi-Yau manifolds (algebro-geometric aspects), Complex multiplication and abelian varieties Periods and special functions in transcendence | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Working over an algebraically closed characteristic \(p>0\) field, let \(\overline{\mathcal{M}}_g\), for \(g \geq 2\), be the Deligne-Mumford compactification of the moduli space of smooth genus \(g\) curves. In the article under review, the authors study the \(p\)-rank \(f\) stratum
\[
\overline{\mathcal{M}}^f_g \subset \overline{\mathcal{M}}_g,
\]
for \(0 \leq f \leq g\).
In more precise terms, fix an odd prime \(\ell \not = p\), let \(\overline{\mathcal{T}}_{\ell,g}\) denote the moduli space of admissible stable genus \(g\), \(\mathbb{Z} / \ell \mathbb{Z}\)-curves and let
\[
\overline{\mathcal{T}}^0_{\ell,g} \subset \overline{\mathcal{T}}_{\ell,g}
\]
denote the rank \(0\) stratum.
The authors' main result is that if \(\mathcal{S}\) is an irreducible component of \(\overline{\mathcal{T}}^0_{\ell,g}\), then \(\mathcal{S}\) contains a point that represents a compact type curve which has at least \(\operatorname{dim}(\mathcal{S})+1\) components. This result can be seen as a generalization of the hyperelliptic case, which was treated in [\textit{J. D. Achter} and \textit{R. Pries}, Adv. Math. 227, No. 5, 1846--1872 (2011; Zbl 1219.14033)].
The proof of this result is by induction on the genus. It makes use of certain clutching maps for labeled admissible stable \(\mathbb{Z} / \ell \mathbb{Z}\) curves of compact type. This approach builds on that of \textit{J. D. Achter} and \textit{R. Pries} [Math. Ann. 338, No. 1, 187--206 (2007; Zbl 1129.11027)].
Finally, the authors study the special case that \(\ell = 3\) in detail. stable curves; cyclic covers; moduli; \(p\)-rank Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Coverings of curves, fundamental group, Automorphisms of curves, Curves over finite and local fields, Algebraic moduli of abelian varieties, classification The boundary of the \(p\)-rank \(0\) stratum of the moduli space of cyclic covers of the projective line | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 abelian varieties defined over function fields of curves in positive characteristic \(p\), focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the \(L\)-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension \(g\), exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves. abelian varieties; Mordell-Weil groups; Artin-Schreier curves Abelian varieties of dimension \(> 1\), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Rational points, Elliptic curves over global fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties Arithmetic of abelian varieties in Artin-Schreier 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 Let \(D\) be a smooth Cartier divisor on a smooth quasi-projective surface \(S\). The Hilbert scheme \((S \setminus D)^{[n]}\) of \(n\) points on \(S \setminus D\) is not proper, but \textit{J. Li} and \textit{B. Wu} have constructed a compactification relative to \(D\) [``Good degeneration of Quot-schemes and coherent systems'', Preprint, \url{arXiv:1110.0390}], called the relative Hilbert scheme. The author uses the moduli stack of stable ideal sheaves and the stack of expanded degenerations of \textit{J. Li} [J. Differ. Geom. 57, 509--578 (2001; Zbl 1076.14540)] to produce the generating function for the normalized Poincaré polynomial of the relative Hilbert scheme of points analogous to the generating function for the Hilbert scheme of \(n\) points given by \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann. 296, No. 2, 235--245 (1993; Zbl 0789.14002)]. When \(S = \mathbb P^2\) and \(D \subset \mathbb P^2\) is a line, the cohomology groups of the relative Hilbert scheme are computed and it is shown that the natural map from the Chow group to the Borel-Moore homology is an isomorphism. Hilbert scheme of points; relative Hilbert scheme; Poincaré polynomial Iman Setayesh, Relative Hilbert scheme of points, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.) -- Princeton University. Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Relative 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 \(X\) be a complete nonsingular irreducible algebraic curve of genus \(g \) defined over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(\pi_1 (X)\) be its algebraic fundamental group, and \(\Gamma_g\) the topological fundamental group of (any) compact Riemann surface of genus \(g\). It is well known since Grothendieck, that a finite group \(G\) of order prime to \(p\) can be realized as the Galois group of an unramified cover of \(X\) if and only if it is a quotient of \(\Gamma_g\). Other information on the structure of \(\pi_1 (X)\) comes from the Hasse-Witt invariant of \(X\), i.e. the \(\mathbb{F}_p\)-dimension of the \(p\)-torsion subgroup of the Jacobian variety of \(X\).
In this paper the author counts the number of unramified Galois coverings of \(X\) whose Galois group is isomorphic to an extension of a group \(G\) of order prime to \(p\) by a finite group \(H\), which is an irreducible \(\mathbb{F}_p [G]\)-module. This counting depends on some invariants attached to a Galois cover \(Y \to X\) of group \(G\), called generalized Hasse-Witt invariants, arising in the canonical decomposition of the \(p\)-torsion space of the Jacobian variety \(J_Y\) of \(Y\). -- Some particular cases had already been treated by \textit{S. Nakajima} [in: Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 69-88 (1983; Zbl 0529.14016)] and \textit{H. Katsurada} [J. Math. Soc. Japan 31, 101-125 (1979; Zbl 0401.14004)], whose results are generalized in the present paper. algebraic curve; algebraic fundamental group; topological fundamental group; Hasse-Witt invariant; Jacobian variety; number of unramified Galois coverings Pacheco, A., Unramified Galois coverings of algebraic curves, J. number theory, 53, 211-228, (1995) Coverings of curves, fundamental group, Inverse Galois theory, Coverings in algebraic geometry Unramified Galois coverings of algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian variety of an algebraic curve \(C\) is a connected component of the moduli space of line bundles on \(C\). Some subschemes determined by \(h^0(L)\) were classically called the Brill-Noether loci. This article gives a survey of several topics on similar subschemes of the moduli space of vector bundles on \(C\). To be precise, for a line bundle \(\xi\) on \(C\), let \(M_C(2,\xi)\) be the moduli space of stable vector bundles \(E\) of rank two on \(C\) with \(c_1(E)=\xi\). For the canonical bundle \(K\), let \(M_C(2,K,n)\) be the subscheme of \(M_C(2,K)\) consisting of \(E\)'s with \(h^0(E)\geq n+2\). Given a vector bundle \(F\) of rank two, let \(M_C(2,K:\nu F)\) be the subscheme of \(M_C(2,K \otimes\text{det} F)\) consisting of \(E\)'s with \(\dim\Hom (F,E)\geq\nu\).
The author surveys several known results on these vector bundle versions of Brill-Noether loci. For example, if \(g(C)=7\) and if there is no \(g^1_4\) on \(C\), then \(M_C(2,K,3)\) is a smooth Fano threefold of Picard number one and of genus 7; if \(C\) is a smooth plane curve of degree four, \(M_C(2,K:3F)\) is a smooth Fano threefold of Picard number one and of genus 9, if \(F\) is of odd degree and if \(S^2\geq 3\) for any section \(S\) of the \(\mathbb{P}^1\)-bundle \(\mathbb{P}(F)\) over \(C\). Finally, let \(C\) be a curve of genus \(g\geq 11\) with \(g\equiv 3\bmod 4\), and assume that \(C\) can be embedded in some \(K3\) surface and that \(C\) is general among such curves. Then \(T:=M_C(2,K,(g-1)/2)\) is a \(K3\) surface. Moreover, any \(K3\) surface \(X\) containing \(C\) must be isomorphic to a certain subscheme of Brill-Noether type of a certain moduli space of vector bundles on \(T\) (hence \(X\) is uniquely determined by \(C)\). This is based on a duality between \(K3\) surfaces, as certain moduli spaces of vector bundles on each other. line bundles on an algebraic curve; Jacobian variety; Brill-Noether locus; moduli space of stable vector bundles; Fano threefold S. Mukai, Non-abelian Brill-Noether theory and Fano 3-folds [translation of Sūgaku 49 (1997), no. 1, 1-24; MR1478148 (99b:14012)], Sugaku Expositions 14 (2001), no. 2, 125-153. Vector bundles on curves and their moduli, Fano varieties, \(3\)-folds, Algebraic moduli problems, moduli of vector bundles Noncommutativizability of Brill-Noether theory and 3-dimensional Fano variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, a method of using Hurwitz moduli spaces of covers of \({\mathbb P}^1\) for the regular inverse Galois problem is refined and applied to several examples. One of the new ingredients of this paper is to introduce a very general class of rational curves on Hurwitz spaces arising from plane curves, where tools on plane curve singularities, dual curves, Puiseux expansions of algebraic functions etc. can play effective roles to analyze the resulting subfamily of Hurwitz spaces through the associated group theoretical data. This enables the author to find rigid tuples for a wide class of finite groups \(G\) that make \(G\) realized as Galois groups over the rational function field \(\mathbb Q(t)\). Among several illustrative examples to present this method, rigid tuples for the Mathieu groups \(M_{23}\), \(M_{11}\) are detected from the concrete plane curves \(x(y-27/4\cdot(x^3+x^2))=0\). Also, it is shown that the method combined with the theory of middle convolutions (or braid companion functors) yields a rigid tuple for SL\(_5(9)\) from the plane curve \(x(y+x^2+1)(y-(x-1)^2)(y-(x+1)^2)=0\). Hurwitz moduli space; rigid tuples; middle convolution M. Dettweiler, Plane curve complements and curves on Hurwitz spaces, Journal für die Reine und Angewandte Mathematik 573 (2004), 19--43. Coverings of curves, fundamental group, Inverse Galois theory, Families, moduli, classification: algebraic theory, Coverings in algebraic geometry Plane curve complements and curves on Hurwitz 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 Let \(X\) be a smooth projective variety over the field of complex numbers. Grothendieck's General Hodge conjecture asserts that the linear span of the set of cohomology classes supported on algebraic subvarieties of codimension at least \(r\) is the largest Hodge structure contained in \(F^rH^n (X,\mathbb{C})\cap H^n(X, \mathbb{Q})\), where \(F^rH^n (X,\mathbb{C})\) denotes the \(r\)-th part of the Hodge filtration. The usual Hodge conjecture coincides with the special case \(n=2r\) of the general Hodge conjecture. The main result of the paper is a proof of the general Hodge conjecture for any complex abelian variety of CM-type such that the Hodge ring of every power of the abelian variety is generated by divisors or equivalently whose Hodge group coincides with its Lefschetz group. The approach of the proof is to reduce the general conjecture to the usual Hodge conjecture. Several examples of such CM-type abelian varieties are given. general Hodge conjecture; abelian variety of CM-type Transcendental methods, Hodge theory (algebro-geometric aspects), Complex multiplication and abelian varieties, Variation of Hodge structures (algebro-geometric aspects), Algebraic cycles Hodge structures an abelian varieties of CM-type | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a noetherian scheme and \({\mathcal L}\) an invertible \({\mathcal O}_X\)-module; does there exist a Cartier divisor \(D\in\text{Div}(X)\) with \({\mathcal L}\cong{\mathcal O}_X(D)\)? The goal of this short note is to provide an answer and to correct an erroneous counterexample [\textit{R. Hartshorne}, ''Ample subvarieties of algebraic varieties'', Lect. Notes Math. 156 (1970; Zbl 0208.48901); example 1.3].
In the first part of this note we will discuss how the construction of Grothendieck's criterion [see \textit{A. Grothendieck} and \textit{J. Dieudonné}, Publ. Math., Inst. Hautes Étud. Sci. 32, 361 (1967; Zbl 0153.22301)] can be modified in order to obtain the desired counterexample. In the second part we prove a positive result which complements Grothendieck's criterion in the following way:
Let \(T\subset X\) be a finite subset containing \(\text{Ass}({\mathcal O}_X)\); then there is a Cartier divisor \(D\in\text{Div}(X)\) with \({\mathcal L}\cong{\mathcal O}_X(D)\) and support \(\text{Supp}(D)\) disjoint from \(T\) if and only if the restriction of \({\mathcal L}\) to \(T\) is trivial. Schröer, S.: Remarks on the existence of cartier divisors. Arch. math. 75, 35-38 (2000) Divisors, linear systems, invertible sheaves Remarks on the existence of Cartier divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C \rightarrow \mathrm{Spec}(R)\) be a relative proper flat curve over a henselian base. Let \(G\) be a reductive \(C\)-group scheme. Under mild technical assumptions, we show that a \(G\)-torsor over \(C\) which is trivial on the closed fiber of \(C\) is locally trivial for the Zariski topology. Stacks and moduli problems, Étale and other Grothendieck topologies and (co)homologies Local triviality for \(G\)-torsors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The article studies flattening techniques for coherent sheaves in the context of Berkovich's analytic spaces. Such a study has been already done for schemes by \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)] and for complex analytic spaces by \textit{H. Hironaka} [Am. J. Math. 97, 503--547 (1975; Zbl 0307.32011)], but it is new in this context. The methods used by Ducros are inspired by the work on schemes by Gruson and Raynaud and are not related to the methods of Hironaka. The results of the paper only apply to compact Berkovich's analytic spaces over a non-Archimedean field \(k\), and already in this case are quite more complicated than the algebraic ones because of the complexity of the analytic topology.
The paper starts by recalling the basic language of Berkovich's analytic spaces focusing on basic terminology and results about coherent sheaves over such spaces. This makes the paper accessible to non-experts although some previous knowledge of Berkovich's geometry or at least non-Archimedean geometry seems necessary for a full understanding of the contents. The technical part of the paper begins in \S 2. In particular, this section is devoted to the study of the support of coherent sheaves and to the development of counter-parts in Berkovich's geometry of notions and properties discussed in [\textit{M. Raynaud} and \textit{L. Gruson}, Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)]. The subsequent two sections are devoted to three key notions for the proof of the main results: the ideal of coefficients and the décupage associated to morphisms of Berkovich's analytic spaces, and the dévissage associated to a morphism of Berkovich's analytic spaces and a coherent sheaf on its domain. In \S 5, blow-ups of Berkovich's analytic spaces are studied. In particular, their existence is proved and basic properties a provided. In the next section, Theorem 6.6, the main result of the paper, is proved. It is hard to summarize this theorem for its highly technical content, but, roughly, it contains analogs of the main results of that in [loc. cit.] have been obtained for schemes, stated utilizing the notions developed previously by the author. The concluding section of the paper concerns applications. The most strikings of which are related to the study of the images of morphisms between Berkovich's analytic space. Berkovich spaces; flattening by blowing-up; coherent sheaf; blow-up Rigid analytic geometry, Arithmetic problems in algebraic geometry; Diophantine geometry Unscrew, cut, burst and flatten Berkovich spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f:Y \to X\) be a surjective morphism from a smooth projective variety to a smooth projective variety of dimension \(n\), such that \(f\) is smooth over the complement of a normal crossing divisor on \(X\). The following two conjectures represent relative versions of Fujita's freeness conjecture and of a stronger local version due to the author [Math. Ann. 308, 491--505 (1997; Zbl 0909.14001)], respectively.
(1) Let \(H\) be an ample divisor on \(X\); then the locally free sheaf \(R^qf_*\omega_Y \otimes \mathcal O_X(mH)\) is generated by global sections if \(m \geq n+1\) or \(m=n\) and \(H^n\geq 2\).
(2) Let \(L\) be a nef and big invertible sheaf on \(X\) such that \(L^n > n^n\) and \(L^dZ \geq n^d\) for any irreducible subvariety \(Z\) of \(X\) of dimension \(d\), passing through a given point \(x \in X\); then the natural homomorphism \(H^0(X,R^qf_*\omega_Y \otimes L) \to R^qf_*\omega_Y \otimes L \otimes \kappa(x)\) is surjective for any \(q \geq 0\).
Generalizing a strategy toward Fujita's conjecture and its stronger local version developed previously, the author establishes a criterion for the natural homomorphism above to be surjective. In particular this proves conjecture (2) for \(n \leq 3\) and conjecture (1) for \(n=4\). The main tool is an extension of the Kollár vanishing theorem (the relative version of the Kodaira vanishing theorem) to the setting of \(\mathbb Q\)-divisors. local freeness; higher direct image of dualizing sheaves; relative vanishing theorem; \(\mathbb Q\)-divisors; parabolic structure Divisors, linear systems, invertible sheaves, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vanishing theorems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry On a relative version of Fujita's freeness 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 Let \(G\) be an algebraic group scheme of finite type over a locally noetherian base scheme \(L\) and assume that \(G\) acts morphically on the algebraic space \(X\) of finite type over \(L\). Following \textit{D. Mumford}'s ``Geometric invariant theory'' (1965; Zbl 0147.39304) one defines the notions of geometric and (uniform) categorical quotients for the action of \(G\) on \(X\). As a consequence of their main result the authors obtain: If \(G\) is flat over \(L\) and acts properly on \(X\) with finite stabilizer, then there is a geometric and uniform categorical quotient \(q : X \to Y\) for the action of \(G\) on \(X\), where \(Y\) is a separated algebraic space over \(Y\). The presented approach requires the more general setting of groupoids; examples of such groupoids are obtained by restricting the equivalence relation defined by the action of \(G\) on \(X\) to a not necessarily \(G\)-stable subspace \(W\) of \(X\). In these terms the main result reads as follows:
Every flat groupoid \(\jmath : R \to X \times X\) with finite stabilizer has a geometric and uniform categorical quotient. If \(\jmath\) is finite, then the quotient space is separated.
As the authors outline, the concept of their proof is to choose appropriate subspaces \(W\) of \(X\) such that the restriction \(R'\) of \(R\) to \(W\) is quasifinite and then reduce the problem to finding quotients for \(R' \to W \times W\). The main result of the article under review generalizes various previous results on geometric quotients, see e.g. \textit{H. Popp} [Invent. Math. 22, 1-40 (1973; Zbl 0281.14011)] and, more recently, \textit{E. Viehweg} [``Quasi-projective moduli of polarized manifolds'' (1995; Zbl 0844.14004)] and \textit{J. Kollár} [Ann. Math., II. Ser. 145, No. 1, 33-79 (1997; see the preceding review)]. algebraic group actions; geometric quotients; algebraic spaces; categorical quotient Seán Keel & Shigefumi Mori, ``Quotients by groupoids'', Ann. Math.145 (1997) no. 1, p. 193-213 Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Generalizations (algebraic spaces, stacks), Geometric invariant theory Quotients by groupoids | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 discuss an algorithm to compute bases for the space \(\mathcal{L} (G)\), provided \(G\) is a rational divisor over a non-singular absolutely irreducible algebraic curve. The algorithm is founded on the Brill-Noether algorithm by using the theory of Hamburger-Noether expansions, and it is given in terms of symbolic computation. As a byproduct, we introduce a method to compute the Weierstrass semigroup at \(P\) together with functions for each value in this semigroup, provided \(P\) is a rational point of this curve. This methods are nice applications of the classical adjunction theory.
On the other hand, we discuss an alternative method for the computation of Weierstrass semigroups and the corresponding functions on the basis of the Abhankar-Moh theorem, for the case of a plane curve with only one branch at infinity. This method requires the precomputation of a certain integral basis. Both alternative methods be applied to the effective construction (Riemann-Roch problem) and decoding (Weierstrass semigroups) of Algebraic Geometry codes. The second method can be also regarded as a kind of adjunction procedure, and we compare both alternatives from a general point of view. coding theory; AG codes; Brill-Noether algorithm; Weierstrass semigroup Campillo, A.; Farrán, J.: Adjoints and codes, Rend. sem. Mat. univ. Politec. Torino 62, 209-223 (2004) Computational aspects of algebraic curves, Applications to coding theory and cryptography of arithmetic geometry, Symbolic computation and algebraic computation, Geometric methods (including applications of algebraic geometry) applied to coding theory Adjoints and codes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the hyperelliptic curve defined by the equation \(y^ 2=1+\mu x^ N\) over the ring W(k) of Witt vectors of a finite field k (where \(\mu\) is a unit in W(k) and \(N>2\) is prime to \(p=char(k)\neq 2)\). To investigate the formal completion of the Jacobian \(J_ p\) of the reduction \(C_ p\) of C mod pW(k), the authors introduce the ``structural cycles'' \(\gamma_ t,\quad t\in \{1,...,[(N-1)/2]\},\) defined by elementary arithmetic properties of N and p. For each structural cycle \(\gamma\) they define a W(k)-module \(TR_{\gamma}\) (with actions of Frobenius F and Verschiebung V) by means of the p-adic gamma function. The main result is that \(TR:=\oplus_{\gamma}TR_{\gamma} \) is isomorphic as an W(k)[F,V]-module to the connected component of the dual of the Dieudonné-module of \(J_ p\). The isogeny type of \(TR_{\gamma}\) is also determined, as well as the characteristic polynomial of the Frobenius action on \(TR_{\gamma}\) and on the p- divisible group associated to \(J_ p\). This last formula uses the work of \textit{B. H. Gross} and \textit{N. Koblitz} [Ann. Math., II. Ser. 109, 569- 581 (1979; Zbl 0406.12010)]. The paper generalizes results that were obtained by \textit{T. Honda} [Osaka J. Math. 3, 131-133 (1976; Zbl 0345.12101)]. hyperelliptic curve; Witt vectors; Jacobian; p-adic gamma function; Dieudonné-module; Frobenius action; p-divisible group Ditters, On the connected part of the covariant Tate p-divisible group and the {\(\zeta\)}-function of the family of hyperelliptic curves y2 = 1 + {\(\mu\)}xN modulo various primes, Math. Z. 200 pp 245-- (1989) Formal groups, \(p\)-divisible groups, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic ground fields for curves On the connected part of the covariant Tate p-divisible group and the \(\zeta\)-function of the family of hyperelliptic curves \(y^ 2=1+\mu x^ N\) modulo various primes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0717.00010.]
In the present paper, the author continues his investigations on the numerical properties of the relative dualizing sheaf of an arithmetic surface, i.e., of the (unique) regular and minimal model of a smooth, geometrically connected algebraic curve of genus at least one over an algebraic number field. He extends his previous results [cf. Séminaire sur les pinceaux de courbes de genre au moins deux, Astérique 86, 44-78 (1981; Zbl 0517.14006)] in two directions.
Namely, in the first part of the paper, he deals with the case of elliptic curves over a number field and describes, from the viewpoint of Arakelov theory, their isogenies, Green's functions, Arakelov degree of the relative dualizing sheaf and their torsion points. Moreover, this part contains an explicit description of the values of the Arakelov-Green function at the 2-torsion points of Frey curves [cf. \textit{G. Frey}, J. Reine Angew. Math. 331, 185-191 (1982; Zbl 0474.14011)] and, at the end, a discussion of the so-called discriminant conjecture for elliptic curves over number fields.
The second part of the article deals with curves of genus at least two and the self-intersection of their relative dualizing sheaves. The author verifies a conjecture of F. Bogomolov in a special case, and that by relating it to the non-nullity of the self-intersection of the relative dualizing sheaf, and concludes the paper by a discussion of some related open problems and conjectures. These problems concern upper bounds for the self-intersection of the relative dualizing sheaf and the Néron- Tate height of two points in the Jacobian of an arithmetic surface of genus \(g\geq 2\). self-intersection of dualizing sheaves; elliptic curves; Arakelov theory; Arakelov-Green function; Frey curves; genus; Néron-Tate height; Jacobian Szpiro, L. 1990.Sur les propriétés numériques du dualisant relatif d'une surface arithmétique, The Grothendieck Festschrift Vol. III, 229--246. Boston: Birkhäuser. Arithmetic varieties and schemes; Arakelov theory; heights, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Elliptic curves, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves Sur les propriétés numériques du dualisant relatif d'une surface arithmétique. (On the numerical properties of the relative dualizing sheaf of an arithmetic surface) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a compact Riemann surface of genus \(g\geq3\), and let \(J(X)\) be the Jacobian variety of \(X\). In [\textit{G. Ceresa}, Ann. Math. (2) 117, 285--291 (1983; Zbl 0538.14024)], it is shown that for a generic \(X\), the algebraic cycle \(X-X^{-}\) is not algebraically equivalent to zero in \(J(X)\). There are, however, few explicit nontrivial examples of such a curve. In this paper the author considers the case of the Fermat sextic \(F(6)\) defined by \(x^{6}+y^{6}=1\), and shows that the algebraic cycle \(F(6)-F(6)^{-}\) is not algebraically equivalent to zero in \(J(F(6))\). A main ingredient of his proof is the harmonic volume which is defined as follows. Let \(R\) be a discrete subring of \(\mathbb{C}\) and suppose that all the entries of the period matrix of \(X\) can be reduced to elements of \(R\). Let \(H_{R}^{1,0}\) denote the space of holomorphic one-forms on \(X\) with \(R\)-periods, and let \(\{K_{1},\cdots,K_{2g}\}\) be a basis of \(H_{1}(X,\mathbb{Z})\). The harmonic volume is a homomorphism \(I_{R}:(H_{R}^{1,0})^{\otimes_{R}3}\rightarrow\mathbb{C}/R\) defined by \(I_{R}(\omega_{1}\otimes\omega_{2}\otimes\omega_{3})=\sum_{r=1}^{2g}a_{r}\int_{C_{r}}\omega_{1}\omega_{2}\: \mod R\), where \(C_{r}\) is a loop in \(X\) at a fixed base point whose homology class is \(K_{r}\), the Poincaré dual of \(\omega_{3}\) is equal to \(\sum_{r=1}^{2g}a_{r}K_{r}\), and the integral \( \)\(\int_{C_{R}}\omega_{1}\omega_{2}\) is Chen's iterated integral [\textit{K.-T. Chen}, Trans. Am. Math. Soc. 156, 359--379 (1971; Zbl 0217.47705)]. By [\textit{B. Harris}, Nankai Tracts in Mathematics 7. River Edge, NJ: World Scientific (2004; Zbl 1063.14010)], one knows that if the algebraic cycle \(X-X^{-}\) is algebraically equivalent to zero in \(J(X)\), then \(2I_{R}(\omega)\equiv0\:\mod R\) for any \(\omega\in(H_{R}^{1,0})^{\otimes_{R}3}\). Employing this result he conludes the proof by finding some element \(\omega\in(H_{R}^{1,0})^{\otimes_{R}3}\) such that \(2I_{R}(\omega)\neq0\: \mod R\) for the Fermat sextic. algebraic cycles; Jacobian varieties; algebraic equivalence Tadokoro Y.: A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic. Tsukuba J. Math. 33(1), 29--38 (2009) Algebraic cycles, Jacobians, Prym varieties A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 upon my lectures at the Tata Institute from November 1975 to March 1976 and further oral communication between me and the note taker, \textit{Balwant Singh}. The notes are divided into two parts. In \S 8 of part one we prove the fundamental theorem on the structure of the coordinate ring of a meromorphic curve and its value group. We then give some applications of the fundamental theorem, the principal one among them being the epimorphism theorem. The proof of the main lemmas (\S 7) presented here is a simplified version of the original proof of the author and \textit{T.-T. Moh} [J. Reine Angew. Math. 260, 47--83 and 261, 29--54 (1973; Zbl 0272.12102)].
In part two we record some progress on the Jacobian problem, which is as yet unsolved. The results presented here were obtained by me during 1970/71. Partial notes on these were prepared by \textit{M. van der Put} and \textit{W. Heinzer} at Purdue University in 1971.
[For an updated version of the book under review including these notes see the author, Proc. Indian Acad. Sci., Math. Sci. 104, No. 3, 515--542 (1994; Zbl 0812.13013)]. coordinate ring of a meromorphic curve; epimorphism theorem; Jacobian problem S.S. Abhyankar , '' Expansion technics in Algebraic Geometry ,'' Tata Institute of Fundamental Research, Bombay, 1977. Relevant commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra Lectures on expansion techniques in algebraic geometry. With notes by Balwant Singh | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 nodal curve, i.e., \(X\) is reduced, possibly reducible, and has at most nodes as singularities. Let \(g\) be the arithmetic genus of \(X\). The compactified Picard variety \(\overline{P^{g-1}_X}\) of degree \(g-1\) is the finite union of irreducible \(g\)-dimensional components each of which contains as an open subset a copy of the generalized Jacobian of \(X\). It is known that the compactified Picard variety of any nodal curve has a polarization, the theta divisor \(\Theta(X)\). The author studies the theta divisor of the compactified Jacobian of a nodal curve \(X\). The first main result of the paper describes the irreducible components of \(\Theta(X)\) and establishes that every irreducible component of the compactified Jacobian contains a unique irreducible component of the theta divisor, unless \(X\) has some separating node. The proof uses the Abel map \(X^{g-1} \rightarrow \mathrm{Pic}^{g-1}\). It turns out that the theta divisor coincides with the closure of the image of the Abel map for every stable multidegree.
Another goal of the paper is the geometric interpretation of \(\Theta(X)\) and the precise description of the objets which are parametrized by \(\Theta(X)\). In particular, in this direction it is proved that a stratification of \(\overline{P^{g-1}_X}\) induces the canonical stratification on \(\Theta(X)\). This stratification of \(\Theta(X)\) allows to describe the partial normalizations of \(X\) in terms of effective line bundle.
The author applies the results and the techniques of the paper to generalize to singular curves the characterization of smooth hyperelliptic curves via the singular locus of their theta divisor. Nodal curve; line bundle; compactified Picard scheme; theta divisor; Abel map; hyperelliptic stable curve Caporaso L.: Geometry of the theta divisor of a compactified Jacobian. J. Eur. Math. Soc. 11, 1385--1427 (2009) Jacobians, Prym varieties, Special divisors on curves (gonality, Brill-Noether theory), Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Geometry of the theta divisor of a 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 Let \(M_{0,n}\) be the moduli space of \(n\)-pointed smooth curves of genus zero and \(\bar{M}_{0,n}\) its Deligne-Mumford compactification. \textit{B. Hassett} [Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] introduced the notion of moduli space of \textit{weighted} stable pointed curves: this is a family of compactifications of \(M_{g,n}\), parametrized by choices of weights \(a_1,\ldots,a_n \in [0,1]\), such that a subset of the markings may coincide if and only if the total weight of these markings is at most \(1\). When all weights equal \(1\) one recovers the Deligne-Mumford compactification.
The paper under review considers the ``weighted'' compactification of \(M_{0,m+n}\) in the case that the first \(m\) markings have weight \(1\) and the last \(n\) markings have weight \(\epsilon \ll 1\), so that any subset of the last \(n\) markings may coincide. This space is denoted he \(\bar M_{0,m| n}\). To get a Deligne-Mumford stack one assumes that \(m \geq 2\) and \(m+n \geq 3\). The main result is the calculation of the \(S_m \times S_n\)-equivariant Poincaré polynomial of \(\bar M_{0,m| n}\).
When \(n=0\) this recovers a result of \textit{E. Getzler} [Prog. Math. 129, 199--230 (1995; Zbl 0851.18005)] and the method of proof is a generalization of Getzler's. Let me briefly summarize the idea of Getzler. First he calculates the \(S_n\)-equivariant cohomology of \(M_{0,n}\). Then he uses that \(\bar M_{0,n}\) has a stratification in which all strata are products of smaller moduli spaces \(M_{0,n_i}\), so that one basically needs to sum the contributions of all strata. He does this by encoding the data in terms of symmetric functions, using the bijection between symmetric functions and virtual representations of the symmetric group. Since the strata of \(\bar M_{0,n}\) correspond to trees, he needs to compute a sum over trees, which can then be encoded in terms of an operation on symmetric functions which is an analogue of the classical Legendre transform.
In the situation of the paper under review, the author instead begins by computing the \(S_m \times S_n\)-equivariant Poincaré polynomial of \(M_{0,m | n}\), which denotes the Zariski open subset of \(\bar M_{0,m | n}\) parametrizing smooth curves. The equivariant Poincaré polynomials of \(M_{0,m | n}\) are now indexed instead by a bisymmetric function. Then \(\bar M_{0,m| n}\) has a stratification whose strata are all products of smaller spaces \(M_{0,m_i | n_i}\), and the strata are now indexed by \textit{bicolored} trees -- more specifically, trees whose legs and internal edges and legs can have two colors (corresponding to markings of weight \(1\) resp. \(\epsilon\)), where all internal edges correspond to weight \(1\). The operation of summing over such bicolored trees is interpreted by an operation that the author calls the ``partial Legendre transform'' of bisymmetric functions. moduli of curves; poincare polynomial; operads; tensor species Families, moduli of curves (algebraic), Representations of finite symmetric groups, Fine and coarse moduli spaces Equivariant cohomology of certain moduli of weighted pointed rational 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 \(A\) be either a nonsingular affine variety or a projective space \(\mathbb P^n\) over the field of complex numbers. Two closed subvarieties of \(A\) are said to be \textit{geometrically linked} if they have no common component and their union is a complete intersection in \(A\). If one of the varieties, say \(X\), is fixed and if the complete intersection is chosen in as general a way as possible containing \(X\), then the complement \(Y\) is called a \textit{generic link} of \(X\). The object of this paper is to study how singularities behave under generic linkage. The author gives a description of the Grauert-Riemenschneider canonical sheaf of \(Y\) in terms of the multiplier ideal sheaves associated to \(X\), and uses it to study the singularities of \(Y\). His work generalizes previous results of Chardin and Ulrich, among others. He gives several applications of his main theorem, for instance to the study of rational singularities and to the study of long canonical threshold under generic linkage. Finally, he applies it to generalize known results by \textit{T. De Fernex} and \textit{L. Ein} [Am. J. Math. 132, No. 5, 1205--1221 (2010; Zbl 1205.14020)], and by \textit{M. Chardin} and \textit{B. Ulrich} [Am. J. Math. 124, No. 6, 1103--1124 (2002; Zbl 1029.14016)], on the Castelnuovo-Mumford regularity bound for a projective variety. generic link; singularity; multiplier ideal; Castenlnuovo-Mumford regularity; log canonical threshold Niu, Wenbo, Singularities of generic linkage of algebraic varieties, Amer. J. Math., 136, 6, 1665-1691, (2014) Linkage, Singularities of surfaces or higher-dimensional varieties Singularities of generic linkage of algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a germ \(f = 0\) of an isolated plane curve singularity defined by \({f \in \mathbb{C}\{X, Y\}}\) we consider the Jacobian Newton polygon \({\nu_\mathbf{J}(f)}\) introduced by Bernard Teissier. For two such germs \(f = 0\), \(g = 0\) we study the case \({\nu_\mathbf{J}(f) = \nu_\mathbf{J}(g)}\). When \(f\) and \(g\) are irreducible then the germs \(f = 0\), \(g = 0\) are equisingular (Merle's result). The same is true for \(f\), \(g\) unitangent and nondegenerate in the Kouchnirenko sense (author's result). We generalize these theorems. We formulate our result in terms of the Eggers tree. curve singularity; Newton polygon Milnor fibration; relations with knot theory, Singularities of curves, local rings Eggers tree and Jacobian Newton polygon | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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} [Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007)] has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.
There are no proofs in this paper; its intention is to be purely expository and motivational. For details and proofs, we refer to the bibliography at the end of the paper. Deligne conjecture; generalized Jacobians; generalized Albanese varieties; Hodge conjecture Niranjan Ramachandran, From Jacobians to one-motives: exposition of a conjecture of Deligne, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 215 -- 234. Transcendental methods, Hodge theory (algebro-geometric aspects), Picard schemes, higher Jacobians From Jacobians to one-motives: exposition of a conjecture of Deligne. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(S_n\) denote the variety of infinitely near points of order \(n\) to points of the projective plane. The author studies the natural action of \(G := \text{PGL}(3, {\mathbb{C}})\) on \(S_n\), using the differential invariants introduced by G.H. Halphen. It is shown that the field of invariant rational functions of \(S_n\) is purely transcendental over \({\mathbb{C}}\). This is applied to obtain rationality for moduli spaces of pointed plane curves of a given degree. For the cases \(n=8,9\) an explicit construction of open \(G\)-stable subsets of \(S_n\) that admit a quotient by the action of \(G\) is presented. infinitely near points; algebraic group actions; field of invariant rational functions; rationality of moduli spaces of pointed plane curves Mazouni, A, Quotient de la variété des points infiniment voisins d'ordre 9 sous l'action de \(PGL_{3}\), Bull. SMF, 124, 425-455, (1996) Families, moduli of curves (algebraic), Group actions on varieties or schemes (quotients), Rational points, Infinitesimal methods in algebraic geometry, Rational and unirational varieties, Geometric invariant theory Quotient of the variety of infinitely near points of order 9 under the action of \(\text{PGL}_ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper obtains some arithmetic results for abelian varieties over function fields of one variable over finitely generated fields.
Let \(k\) be a field of characteristic \(0\), and let \(C\to \mathrm{Spec}\, k\) be a smooth, separated and geometrically connected algebraic curve over \(k\), and let \(K=k(C)\) be the function field of \(C\). Let \(\mathcal{A}\to C\) be an abelian scheme over \(C\), and let \(A=A_K=\mathcal{A}\times_C \mathrm{Spec}\, K\). Then \(A\) is an abelian variety over \(K\). Denote by \(A(K)\) the group of \(K\)-rational points on \(A\). The profinite Selmer group Sel\((A)\) and the Shafarevich-Tate group Ш\((A)\) are then defined (similar to the number field case).
Assuming that \(k\) is finitely generated over \(\mathbb{Q}\), the paper introduces the new invariants for \(A\), namely, the discrete Selmer groups, and discrete Shafarevich-Tate groups. These groups are shown to be finitely generated \(\mathbb{Z}\)-modules.
The main result is formulated in the following
Theorem: Let \(k\) be a finitely generated field over \(\mathbb{Q}\). Assume that an abelian variety \(A\) is isotrivial. Then the discrete Shafarevich-Tate group vanishes, and the discrete Selmer group coincides with the Mordell-Weil group of \(A\).
The main ingredient for proving this theorem is a specialization theorem for first Galois cohomology groups. abelian variety; function field; finitely generated field; Selmer group; Shafarevich-Tate group; discrete Selmer group; discrete Shafarevich-Tate group; Mordell-Weil group Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties On the arithmetic of abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0741.00019.]
This article provides a survey on some old and new results relating complex analytic and arithmetic properties of algebraic varieties. The starting point is the observation due to the author himself [cf. Invent. Math. 98, No. 1, 115-138 (1989; Zbl 0662.14019)] that Mordell's conjecture (on the finiteness of \(k\)-rational points in algebraic curves of genus \(g>1\) over a number field \(k\)) can be confirmed by a method which is different from Faltings' original approach [cf. \textit{G. Faltings}, Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)]. The author's method is based on the fact that complex curves are also characterized by various analytic hyperbolicity properties, and that their Nevanlinna theory (i.e., value distribution theory for holomorphic mappings) can be translated to the arithmetical number field case. The present article gives a sketch of this link and discusses, besides a proof of the Mordell conjecture, some very recent results in this direction. Among those are Faltings' generalization of the author's method to abelian varieties over number fields [cf. \textit{G. Faltings}, Ann. Math., II. Ser. 133, No.3, 549-576 (1991; Zbl 0734.14007)], E. Bombieri's simplification of the author's proof [cf. \textit{E. Bombieri}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No.4, 615-640 (1990; Zbl 0722.14010)] and concludingly, a list of conjectures and open problems concerning the link between arithmetic and hyperbolic geometry. curves over algebraic number fields; Nevanlinna theory; Mordell conjecture; hyperbolic geometry Rational points, Arithmetic ground fields for curves, Hyperbolic and Kobayashi hyperbolic manifolds, Value distribution of meromorphic functions of one complex variable, Nevanlinna theory, Value distribution theory in higher dimensions, Arithmetic theory of algebraic function fields Arithmetic and hyperbolic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(D\) be a reduced divisor on a smooth projective curve \(X\). Let \(G\) denote a complex reductive algebraic group and \(N>0\) a fixed integer. A ramified \(G\)-bundle over \(X\) of type \(N\) with ramification over \(D\) is a smooth variety \(E\) with a proper action of \(G\), together with a morphism \(h: E \to X\) such that \((h,E,X)\) is a geometric quotient, the restriction of \(h\) is a principal \(G\)-bundle over \(X-D\) and the (finitely many) nontrivial isotropy subgroups are cyclic groups whose orders divide \(N\). The authors establish a bijective correspondence between ramified \(G\)-bundles over \(X\) and principal \(G\)-bundles with Galois group action over a suitable Galois cover of \(X\). The valuative criterion for properness for semistable ramified \(G\)-bundles is proved.
Let PVect \((X,D)\) (respectively, PVect \((X,D,N)\)) denote the category of parabolic vector bundles over \(X\) with parabolic structure over \(D\) (respectively, with weights of the form \(a_i/N, \, a_i\) integers). A parabolic \(G\)-bundle is defined as a \(G\)-functor \(f\) from the category Rep \((G)\) of finite dimensional representations of \(G\) to the category PVect \((X,D)\) with certain properties [Tohoku Math. II. Ser. 53, 337-367 (2001; Zbl 1070.14506)]. The main result of the present paper is the following.
Theorem. There is a bijective correspondence between ramified \(G\)-bundles \(E\) over \(X\) of type \(N\) ramified over \(D\) and parabolic \(G\)-functors \(f_E: \) Rep \((G) \to \) PVect \((X,D,N)\). This correspondence preserves (semi)stability. Other different approaches to parabolic \(G\)-bundles are studied by \textit{U. N. Bhosle, A. Ramanathan} [Math. Z., 202, No.~2, 161--180 (1989; Zbl 0686.14012)] and \textit{C. Teleman, C. Woodward} [Ann. Inst. Fourier, Grenoble, 51, 3, 713--748 (2003; Zbl 1041.14025)]. principal bundles; parabolic bundles; curves V. Balaji, I. Biswas, and D. S. Nagaraj, Ramified \?-bundles as parabolic bundles, J. Ramanujan Math. Soc. 18 (2003), no. 2, 123 -- 138. Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic groups Ramified \(G\)-bundles as 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 This short paper gives an almost self-contained proof of the following torsion-free and vanishing theorem of \textit{F. Ambro} [Proc. Steklov Inst. Math. 240, 214--233 (2003; Zbl 1081.14021)]: Let \(Y\) be a smooth projective variety, and \(B\) a boundary \(\mathbb{Q}\)-divisor with simple normal crossing support. Let \(f : Y \to X\) be a projective morphism and \(L\) a Cartier divisor on \(Y\) for which \(L - (K_Y + B)\) is \(\mathbb{Q}\)-equivalent to an \(f\)-semi-ample divisor \(H\). Then
(1) every non-zero local section of \(R^q f_{\ast} \mathcal{O}_Y(L)\) contains in its support the image of a log canonical center of \((Y, B)\) (or \(f(Y)\) itself);
(2) if \(H\) is \(\mathbb{Q}\)-equivalent to the pullback of an ample \(\mathbb{Q}\)-Cartier divisor from \(X\), then \(H^p(X, R^q f_{\ast} \mathcal{O}_Y(L)) = 0\) for \(p > 0\) and \(q \geq 0\).
This result is part of the general log minimal model program for log canonical pairs by F. Ambro and the author.
The main step in the proof is a generalization of Kollár's injectivity theorem, for which the author provides a self-contained argument using the \(E_1\)-degeneration of the Hodge-de Rham spectral sequence for the log complex from \textit{P. Deligne} [Matematika, Moskva 17, No.5, 3--56 (1973; Zbl 0282.14001)]. As an application of his result, the author proves a version of the Kodaira vanishing theorem for log canonical pairs, and an extension theorem from log canonical centers. Vanishing theorem; Torsion-freeness; Injectivity theorem; Hodge theory Fujino, O, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Jpn Acad., 85, 95-100, (2009) Minimal model program (Mori theory, extremal rays), Vanishing theorems in algebraic geometry, Vanishing theorems On injectivity, vanishing and torsion-free theorems for algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main objective of this expository paper is to discuss recent results and problems on classical and Drinfel'd modular curves with regards to the \(d\)-gonality problem and other related issues.
Let \(d\) be a positive integer. A smooth projective curve \(C\), defined over a field \(k\) of characteristic different from 2 and 3, is said to be \(d\)-gonal if there exists a morphism \(f: C \rightarrow {\mathbb P}_{\bar k}^1\) of degree \(d\). If \(K = k(C)\) is the function field of \(C\), the function field analogue of the strong Uniform Boundedness Conjecture (sUBC) states that the set of all isomorphism classes of finite abelian groups occurring as the full non-\(p\)-parts of torsions in the Mordell-Weil groups of non-constant elliptic curves over \(K\), with an extension \({\overline{k}} ({\mathbb P}^1) \hookrightarrow {\overline{k}} (C)\) of degree \(d\), is finite. When a non-constant elliptic curve \(E\), defined over \(K\), has a torsion element of order \(N\) in \(E(K)\), the problem of sUBC is reduced to the problem of \(d\)-gonality of \(X_0(N)\). Three approaches to the \(d\)-gonality problem for modular curves were discussed: an algebraic approach generalizing an earlier method of Ogg, a class number approach, and an analytical approach in characteristic zero.
Some connections between the \(d\)-gonality problem and other questions, such as the modular parametrization of an elliptic curve, the Coleman-Kaskel-Ribet conjecture for modular curves, as well as Seshadri's constant, are also briefly discussed.
Analogues of the \(d\)-gonality problem in the case of Drinfel'd modular curves and its connections with other questions exist and are also introduced and discussed. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Elliptic curves over global fields, Drinfel'd modules; higher-dimensional motives, etc., Families, moduli of curves (algebraic) Modular curves and some related issues | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a non singular complete curve, of genus \(g\geq 1\), defined over an algebraically closed field k. Let F be a locally free sheaf of rank r. Consider the family of coherent sheaves G which are quotients of F. Such a family can be provided with a structure of k-scheme [see \textit{A. Grothendieck}, Sémin. Bourbaki 14 (1961/62), Exposé 232 (1962; Zbl 0238.14014)] which we shall denote by \(Quot_ F\). Moreover given a very ample divisor over X, one can define for every coherent sheaf G its Hilbert polynomial \(P(n)=\chi (X,G(n))\). The family of quotients of F consisting of coherent sheaves G with the same Hilbert polynomial P, has a structure of a projective scheme over k, which will be denoted by \(Quot^ P_ F\); one has \(Quot_ F=\coprod_{P}Quot^ P_ F.\) The schemes \(Quot^ P_ F\) are related to the varieties of moduli of locally free sheaves on X.
In the particular case where \(F={\mathcal O}_ X\) and \(P=d\) is the constant polynomial then \(Quot^ d_{{\mathcal O}_ X}=S^ d(X)\) and we have the well known natural morphism \(\Phi: Quot^ d_{{\mathcal O}_ X}=S^ d(X)\to Pic^ d_ X.\) This construction can be extended, in a natural way, to the case of an arbitrary locally free sheaf \({\mathcal F}\). If \({\mathcal G}\) is a quotient of \({\mathcal F}\) with its Hilbert polynomial P then we get a rational morphism \(\Phi: Quot^ P_{{\mathcal F}}\to M,\) where M is a moduli variety of locally free sheaves. The morphism \(\Phi\) is, in the general case, more complicated than in the case \(F={\mathcal O}_ X\) because \(Quot^ P_{{\mathcal F}}\) is, in general, not smooth, neither irreducible nor reduced. The main object of this paper is to show that if F is a ''generic'' locally free sheaf (in characteristic \(0\)) then the scheme \(Quot^ P_{{\mathcal F}}\) is smooth and is of ''good'' dimension for all P. In fact, in some cases one can show that the scheme \(Quot^ P_{{\mathcal F}}\) is irreducible for any P. quot schemes; Hilbert polynomial; moduli of locally free sheaves Ghione, F.: Quotient schemes over a smooth curve. Napoli Publ. Ist. Mat.33, (1981/82) Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Quot schemes over a smooth 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 We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over \(\overline{\mathbb{Q}}\). This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of \(D\)-group schemes attached to abelian schemes over algebraic curves over \(\overline {\mathbb{Q}}\). We also derive the Grothendieck period conjecture for cycles of codimension 1 in abelian varieties over \(\overline{\mathbb {Q}}\) from a classical transcendence theorem à la Schneider-Lang. algebraization; transcendence; \(D\)-group schemes; Abelian schemes Varieties over finite and local fields, Transcendence (general theory), Differential algebra, Formal neighborhoods in algebraic geometry, de Rham cohomology and algebraic geometry, Arithmetic ground fields for abelian varieties Algebraization, transcendence, and \(D\)-group schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the following extension to families of open curves of Manin's finiteness theorem for rational points of algebraic curves over function fields [\textit{Yu. I. Manin}, Transl., Ser. 2, Am. Math. Soc. 50, 189-234 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)], well-known as the function-field analog of the Mordell conjecture:
Theorem 1. Let \(f: X\to S\) be a family of curves of general type, and let \(\bar f: \bar X\to \bar S\) be its compactification. If the set \({\mathcal H}(S,f)\) of holomorphic sections of f is infinite, then there is a (shared) bimeromorphic trivialization of the families f and \(\bar f,\) mapping all but perhaps a finite number of sections in \({\mathcal H}(S,f)\) to constant sections and the horizontal curve \(A_ h(\bar f)\) to the union of a finite set of constant sections.
Here S denotes a smooth complex curve, Zariski-open in its smooth compactification \(\bar S,\) and \(A_ h(\bar f)\) is the complement of X in the (reduced irreducible normal complex compact) surface \(\bar X.\) Especially, the cases of families of punctured elliptic curves present a function field analog of the Siegel-Mahler theorem for elliptic curves over number fields [see \textit{S. Lang}, Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)]. For the proof the author needs the techniques of resolution of singularities, semistable reduction and the meromorphic domination of family pairs \((f,\bar f)\) by pairs \((g,\bar g)\) with the property of ``relative hyperbolic embeddedness''. The latter is due to the author in his previous paper [``Hyperbolicity criteria and families of curves'', Teor. Funkts., Funkts. Anal. Prilozh. 52, 40-54 (1989)]. An important role plays the existence of a complex structure (Douady) on the space of sections. Then compactness of the space of sections on the open part (proposition 2.1) and local finiteness (lemma 4.4) are verified and used to find a trivialization of the given family of curves. In the last section 5 there is given an alternative proof by reduction to Manin's theorem following an idea of Parshin. families of open curves; function-field analog of the Mordell conjecture Arithmetic ground fields for curves, Families, moduli of curves (analytic), Rational points, Arithmetic varieties and schemes; Arakelov theory; heights A function-field analog of the Mordell conjecture: A non-compact version | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theory of unipotent fundamental groups of varieties was initiated in [\textit{P. Deligne}, in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 79--297 (1989; Zbl 0742.14022)]. Deligne considered different realizations (étale, de Rham, etc.) of the unipotent fundamental group and established comparison isomorphisms. This suggested the existence of a theory of motivic unipotent fundamental groups of varieties.
Let \(X\) be a smooth curve over an algebraically closed field \(k\), and let \(p\) be a prime different from the characteristic of \(k\). The author constructs explicitly a universal pro-unipotent smooth \(p\)-adic étale sheaf \(\mathbb{L}\). Consider a \(k\)-point \(x \in X(k)\). The stalk of \(\mathbb{L}\) at \(x\), denoted \(A = \mathbb{L}[x]\), is isomorphic to the endomorphism algebra \(\mathrm{End}(\mathbb{L})\) and has the property that it can be endowed with a co-commutative Hopf algebra structure. The étale unipotent fundamental group of \(X\) is then \(G = \mathrm{Spec} (A^\vee)\).
The main result of this paper gives sufficient conditions for the existence of \(\mathbb{L}\) in terms of algebraic cycles, or in other words for \(G\) to be of motivic origin. These conditions are very explicit: there exists a sequence of cycles \((a_n)_{n>0}\) such that (1) \(a_n\) is a codimension \(n\) cycle on \(X\times J^n \times \mathbb{A}^{n-1}\), where \(J\) is the Jacobian of \(X\); (2) \(a_1\) is the divisor associated to the Poincaré bundle on \(X\times J\); (3) the cycles \(a_n\) satisfy a certain equation called the Maurer-Cartan equation.
The author then establishes that those sufficient conditions hold when \(X\) is an elliptic curve minus the origin or the projective line minus two points. However, it is remarked that such a sequence cannot exist for all curves of genus bigger than one. unipotent fundamental groups; algebraic cycles; Maurer-Cartan equation; Massey products Algebraic cycles, Motivic cohomology; motivic homotopy theory Algebraic cycles satisfying the Maurer-Cartan equation and the unipotent fundamental group of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathbb{P}^ 3\) be the projective space of dimension 3 over the complex numbers. For \(d\geq 4\) we denote by \(\mathbb{P}^ N=\mathbb{P}^{{d+3\choose 2}- 1}\) the projective space whose points correspond to surfaces of degree \(d\) in \(\mathbb{P}^ 3\) and by \(S(d)\subseteq\mathbb{P}^ N\) the open subset consisting of points corresponding to smooth surfaces. By the Noether- Lefschetz theorem, there is a countable set of proper irreducible closed subvarieties of \(S(d)\) such that for every point \(s\) outside the union of these subvarieties, the corresponding surface \(S\) has \(\text{Pic} S\cong\mathbb{Z}\) generated by \({\mathcal O}_ S(1)\). The union of the mentioned subvarieties, i.e., the locus of surfaces with Picard group different from \(\mathbb{Z}\), is called the Noether-Lefschetz locus and denoted NL\((d)\).
It is known that the codimension \(c\) of a component of the Noether- Lefschetz locus NL\((d)\) satisfies \(d-3\leq c\leq{d-1\choose 3}\). We prove that for \(d\geq 47\) and for every integer \(c\in\bigl[{9\over 2}d^{{3\over 2}},{d-1\choose 3}\bigr]\) there exists a component of NL\((d)\) with codimension \(c\). This is done with families of surfaces of degree \(d\) in \(\mathbb{P}^ 3\) containing a curve lying on a cubic or on a quartic surface or a curve with general moduli. Moreover we produce an explicit example, for every \(d\geq 4\), of components of maximum codimension \(d-1\choose 3\), thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem). codimension of component of Noether-Lefschetz locus Ciliberto, C.; Lopez, A., \textit{on the existence of components of the Noether-Lefschetz locus with given codimension}, Manuscripta Math., 73, 341-357, (1991) Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry, Special surfaces On the existence of components of the Noether-Lefschetz locus with given codimension | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 monograph is essentially about the representation theory of a sheaf of reductive Lie algebras on a generalization \(J(X;L,d)\) of the classical Jacobian to smooth projective surfaces \(X\), whose closed points are pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) torsion-free sheaves of rank 2 on \(X\), \(c_1(\mathcal{E})=L\), \(L\) a fixed divisor on \(X\), \(c_2(\mathcal{E})=d \geq 0\), \(e\) a global section of \(\mathcal{E}\) with homothety class \([e]\), \(Z_e = (e=0)\), and the sheaf of Lie algebras in question is obtained from reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\). Among the many applications developed in this work (analog of a Lie algebraic aspect of the classical Jacobian, analog of a variation of Hodge structure à la Griffiths, analog of an infinitesimal Torelli theorem, toric geometry, action of affine Lie algebras on the direct sum of cohomology rings of Hilbert schemes, \dots) chief among those is probably the connection with Langlands Duality, and it is very much in this spirit that this work was initiated. It is also having Langlands duality in mind that we recommend reading this work lest the reader be quickly sidetracked by peripheral results that are certainly bold and tantalizing connections to various subfields of Algebraic Geometry and Representation Theory, but which unfortunately are not fully exploited and somewhat obscure the author's original goal, that of providing new insights into the Langlands program.
Two connected results of the author that are worthy of attention are for one thing that \(J(X;L,d)\) yields a finite collection \(\mathcal{V}\) of quasi-projective subvarieties of \(X^{[d]}\), every element \(\Gamma\) of which determines a finite collection of nilpotent orbits in \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(d[\Gamma] \leq d\) intrinsically associated to \(\Gamma\), and second that the same \(\Gamma\)'s determine a finite collection \( ^L R(\Gamma)\) of irreducible representations of the Langlands dual group \( ^L \mathfrak{sl}_{d[\Gamma]}(\mathbb{C}) = \mathrm{PGL}_{d[\Gamma]}(\mathbb{C})\). On a certain subset \(\breve{J}\) of \(J(X;L,d)\), \(H^0(\mathcal{O}_{Z_e})\) has a certain direct sum decomposition, and with the ring structure on \(H^0(\mathcal{O}_{Z_e})\), we get a reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\), the semisimple part of which is denoted by \(\mathcal{G}(\mathcal{E}, [e])\). We also have a morphism of schemes \(J(X;L,d) \rightarrow{\pi} X^{[d]}\) that sends a point \((\mathcal{E}, [e])\) to \([Z_e]\). One can attach a nilpotent element \(D^+(\nu)\) of \(\mathcal{G}(\mathcal{E}, [e])\) to every tangent vector \(\nu\) of \(\breve{J}\) along the fibers of \(\pi\), at a point \((\mathcal{E}, [e])\). If one denotes by \(T_{\pi}(\mathcal{E}, [e])\) the space of all such vectors of \(\breve{J}\) at \((\mathcal{E}, [e])\), one obtains a linear map
\[
D^+_{(\mathcal{E}, [e])}: T_{\pi}(\mathcal{E}, [e]) \rightarrow \mathcal{N}(\mathcal{G}(\mathcal{E}, [e])).
\]
The nilpotent cone of \(\mathcal{G}(\mathcal{E}, [e])\) being partitioned into a finite set one gets the first result.
The loop version of this map has values in the infinite Grassmannian \(\mathrm{Gr}(\mathcal{G}(\mathcal{E}, [e]))\) of \(\mathcal{G}(\mathcal{E}, [e])\) and one obtains a loop version of the first result where now \(\Gamma\)'s determine a finite collection of orbits in \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), and taking the intersection cohomology complexes of those orbits one passes to the category of perverse sheaves on \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), from which the second result follows after making use of the geometric version of the Satake isomorphism of [\textit{V. Ginzburg}, ``Perverse Sheaves on a Loop Group and Langlands Duality'', \url{arXiv:alg-geom/9511007}] and \textit{I. Mirkovic} and \textit{K. Vilonen} [Ann. Math. (2) 166, No. 1, 95--143 (2007; Zbl 1138.22013)].
One other result that is worthy of attention is the following. \(J(X;L,d)\) determines a finite collection \(\mathcal{P}(X;L,d)\) of perverse sheaves on \(X^{[d]}\), intersection cohomology complexes associated to local systems \(\mathcal{L}_{\lambda}\) on \(\Gamma\), \(\Gamma \in \mathcal{V}\), \(\mathcal{V}\) as in the first result above, \(\mathcal{L}_{\lambda}\) corresponding to a representation \(\pi_1(\Gamma, [Z]) \rightarrow \mathrm{Aut}(H^{\bullet}(B_{\lambda}, \mathbb{C}))\), \(B_{\lambda}\) a Springer fiber over the nilpotent orbit \(O_{\lambda}\) of \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(\lambda\) a partition of \(d[\Gamma]\). \(\mathcal{P}(X;L,d)\) gives rise to a distinguished collection \(C(X;L,d)\) of irreducible perverse sheaves on \(X^{[d]}\) and one denotes by \(\mathcal{A}(X;L,d)\) the abelian category of finite direct sums of \(C[n]\), \(n \in \mathbb{Z}\), \(C \in C(X;L,d)\). \(J(X;L,d)\) also comes equipped with a Cartier divisor \(\Theta(X;L,d)\) parametrizing pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) not locally free, and letting \(J^0(X;L,d) = J(X;L,d)\setminus \Theta(X;L,d)\), \(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}\) the sheaf of relative differentials of \(J^0\) over \(X^{[d]}\), then the author proves there is a natural map \(H^0(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}) \rightarrow \mathcal{A}(X;L,d)\), which one can see as a deformation theoretic result instrumental in reformulating the classical Langlands correspondence into the geometric Langlands correspondence. Jacobian; Hilbert scheme of points; period map; Torelli problem; Springer resolution; Langlands duality; perverse sheaves; Griffiths period domain; affine Lie algebras; variation of Hodge structure Parametrization (Chow and Hilbert schemes), Variation of Hodge structures (algebro-geometric aspects), Torelli problem, Families, moduli, classification: algebraic theory, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Nonabelian Jacobian of projective surfaces. Geometry and 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 This paper is part of the author's Geometric Arithmetic Program and is a continuation of his previous studies of nonabelian zeta functions [see Ohbuchi, Akira (ed.) et al., Proceedings of the symposium on algebraic geometry in East Asia, Kyoto, Japan, August 3--10, 2001. River Edge, NJ: World Scientific, 245--262 (2002; Zbl 1086.14017)].
In Chapter I, the author introduces a new type of zeta functions (non-abelian zeta functions) for curves defined over finite fields using the corresponding moduli spaces of semi-stable vector bundles. Based on the vanishing theorem (duality, Riemann-Roch theorem) for cohomologies of semi-stable vector bundles the author shows that these new zeta functions are indeed rational and satisfy certain functional equations. Make use these results, in particular, the rationality, the author introduces global non-abelian zeta functions for curves defined over number fields via the Euler product formalism. The main result concerning these global zeta functions is about a convergence region of the corresponding Euler products. Key ingredients of the proof is a result of Siegel about quadratic forms, a very precise structure description of the above mentioned local zeta functions, the Clifford lemma for semi-stable vector bundles, and Weil's theorem on the Riemann hypothesis for Artin's zeta functions.
As for the justification, the author shows that when only line bundles are involved (so moduli spaces of semi-stable bundles are nothing but the standard Picard groups), the above zeta functions, local and global, coincide with the classical Artin zeta functions defined over finite fields and the Hasse-Weil zeta functions for curves defined over number fields, respectively. As a concrete example, the author computes rank two zeta functions for two curves of genus two by studying Weierstrass points and non-abelian Brill-Noether loci in terms of what is called their infinitesimal structures.
In Chapter II, the author introduces a new type of \(L\)-functions (non-abelian \(L\)-functions) for curves over finite fields using integrations of Eisenstein series over a much more general type of moduli spaces. Based on the theory of Eisenstein series of Langlands and Morries the author establishes meromorphic continuation, functional equations and studies singularities of these non-abelian \(L\)-functions. Semi-stable and stable vector bundles on regular projective curves; moduli space of stable bundles; local non-abelian zeta functions for curves defined over finite fields (rationality and functional equations); global non-abelian zeta functions for curves defined over number fields; non-abelian L--functions for function fields (rationality and functional equations) Weng, L.: Non-abelian L function for number fields Zeta and \(L\)-functions in characteristic \(p\), Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Vector bundles on curves and their moduli Non-abelian zeta functions for function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, a \(S\)-semistable curve \(X/S\) means a flat projective morphism \(X\rightarrow S\) of schemes (over \(k\)) such that each geometric fiber is a connected curve with at worst ordinary double points. Consider a faithful action of a finite group \(G\) on \(X\rightarrow S\). Then the quotient \(X/G\rightarrow S\) may not be semistable. To specialize this question, consider a normal subgroup \(H\vartriangleleft G\). Then there is an induced action of \(G/H\) on the quotient \(X/H\) which makes the quotient map equivariant with respect to \(G\rightarrow G/H.\) The authors wish to compare the \(G\)-equivariant deformation functor \(\text{Def}_G(X)\) of \(X\) with the corresponding \(G/H\)-equivariant deformation functor \(\text{Def}_{G/H}(X/H)\). The case \(G=H\) already gives information to the first question. Provided that for any base change \(T\rightarrow S\) the natural morphism \((X/G)\times_S T\rightarrow(X\times_S T)/G\) is an isomorphism (base change and quotients are commuting operations), there exists a morphism between the two deformation functors:
\[
\text{Def}_H(X)\overset{\text{res}}\leftarrow\text{Def}_G\overset{\text{Ind}}\rightarrow\text{Def}_{G/H}(X/H).
\]
It is known that base change and quotients commute if \(G\) acts freely, and if the order of \(G\) is invertible in the structure sheaf of \(S\).
One of the main results of the article is a new condition under which base change and quotients commute: If \(G\) is a finite group of \(S\)-automorphisms of \(X/S\) and if the action of \(G\) is free on an open dense set on any geometric quotient, then base change and quotients commutes, and moreover \(X/G\) is semistable.
Suppose we have a semistable curve over a one dimensional regular base. Assume that the generic fibre is smooth, and that the curve is acted upon by a finite group \(G\). Then the group \(G\) will not act freely on on any open dense set of the special fiber. The hypothesis is restored by letting the group itself degenerate. This leads to a local version of the result above for a finite, flat \(S\)-group scheme \(\mathbf G\) acting on a smooth affine \(S\)-curve.
This is the idea of the article, but the method for proving this leads to lots of beautiful mathematics: An approximation to the ``quotients'' \(A^{\mathbf G}\) is the Kleiman-Lønsted algebra \(\Sigma_R^{\mathbf G}(A)\) associated to a finite flat group scheme action. \(\Sigma_R^{\mathbf G}(A)\) and \(A^{\mathbf G}\) have the same maximal spectrum, and if the action of the constant group \(G\) is free, then they are equal. The questions above are solved for the Kleiman-Lønsted algebra, and then generalized to the original question. In particular this applies to the case of quotients of semistable curves, where it gives nice results. After proving that restriction and induction morphisms exists, it is commented that a local-global principle reduces the study of the deformation functor \(\text{Def}_G(X)\) to the study of the local deformation functor at singular points and wildly ramified points. This is an interesting tread to follow, and the authors have a paper in preparation. The final example in the article illustrates the local theory. Kleiman-Lønsted algebra; equivariant deformations [3] --, `` Problem of formation of quotients and base change {'', \(Manuscripta Math.\)115 (2004), no. 4, p. 467-487. &MR 21} Geometric invariant theory, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Group actions on varieties or schemes (quotients) Problem of formation of quotients and base change | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author generalizes the Thom-Sebastiani Theorem to the \(\ell\)-adic setting in this article. Let \(f: (\mathbb{C}^{m+1},0)\to (\mathbb{C},0)\) and \(g: (\mathbb{C}^{n+1},0)\to (\mathbb{C},0)\) be germs of holomorphic functions having \(0\) as an isolated critical point with value \(0\), the classical Thom-Sebastiani Theorem describes the vanishing cycles group of \(f\oplus\) at \(0\) as a tensor product:
\[
\Phi^m(f)\otimes \Phi^n(g)\overset{\sim}{\rightarrow} \Phi^{m+n+1}(f\oplus g).
\]
In the \(\ell\)-adic setting (in positive characteristic), Deligne observed that the tensor product should be replaced by a certain convolution product, and the author uses Deligne's theory of nearby cycles over general bases to prove algebraic variants and generalizations. The main ingredient is a Künneth formula for nearby cycles. A short proof of this formula is given by Weizhe Zheng in the appendix. In the last section, the author studies the tame sheaves on \(\mathbb{A}^1\) in detail and the the relation between the tensor and convolution products in both global and local cases. Thom-Sebastiani Theorem; Künneth formula; nearby cycle; vanishing topos; convolution Illusie, L.: Around the Thom-Sebastiani theorem, with an appendix by Weizhe Zheng. Manuscr. Math. (2016). 10.1007/s00229-016-0852-0 Étale and other Grothendieck topologies and (co)homologies, Exponential sums, Grothendieck topologies and Grothendieck topoi, Deformations of complex singularities; vanishing cycles, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Around the Thom-Sebastiani theorem, with an appendix by Weizhe Zheng | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Motivated by the Mordell--Lang conjecture on (semi)abelian varieties (theorems of Faltings and Vojta) and Skolem--Mahler--Lech theorems on return sets of linear recurrences, \textit{D. Ghioca} and \textit{T.J. Tucker} [J. Number Theory 129, No. 6, 1392--1403 (2009; Zbl 1186.14047)] proposed a (rank-one) dynamical Mordell--Lang conjecture. The author proves this conjecture for birational polynomials on \(\mathbb{A}^2\).
More specifically, his main theorem (Theorem A) proves the following:
Theorem. Let \(K\) be an algebraically closed field of characteristic \(0\), \(f:\mathbb{A}^2\rightarrow \mathbb{A}^2\) be a birational polynomial morphism defined over \(K\), \(C\) be a curve in \(\mathbb{A}^2\) defined over \(K\), and \(P\in \mathbb{A}^2(K)\). Then the set \(\{n\in \mathbb{N}: f^n(P)\in C\}\) is a finite union of arithmetic progressions, where \(f^n\) denotes the \(n\)-th iterate of \(f\) and where an arithmetic progression can have a common difference of \(0\).
The central case is when the dynamical degree \(\lambda(f) > 1\), in which case his theorem holds in any characteristic. Here, the major ingredient of the proof is the compactification of \(\mathbb{A}^2\) provided in [\textit{C. Favre} and \textit{M. Jonsson}, Ann. Math. (2) 173, No. 1, 211--249 (2011; Zbl 1244.32012)]: a smooth projective surface \(X\) to which an iterate of \(f\) extends as an algebraically stable map, such that there is a fixed point \(Q\in X\setminus \mathbb{A}^2\) satisfying \(f(X\setminus \mathbb{A}^2) = \{Q\}\). Then a sufficient condition (Theorem 1.2) for periodicity under such \(f\) shows that we may assume \(C\setminus \mathbb{A}^2 = \{Q\}\). In this case, Proposition 7.5 shows that all but bounded-height points of \(C(\bar{\mathbb{Q}})\) are in an attracting basin of \(Q\) with respect to some absolute value. The author then uses Theorem 1.1, which is a local dynamical Mordell--Lang conjecture: if \(f^n(P)\) tends to \(Q\) with respect to some absolute value, then \(C\) is fixed or \(f^n(P) = Q\) for some \(n\). This finishes the proof of Theorem A. To prove Theorems 1.1 and 1.2, the author takes advantage of birationality to extend \(f\) to normalizations of \(C\) and invokes some intersection theory, as well as analyzing how the dynamical Mordell--Lang properties are preserved inside a birational class.
Note that the author has now posted a generalization to all (not necessarily birational) polynomials on \(\mathbb{A}^2\) [``The Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane'', Preprint, \url{arXiv:1503.00773}]. dynamical Mordell--Lang conjecture; birational polynomials; Favre--Jonsson compactification J. Xie, Dynamical Mordell-Lang Conjecture for birational polynomial morphisms on \(\AECFbA^2\), to appear in Math. Ann. Rational and birational maps, Arithmetic dynamics on general algebraic varieties, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps Dynamical Mordell-Lang conjecture for birational polynomial morphisms on \(\mathbb A^2\) | 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 \(R=k[x_1,\dots,x_n]\) denote a polynomial ring and let \(h:\mathbb{N} \to\mathbb{N}\) be a numerical function. Consider the set of all graded Artin level quotients \(A=R/I\) having Hilbert function \(\underline h\). This set (if nonempty) is naturally in bijection with the closed points of a quasiprojective scheme \({\mathfrak L}^\circ (\underline h)\). The object of this note is to prove some specific geometric properties of these schemes, especially for \(n=2\). The case of Gorenstein Hilbert functions (i.e., where \(A\) has type 1) has been extensively studied, and several qualitative and quantitative results are known. Our results should be seen as generalizing some of them to the non-Gorenstein case. We derive an expression for the tangent space to a point of \({\mathfrak L}^\circ(\underline h)\). In the case \(n=2\), we give a geometric description of a point of \({\mathfrak L}^\circ (\underline h)\) in terms of secant planes to the rational normal curve, which generalizes the one just given for \(t=1\). We relate this description to Waring's problem for systems of algebraic forms and solve the problem for \(n=2\). In the last section we prove a projective normality theorem for a class of schemes \({\mathfrak L}(i,r)\) using spectral sequence techniques. The results are largely independent of each other, so they may be read separately. DOI: 10.1307/mmj/1049832900 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Classical problems, Schubert calculus On parameter spaces for Artin level algebras. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Many authors have studied the Hilbert scheme \(\text{Hilb}^d (X)\) parametrizing closed subschemes \(Z \subset X\) of length \(d\) on a variety \(X\). \textit{J. Fogarty} showed that \(\text{Hilb}^d (X)\) is smooth and irreducible when \(X\) is a smooth connected surface [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], but for \(n>2\) and \(d \gg 0\) \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)] proved that \(\text{Hilb}^d (\mathbb A^n)\) is reducible by constructing \textit{elementary} components \(Z \subset \text{Hilb}^d (\mathbb A^n)\), those parametrizing subschemes supported at a single point.
The author proves that if \(R \subset \mathbb A^n\) is supported at the origin, then the corresponding point \([R] \in \text{Hilb}^d (\mathbb A^n)\) lies on an elementary component if \(R\) has \textit{trivial negative tangents}, meaning that the tangent map \(\langle \partial_1, \dots, \partial_n \rangle \to \text{Hom}(I_R,{\mathcal O}_R)_{<0}\) of the orbit of \([R]\) under translation is surjective. Conversely, if \(Z \subset \text{Hilb}^d (\mathbb A^n)\) is a generically reduced elementary component and char \(k =0\), then the general point \([R] \in Z\) has trivial negative tangents (it is unknown whether there exist generically non-reduced components).
The main tools in the proof are obstruction theory and an extension of the decomposition theorem of \textit{A. Bialynicki-Birula} [Ann. Math. (2) 98, 480--497 (1973; Zbl 0275.14007)] from smooth proper varieties to the singular non-proper Hilbert schemes \(\text{Hilb}^d (\mathbb A^n)\).
The author uses his criterion to construct infinitely many smooth points of distinct elementary components of \(\text{Hilb}^d (\mathbb A^4)\); these examples show that the Gröbner fan need not distinguish components of \(\text{Hilb}^d (\mathbb A^n)\). He also reduces the question of whether \(\text{Hilb}^d (\mathbb A^n)\) is reduced to a testable conjecture involving the trivial negative tangents condition. Hilbert scheme of points; elementary components Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Deformations and infinitesimal methods in commutative ring theory Elementary components of Hilbert schemes 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 Consider a flat family \(C'\to \Delta'\) of smooth curves of genus \(g>2\) over the punctured disk \(\Delta'=\Delta-\{0\}\) and a stable extension \(C\) of \(C'\) over \(0\). The corresponding extension of the symmetric products \(C^{(2)}\) is not necessarily stable over the special point. The author describes a geometric modification of the family \(C^{(2)}\) over \(\Delta\) to a stable family. The first step consists in replacing \(C^{(2)}\) with the relative Hilbert scheme Hilb\(_2(C/\Delta)\). This gives, over \(0\), a surface with semi-log canonical singularities. A further modification yields a family whose special fiber \(S_0\) also has a dualizing sheaf \(\omega\) with some ample, locally free power.
As a consequence, the author constructs an isomorphism between the moduli space of stable curves \(C\) and the component of the moduli space of surfaces which contains the product \(C^{(2)}\). Van Opstall, M.: Stable degenerations of symmetric squares of curves, Manuscripta math. 119, No. 1, 115-127 (2006) Families, moduli, classification: algebraic theory, Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes) Stable degenerations of symmetric squares of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{M}_g\) be the moduli space of Deligne-Mumford stable genus \(g\) curves. One can compactify the universal moduli space of slope semi-stable vector bundles over \(\overline{M}_g\) by past results of \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] and \textit{R. Pandharipande} [J. Am. Math. Soc. 9, No. 2, 425--471 (1996; Zbl 0886.14002)]. Recently a series of new compactifications of \(M_g\) have been constructed from the viewpoint of the log minimal model program (called the Hassett-Keel program for \(\overline{M}_g\), see [\textit{B. Hassett} and \textit{D. Hyeon}, Trans. Am. Math. Soc. 361, No. 8, 4471--4489 (2009; Zbl 1172.14018); Ann. Math. (2) 177, No. 3, 911--968 (2013; Zbl 1273.14034)] for the first two steps). It is thus natural to ask if there exists a modular compactification of the universal moduli space of vector bundles over a new birational model of \(\overline{M}_g\). In this paper the author constructs a compactified universal moduli space of slope semi-stable sheaves on several beginning models out of the Hassett-Keel program. Moreover, the author gives a complete description for the compactification in the case of pseudo-stable curves (that replace an ellptic tail by a cuspidal singularity as the first step of the Hassett-Keel program). In order to obtain these results, the author studies general Gorenstein curves as well as related GIT problems. moduli space of curves; moduli space of vector bundles; log minimal model program Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic), Vector bundles on curves and their moduli Compactifications of universal moduli spaces of vector bundles and the log-minimal model program on \(\overline{M}_g\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\Gamma\) be the fundamental group of an algebraic curve \(X\) of genus \(g\) with \(n\) points removed, and let \(G\) be a connected reductive algebraic group over the complex numbers or a connected compact Lie group. The article states some facts about the fundamental group of the moduli space of representations \(\mathfrak{X}_\Gamma(G)=\Hom(\Gamma,G)/G\), where we identify representations by conjugation. They present two results on its fundamental group depending on the number of the points removed. In case that \(n\geq 1\), i. e. when some points are removed, authors prove that the group \(\pi_1(\mathfrak{X}_\Gamma(G))\) is isomorphic to \(\pi_1(G/[G,G])\); in case \(n=0\), which is the compact case, they states only for cases \(G=\mathrm{GL}(n,\mathbb{C})\) and \(U(n)\), where \(\pi_1(\mathfrak{X}_\Gamma(G))\cong \mathbb{Z}^{2g}\), and for cases \(G=\mathrm{SL}(n,\mathbb{C})\) and \(G=\mathrm{SU}(n)\), where \(\pi_1(\mathfrak{X}_\Gamma(G))=0\). While the first result is a generalization of a previous theorem published by the same authors (see Theorem B in [\textit{S. Lawton} and \textit{D. Ramras}, New York J. Math. 21, 383--416 (2015; Zbl 1339.57002)]), the second one is a partial result of one published earlier (see Theorem 1.1 in [\textit{I. Biswas} et al., Math. Z. 281, No. 1--2, 415--425 (2015; Zbl 1349.14042)]). In any case, the techniques presented in this paper are very different than the previous used.
For the first statement (the non-compact case), the proof takes a bit more than three pages (recall that in the mentioned paper they uses more than twenty pages to prove a partial result). The key of this proof lies on the long exact sequence in homotopy \(\pi_1(\mathfrak{X}_\Gamma([G,G])\to \pi_1(\mathfrak X_\Gamma(G))\to \pi_1(\mathfrak{X}_\Gamma(G/[G,G]))\to \pi_0(\mathfrak{X}_\Gamma([G,G]))\to 0\). Once they prove \(\mathfrak{X}_\Gamma([G,G])\) is a connected simply-connected space, the result follows. They do several steps to establish this assertion: first, they consider the subset \(\Hom^g(\Gamma,G)\subset \Hom(\Gamma,G)\) of \textit{good} representations (those satisfy \(Z(\rho(\Gamma))= Z(G)\)), and they prove that the natural projection map \(\Hom^g(\Gamma,G)\to \mathfrak{X}_\Gamma(G)\) is a \(PG\)-bundle, where \(PG\) is a parabolic subgroup (see Lemma~2.2). Based in this property, they show that the map \(\Hom(\Gamma,G)\to \mathfrak{X}_\Gamma(G)\) is \(\pi_1\)-surjective. Then, since \(\pi_1([G,G]^r)\to \pi_1(\mathfrak{X}_\Gamma([G,G])\) sends to zero all map, \(\pi_1\)-surjectivity implies the simply-connectivity.
A different technique is applied to prove the second result. Here, authors take into account the moduli space of Higgs bundles. They notice that the moduli space of Higgs bundles of rank \(n\) on \(X\), denoted by \(\mathcal{M}_H(X,n)\), (resp. with trivial determinant, denoted by \(\mathcal{N}_H(X,n)\)) is isomorphic to \(\mathfrak{X}_\Gamma(\mathrm{GL}(n,\mathbb{C}))\) (resp. \(\mathfrak{X}_\Gamma(\mathrm{SL}(n,\mathbb{C}))\)). They states that the map \(\mathcal{M}_H(X,n)\to \mathrm{Pic}^0(X)\) induces an isomorphism between their fundamental groups, so that \(\pi_1(\mathcal{M}_H(X,n))\cong \mathbb{Z}^{2g}\). character variety; moduli space; fundamental group; Higgs bundle Biswas, I., Lawton, S.: Fundamental group of moduli spaces of representations. Geom. Dedic. (to appear). arXiv:1405.3580 Algebraic moduli problems, moduli of vector bundles, Group actions on varieties or schemes (quotients), Homotopy theory and fundamental groups in algebraic geometry Fundamental group of moduli spaces of 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 The aim of this thesis is to give some consequences of the Faltings theorem on isogenies of abelian varieties over number fields: Abelian varieties \(A,B\) over a number field \(K\) are isogenous over \(K\) if and only if \(\nu\)-factors of their \(L\)-functions are equal, for almost every discrete valuation \(\nu\) of the field \(K\).
At first, we study the problem:
\[
L(A,K,s) =L(B,M,s),
\]
where \(A,B\) are abelian varieties over Galois number fields \(K,M\) respectively. Using Faltings theorem and the properties of the Weil functor of the restriction of scalars, we obtain some general results. Also we show that the problem can be reduced to the case of simple abelian varieties. In the sequel, using Serre's result on supersingular reduction of elliptic curves, we solve the problem in the case of elliptic curves defined over \(\mathbb{Q}\).
After that we study the problem of linear independence of \(L\)-functions of abelian varieties. Using Faltings theorem and some elementary facts, we show that a family of different \(L\)-functions of simple abelian varieties over a fixed Galois number field, is linearly independent over \(\mathbb{C}\). Finally, we show that Faltings theorem, in the case of elliptic curves over \(\mathbb{Q}\), can be treated as a generalization of the weak Hasse principle. In fact, we give a reformulation of the relation of equivalence of ternary quadratic forms. After that, we transmit a new relation on ternary cubic forms over \(\mathbb{Q}\) representing zero and point out that the weak Hasse principle, in this case, is in fact, Faltings' theorem. arithmetic of elliptic curves; linear independence of \(L\)-functions; Faltings theorem; isogenies of abelian varieties; equivalence of ternary quadratic forms Elliptic curves over global fields, Elliptic curves, Isogeny Contribution to arithmetic of elliptic curves and abelian varieties. (Abstract of thesis) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 issue of extending a given Galois group is conveniently expressed in terms of embedding problems. If the kernel is an abelian group, a natural method, due to Serre, reduces the problem to the computation of an étale cohomology group, that can in turn be carried out thanks to Grothendieck-Ogg-Shafarevich formula. After introducing these tools, we give two applications to fundamental groups of curves. fundamental groups of curves; embedding problems; Grothendieck-Ogg-Shafarevich formula Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group Extension of Galois groups by solvable groups, and application to fundamental groups 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 Given a projective scheme \(X\) and a coherent sheaf \(E\) on \(X\), the scheme \(\text{Quot}(E/X)\), introduced by \textit{A. Grothendieck} [in Sémin. Bourbaki 1960/1961, Exp. 221 (1961; Zbl 0236.14003)], parametrizes the quasi-coherent quotients of \(E\). The paper under review treats a special case. If \(X\) is a smooth projective surface and \(E\) is locally free of rank \(r\geq 1\), then \(\text{Quot}(E,l)\) parametrizes all quotients \(E\to T\), where \(T\) is zero-dimensional of length \(l\).
The authors prove that \(\text{Quot}(E,l)\) is an irreducible scheme of dimension \(l(r+1)\) and that the fibre of the natural map \(\pi\) from \(\text{Quot}(E,l)\) to \(S^l(X)\) over a point \(\sum_xl_xx\) is also irreducible and of dimension \(\sum_x(rl_x-1)\).
In the case \(r=1\), where \(\text{Quot}(E,l)\) is isomorphic to the Hilbert scheme, the theorem had been proved by J. Fogarty and J. Briançon. For \(r\geq 2\), related results are due to J. Li, D. Gieseker and V. Baranovsky. The theorem cannot be extended to singular or higher-dimensional varieties: It is already false in the first case \(r=1\). quasi-coherent quotients; Hilbert scheme Ellingsrud G and Lehn M, Irreducibility of the punctual quotient scheme of a surface, Ark. Mat. 37(2) (1999) 245--254 Homogeneous spaces and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Irreducibility of the punctual quotient scheme of a surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\phi: X\to Y\) be a proper algebraic map with connected fibres from a connected quasi-projective n-dimensional complex manifold X, \(n\geq 2\), onto a quasi-projective variety Y and let L be an algebraic line bundle on X, which is very ample relatively to \(\phi\). The authors use Reider's technique [\textit{I. Reider}, Ann. Math., II. Ser. 127, No.2, 309-316 (1988; Zbl 0663.14010)] in a local setting to provide results about the adjoint bundle \(K_ X\otimes L^{n-1}\), which generalize those obtained by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)] in the absolute case, i.e. when Y is a point. The authors prove that, unless \(\phi\) exhibits (X,L) as a scroll over a smooth curve, the natural morphism \(\phi^*\phi_*(K_ X\otimes L^{n-1})\to K_ X\otimes L^{n-1}\) is onto. This allows them to construct a normal quasi-projective space \(X'\) and algebraic morphisms with connected fibres \(\Phi: X\to X'\), \(\phi ': X'\to Y\) such that \(\phi =\phi '\circ \Phi\) and \(K_ X\otimes L^{n-1}=\Phi^*{\mathcal L}\), where \({\mathcal L}\) is a line bundle on \(X'\), which is ample and spanned relatively to \(\phi '\). If \(\dim (X')<\dim (X)\) then there is a precise description of \(\phi\), while if \(\dim (X')=\dim (X)\) then \(\Phi\) defines a sort of relative reduction \((X',L')\) of (X,L), up to which, the authors prove that \(K_ X\otimes L^{n-1}\) is very ample relatively to \(\phi\). adjunction; k-spannedness; algebraic line bundle; adjoint bundle Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the relative adjunction mapping | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0686.00006.]
This paper records seminar talks explaining with examples, the author's method of exhibiting finite groups as Galois groups of regular extensions of the field \({\mathbb{Q}}(X)\) of rational functions over the rationals. The basic idea is to construct a (ramified) cover of the complex projective line with the desired group as group of deck transformations, and a rational structure on this cover, by taking the moduli space of such covers and finding a rational point on it. The number r of ramification points is a key parameter here: when \(r=3\), the method implies that the given finite group G can be generated in a very special way by two elements. For larger r, the situation becomes more complicated.
There are some quite general conditions on a group G, together with a faithful representation T of it as permutations of n letters and with an r-tuple \({\mathbb{C}}\) of conjugacy classes of G, that guarantee that G can be exhibited as the Galois group of a regular extension of \({\mathbb{Q}}(X)\) ramified at r rational points. The author constructs a parameter space \({\mathcal H}\), depending on the pair (\({\mathbb{C}},T)\), which parameterizes the covers associated to that pair. \({\mathcal H}\) is a cover of projective r- space with the discriminant locus deleted; for \(r>3\) it is required that there be a \({\mathbb{Q}}\) rational point on the pullback of \({\mathcal H}\) to the r-fold product of the projective line. One of the main results of the paper is the exhibition of a curve cover of the projective line, ramified over only zero, one and infinity (and identified with the projective normalization of the quotient of the upper by a certain finite index subgroup of \(PSL_ 2({\mathbb{Z}}))\) such that there is a rational point in the above pullback if and only if there is a rational point of the curve cover not in a fibre over a ramification point.
The author includes an extended example using \(G=A_ 5\) (the alternating group). number of ramification points; inverse Galois problem; finite groups; Galois groups; cover of the complex projective line; rational point M. D. Fried, ``Arithmetic of \(3\) and \(4\) branch point covers: A bridge provided by noncongruence subgroups of \({\mathrm SL}_ 2(Z)\)'' in Séminaire de theorie des nombres (Paris, 1987--88.) , Progr. Math. 81 , Birkhäuser, Boston, 1990, 77--117. Inverse Galois theory, Galois theory, Ramification problems in algebraic geometry, Coverings in algebraic geometry Arithmetic of 3 and 4 branch point covers. A bridge provided by noncongruence subgroups of \(SL_ 2({\mathbb{Z}})\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a number field, \(\overline{k}\) an algebraic closure and \(C\) a non-singular complete curve defined over \(k\) of genus \(g>1\). If we let \(f\) denote a canonical embedding of \(C\) in its Jacobian variety \(J\), then via \(f\) nontrivial automorphisms of \(C\) induce automorphisms of \(J\), which are stable on the image of \(C\). The author studies the restriction this places on the distribution of \(C(\overline{k})\) in \(J(\overline{k})\) via \(f\). This is done by studying the Neron-Tate height function \(\langle \cdot,\cdot\rangle\) and the corresponding norm \(\| \cdot\| \) on \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), which are associated to a theta divisor on \(\overline{J}=J\times_{\text{Spec} k}\text{Spec} \overline{k}\).
A central result shows there exists a canonically defined scalar product on \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), which is preserved by isomorphisms of Jacobians over \(\overline{k}\) induced by isomorphisms of curves over \(\overline{k}\). In this way a canonically defined height function on a curve of genus \(g>1\) over a number field, which is invariant under isomorphisms over the algebraic closure, is obtained.
As an application of his work the author obtains a new proof of the classical result asserting that the number of fixed points of a non-trivial automorphism of a curve of genus \(g>1\) over a number field is at most \(2g+2\).
Finally he applies his results concretely to plane curves over \(k\) defined by \(C:X^4+Y^4=aZ^4\), \(a\neq 0\), describing the distribution of \(f(C(\overline{k}))\) in \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), and so obtaining a different proof of a result of Dem'yanenko. rational points; Jacobian variety; Neron-Tate height function; fixed points of a nontrivial automorphism; curves of genus \(\neq 1\) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Rational points, Jacobians, Prym varieties Rational points of a curve which has a nontrivial automorphism | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective curve. The authors study properties of a generalization of the Brill-Noether locus: \[ B^k_{n,e}(V):=\{ E\in U^s(n,e) : h^0(c, V\otimes E)\ge k\} \ \ , \] where \(U^s(n,e)\) is the space of stable bundles of rank \(n\), degree \(e\), \(V\) is a fixed vector bundle of rank \(r\) and degree \(d\).
For this notion, the authors quote the paper [\textit{M. Teixidor-I-Bigas} and \textit{L. W. Tu}, Contemp. Math. 136, 327--342 (1992; Zbl 0783.14020)].
During the last decades, there was an intense study of questions related to the classical Brill-Noether locus, as well as about generalizations. In Section 3, ``Preliminaries on twisted Brill-Noether loci'', the paper gives a very well written introduction and states main known results which are used in the following.
Main results: The authors show that, for \(e\) big enough (this is precisely described), \( B^k_{n,e}(V)\) is not empty. Then they show that, in conditions similar to those in studies of Teixidor i Bigas and of Mercat (cf. the quotations in the paper), \( B^k_{n,e}(V)\) has a generically smooth connected component, of expected dimension.
The paper contains other geometrical results and an example of a ``nonempty twisted Brill-Noether locus with negative expected dimension'', which ``contains a component of dimension at least 1''. ``This example shows that even for general \(C\) and general stable \(V\), the twisted Brill-Noether loci can exhibit pathologies.'' Brill-Noether loci; Petri trace map; vector bundles Vector bundles on curves and their moduli, Special divisors on curves (gonality, Brill-Noether theory) Nonemptiness and smoothness of twisted Brill-Noether loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book is based on a course given by the author at the Institut de Mathématiques de Jussieu in Paris in 2004 and 2005. The aim of the course was to introduce graduate students, with a standard knowledge in algebraic geometry, to the main topics about bounded derived categories of coherent sheaves on smooth projective varieties. The book is a good reference for anyone who wants to deal with the subject. Anyway, a minimal knowledge in algebraic geometry is required.
Derived categories were defined by Verdier in order to give the right framework to any kind of derived functor, but they were initially thought as formal objects and they were not investigated as an interesting invariant of smooth projective varieties. Mukai was the first to show a geometrically motivated equivalence between two derived categories of non isomorphic varieties. In his work he introduced a class of functors which have become one of the most powerful tools in managing derived categories and they are now known as Fourier-Mukai transforms. Moreover, the homological mirror symmetry conjecture, stated by Kontsevich, involves directly derived categories and its statement made the interest in such objects grow outside the algebraic geometry framework.
Right now, many things are known on the subject, but a lot of work has still to be done. This book is essential to know the most important results and their relevance in algebraic geometry; almost every proof is given in full detail and exercises make the reader gain a working knowledge. It is a very good starting point to explore open problems related to derived categories, such as for example moduli space problems and birational classification.
The book is organised as follows:
Chapters 1, 2 and 3 explain the foundational material, introducing additive and triangulated categories, exact functors, triangulated autoequivalences, exceptional objects and semiorthogonal decompositions (chapter 1); the derived category of an abelian category, derived functors and spectral sequences (chapter 2); the derived category of coherent sheaves on a scheme and its connections with algebraic geometry, such as derived functors and various formulas and Grothendieck-Serre duality (chapter 3). A little knowledge in algebraic geometry is required.
Chapters 4 to 8 give the main known results on the subject: autoequivalences of derived categories in the ample (anti-)canonical bundle case, point-like objects and ample sequences (chapter 4); Fourier-Mukai transforms and their passage to \(K\)-theory and cohomology (chapter 5); geometrical aspects of such transforms, Kodaira dimension and nefness under derived equivalence, connections between derived equivalence and birationality (chapter 6); equivalence criteria for Fourier-Mukai transforms (chapter 7); spherical and exceptional objects, autoequivalences and braid group actions (chapter 8). In this part, a deeper knowledge of algebraic geometry could help the reader to see these results in a greater picture, but a standard knowledge is enough to understand them.
Chapters 9 to 12 detail more specific cases in which many results are already known: abelian varieties and their autoequivalences, focusing especially on the Poincaré bundle as a Fourier-Mukai kernel and on the SL\(_2\) action (chapter 9); derived equivalences of \(K3\) surfaces and moduli spaces of sheaves (chapter 10); birational transformations as blow-ups, flips and flops (chapter 11); derived categories of surfaces (chapter 12). Any of these four chapters comes with a brief well referenced introduction on the subject in order to make it almost self-contained.
The last chapter (chapter 13) introduces briefly and gives the most recent results on the topics involving open problems on derived categories: McKay correspondence, homological mirror symmetry, D-branes and stability conditions, twisted derived categories. derived categories; Fourier-Mukai transforms; derived functors; triangulated categories; autoequivalences; exceptional objects; spherical objects; abelian varieties; \(K3\) surfaces; moduli spaces; birational maps; Serre functors; derived equivalence; braid group action; McKay correspondence; homological mirror symmetry; stability conditions; twisted derived categories D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Math. Monogr., Clarendon Press, Oxford, 2006. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Fourier-Mukai transforms in 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 [For the entire collection see Zbl 0575.00008.]
Author's preface: The title ''tangency and duality over arbitrary fields'' was given by \textit{A. H. Wallace} to the article in Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956; Zbl 0072.160) in which he pioneered the study of the similarities and differences that appear in the indicated theory, a basic topic in projective algebraic geometry, when the characteristic \(p\) of the ground field is allowed to become positive. The title is also an apt choice for the present work. However, the words ''over arbitrary fields'' were dropped for two reasons: First, the subject has matured to the point where it can, fairly, go without saying that p will be arbitrary. Second and more important, the similarities and differences attendant to p are secondary to the geometry itself. The bulk of material is, moreover, characteristic free, and many of the special considerations required when \(p>0\) highlight features of geometry over the complex numbers that are sometimes taken for granted. A case in point is provided by the central notion of reflexivity. In most situations, instead of assuming \(p=0\), it suffices to assume that the principal varieties are reflexive. A major issue then, when \(p>0\), is to find useful conditions guaranteeing reflexivity.
The present work is an expanded version of the minicourse of 3 lectures given by the author. The work is intended first and foremost to introduce this lovely subject and, in particular, to announce and to introduce a fair number of recent results. An attempt has been made to place the results in context and in perspective, to explain their meaning and significance, and to give a feeling for their proofs. The full details of the proofs, especially if they are available elsewhere, are seldom presented. The presentation is usually expository, rarely formal. There are, however, several mathematical tidbits that are not found elsewhere. Some of these are: a fuller discussion of Wallace's construction of infinitely many plane curves with a given dual curve; a simpler and more conceptual proof of the theorem of generic order of contact, I-(10); a more traditional proof of the reviewer's theorem comparing the ranks of a variety with those of a general hyperplane section and those of a general projection; an account of Landman's unpublished application of Lefschetz theory of the theory of the dual variety; the application of Goldstein's theory of the second fundamental form to the study of the simplicity of a general contact between two varieties one varying; a new derivation of the number, 51, of conics tangent, when \(p=2\), to 5 general conics; and a new study of the limiting behavior of the tangent hyperplanes to a variety degenerating under a homolography. tangency; duality; characteristic p; dual curve; homolography Kleiman, S. , Tangency and duality , in: '' Proc. 1984 Vancouver Conference in Algebraic Geometry '', CMS-AMS Conference Proceedings , pp. 163-226, Vol. 6, 1985. Enumerative problems (combinatorial problems) in algebraic geometry, Projective techniques in algebraic geometry Tangency and duality | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we prove a generalization of \textit{A. Grothendieck}'s Lefschetz theorem for complete intersections [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402); XII 3.5]. \(\pi_1 (X)\) will denote the algebraic fundamental group of a scheme \(X\). If \(k\) is a field, \(\overline k\) will denote an algebraic closure of \(k\). Our result is:
Theorem 1. Suppose that \(k\) is a field, \(W\) is a smooth, geometrically connected subvariety of \(\mathbb{P}^m_k\) of dimension \(n\) and \(Z \subset W\) is a closed subscheme set theoretically defined by the vanishing of \(r\) forms of \(\mathbb{P}^m_k\) on \(W\).
1. If \(r \leq n-1\) then \(Z\) is geometrically connected and there is a surjection \(\pi_1(Z) \to \pi_1(W)\).
2. If \(r \leq n-2\), then \(\pi_1(Z) \cong \pi_1(W)\).
Corollary 2. Suppose that \(k\) is a field and \(Z \subset \mathbb{P}^n_k\) is a closed subscheme set theoretically defined by \(r\) forms.
1. If \(r\leq n-1\) then \(Z\) is geometrically connected.
2. If \(r\leq n-2\), then \(\pi_1(Z) \cong \text{Gal} (\overline k/k)\) where \(\overline k\) is an algebraic closure of \(k\).
The corresponding theorem for the topological fundamental group of a complex projective variety follows from a paper by \textit{H. A. Hamm} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 1, 547-557 (1983; Zbl 0525.14011)] and the theorem of II, 1.2 in the book by \textit{M. Goresky} and \textit{R. MacPherson}, ``Stratified Morse theory'' (1988; Zbl 0639.14012). Their proofs use different methods (Morse theory) and do not extend to positive characteristic. simple connectedness; Lefschetz theorem; algebraic fundamental group Cutkosky, S. D.: Simple connectedness of algebraic varieties, Proc. amer. Math. soc. 125, No. 3, 679-684 (1997) Coverings in algebraic geometry, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Simple connectedness of projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth, projective complex variety. Using the notion of a Bridgeland stability condition (see [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]) on \(\mathcal{D}^b(X)\), the bounded derived category of coherent sheaves on \(X\), and some numerical invariants one can construct moduli spaces of stable objects as Artin stacks, see for example [\textit{M. Lieblich}, J. Algebr. Geom. 15, No. 1, 175--206 (2006; Zbl 1085.14015)] and the references therein. Unfortunately, there is no general method for showing projectivity of these moduli spaces or constructing coarse moduli spaces.
In this article, the authors take a first step in the direction of finding such a general method, by constructing a family of nef divisors on such a moduli space via derived category methods, especially via the use of Fourier-Mukai transforms.
In the case that \(X\) is a \(K3\)-surface, the authors show more, namely that for a stability condition generic with respect to the chosen numerical invariants the divisors they construct are actually ample, and hence provide a proof of the projectivity of such moduli spaces. This generalises results of Hiroki Minamide, Shintarou Yanagida and Kota Yoshioka. Finally, Bayer and Macri also apply their result to give an explanation of the relationship between wallcrossing (i.e. change of the stability condition) and the minimal model program applied to the moduli space. This relationship had already been observed in special cases (for example the Hilbert scheme of points on \(\mathbb{P}^2\)) by various authors, see for instance [\textit{D. Arcara} et al., Adv. Math. 235, 580--626 (2013; Zbl 1267.14023)]. Bridgeland stability conditions; derived category; moduli spaces of complexes; Mumford-Thaddeus principle Bayer, A.; Macrì, E., Projectivity and birational geometry of Bridgeland moduli spaces, J. Amer. Math. Soc., 27, 3, 707-752, (2014) Algebraic moduli problems, moduli of vector bundles, Derived categories, triangulated categories, \(K3\) surfaces and Enriques surfaces, Minimal model program (Mori theory, extremal rays) Projectivity and birational geometry of Bridgeland moduli spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a short survey on the motivic Milnor fibre and motivic vanishing cycles introduced by Denef and Loeser, with emphasis on a motivic analogue of a conjecture of Steenbrink on the spectrum of certain hypersurface singularities.
Consider a smooth algebraic variety \(X\), in this review taken for simplicity over \(\mathbb C\), and a regular function \(f\) on \(X\). Recall [\textit{J. Denef} and \textit{F. Loeser}, J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)] that the motivic Milnor fibre of \(f\) at \(x \in \{f=0\}\subset X\) lives in a suitable Grothendieck ring of algebraic varieties, and that Steenbrink's Hodge spectrum \(sp(f,x)\) of \(f\) at \(x\) can be recovered from it. Suppose now that the singular locus of \(f\) is a curve and take a generic linear form \(g\) vanishing on \(x\). Then \(f+g^N\) has an isolated singularity at \(x\) for \(N\) large enough. There is an explicit formula, conjectured by Steenbrink and proven by Saito, for the difference \(sp(f+g^N)-sp(f,x)\).
The author, \textit{F. Loeser} and \textit{M. Merle} [Duke Math. J. 132, No. 3, 409--457 (2006; Zbl 1173.14301)] proved a motivic analogue and generalization of this formula. For this they first extend the motivic vanishing cycles construction of Denef-Loeser to the whole relative Grothendieck ring over \(X\), and they introduce a motivic iterated vanishing cycle class of \(f\) and \(g\), where now \(f\) and \(g\) are \textsl{arbitrary} regular functions on \(X\). The main theorem of [Zbl 1173.14301] is an equality in the appropriate Grothendieck ring relating these constructions, specializing to a formula for \(sp(f+g^N)-sp(f,x)\). In particular, when \(f\) and \(g\) are as in Steenbrink's original setting, this yields Saito's result. motivic Milnor fibre; iterated vanishing cycles; Hodge spectrum Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Deformations of complex singularities; vanishing cycles, Mixed Hodge theory of singular varieties (complex-analytic aspects), Zeta functions and \(L\)-functions, Singularities in algebraic geometry Motivic vanishing cycles 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 The authors establish the following theorem: Let X be a Noetherian scheme, and let \(\gamma\) be an element of \(H^ 2(X,{\mathcal O}^*_ X)\). Then there is a scheme Y and a proper birational morphism \(\alpha: Y\to X\) such that the cohomology class \(\alpha^*(\gamma)\) is represented by an Azumaya algebra on Y. - The proof is by a relatively brief inductive argument, which the authors describe as a simple version of a proof due to \textit{O. Gabber}.
When second cohomology is a birational invariant (for example, for smooth projective varieties), the theorem comes very close to identifying second cohomology and the Brauer group, and in fact does so when second cohomology is finite. As the authors remark, both these conditions obtain for a nonsingular projective model X of V/G where G is a finite group and V is a faithful complex representation of G (using the result of the first named author that in this case \(H^ 2(X,{\mathcal O}^*)\) is isomorphic to the finite group \(H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}}))\). cohomology class as Azumaya algebra; second cohomology; Brauer group Bogomolov, F. A.; Landia, A. N., \(2\)-cocycles and Azumaya algebras under birational transformations of algebraic schemes, Algebraic geometry (Berlin, 1988), Compositio Math., 0010-437X, 76, 1-2, 1-5, (1990) (Co)homology theory in algebraic geometry, Algebraic cycles, Brauer groups of schemes, Realizing cycles by submanifolds 2-cocycles and Azumaya algebras under birational transformations of algebraic schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth irreducible projective curve of genus \(g\geq 1\) and let \(J(X)\) be the jacobian of \(X\). Let \(I(d):X^{(d)}\to J(X)\) be the natural morphism and consider \(W^ r_ d=\{x\in J(X):\dim(I(d)^{- 1}(x))\geq r\}\). Let \(\rho^ r_ d(g)\) be the Brill-Noether number. The author proves some results concerning the schemes \(W^ r_ d\). For example: ``Suppose \(\dim(W^ r_{d-1})=\rho^ r_{d-1}(g)\geq 0\) and \(\rho^ r_ d(g)<g\). If \(W^ r_{d-1}\) is a reduced (respectively irreducible) scheme, then \(W^ r_ d\) is a reduced (respectively irreducible) scheme''. As an application one proves some dimension theorems for the schemes \(W^ 1_ d\). jacobian; Brill-Noether number; dimension \textsc{M. Coppens,} Some remarks on the schemes \(W^{r}_{d}\), Ann. Math. Pura Appl. (4) \textbf{97} (1990), 183-197. Jacobians, Prym varieties Some remarks on the schemes \(W_ d^ r\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A central theme of Galois module structure theory has been to relate class group invariants defined via arithmetic Galois modules to analytic invariants connected with \(L\)-series. The prototype for results of this kind has been \textit{M. J. Taylor}'s proof [Invent. Math. 63, 41-79 (1981; Zbl 0469.12003)] of Fröhlich's conjecture concerning the relation between the Galois module structure of rings of algebraic integers in tame extensions of number fields and the root numbers of symplectic representations of Galois groups. The object of this paper is to prove, in Theorem 1, a generalization of Taylor's theorem to arithmetic schemes of arbitrary dimension. This result shows that a class group invariant defined via deRham cohomology can be determined from \(\varepsilon\)-factors associated to \(L\)-series. Theorem 1 is definitive up to certain regularity assumptions and technical conditions on finite fibers. It proves the generalization of Fröhlich's conjecture stated in [\textit{T. Chinburg, B. Erez, G. Pappas} and \textit{M. J. Taylor}, Ann. Math. (2) 146, 411-473 (1977; Zbl 0939.14009)], and it realizes one of the main goals described in [\textit{T. Chinburg}, Ann. Math. (2) 139, 443-490 (1994; Zbl 0828.14007)] concerning the generalization of Galois module structure theory to schemes.
In Part I the authors considered the deRham-Euler characteristic
\[
\chi({\mathcal X},G):= \sum_{i=0}^d (-1)^i \chi^P (\lambda^i [\Omega_{{\mathcal X},\mathbb{Z}}^1])\in \text{Cl} (\mathbb{Z} [G]).
\]
The main result of this paper is the following:
Theorem 1. Suppose that \({\mathcal X}\) is a regular equidimensional scheme which is projective and flat over \(\text{Spec} (\mathbb{Z})\) and that \(G\) acts tamely on \({\mathcal X}\). Set \({\mathcal Y}= {\mathcal X}/G\). Suppose further that \({\mathcal Y}\) is also regular and that for each prime \(p\) the special fiber \({\mathcal Y}_p\) is a divisor with strict normal crossings and the multiplicities of its irreducible components are coprime to \(p\). Then:
(i) \(\chi({\mathcal X},G)\) belongs to the kernel subgroup \({\mathbf D}(\mathbb{Z} [G])\).
(ii) \(\chi ({\mathcal X},G)= {\mathbf W}_{{\mathcal X},l}+ \mathbb{R}_{\mathcal X}\).
As explained in Part I, this result is the counterpart for flat schemes over \(\mathbb{Z}\) of the main result of T. Chinburg (loc. cit.), which concerned smooth projective varieties over a finite field. This theorem gives the proof of the higher-dimensional Fröhlich conjecture stated in Part I; there this result was proven when either \(\dim({\mathcal X})= 2\) or when \(G\) acts freely on \({\mathcal X}\). Actually, one expects Theorem 1 to hold without assuming any extra condition on the quotient \({\mathcal Y}\).
Their method of proof is of interest in its own right, since it provides a strategy for computing other linear combinations \(\sum_i \chi^P({\mathcal F}_i)\) of equivariant Euler characteristics of \(G\)-equivariant coherent sheaves \({\mathcal F}_i\). The first step in this method is to use the Gamma and topological filtrations on \(K\)-theory together with moving lemmas to show
\[
\sum_i \chi^P ({\mathcal F}_i)= \sum_j \chi^P ({\mathcal F}_j')
\]
where the \({\mathcal F}_j'\) are coherent sheaves supported on proper closed \(G\)-subschemes \({\mathcal X}_j'\subset{\mathcal X}\). One may then apply a variety of techniques (Lefschetz-Riemann-Roch theorems, resolvent theory, the theory of bi-extensions, crystalline cohomology, etc.) to compute the \(\chi^P({\mathcal F}_j')\). These techniques often become more effective for particular \({\mathcal X}_j'\), e.g. those which are fibral or of low dimension. In carrying out the calculations, it may be very useful to employ Chern character identities. Euler characteristic; Galois module structure; generalization of Taylor's theorem; arithmetic schemes of arbitrary dimension; class group invariant; deRham cohomology; \(\varepsilon\)-factors Chinburg, T.; Pappas, G.; Taylor, M. J.: {\(\epsilon\)}-constants and the Galois structure of de Rham cohomology. II. J. reine angew. Math. 519, 201-230 (2000) Integral representations related to algebraic numbers; Galois module structure of rings of integers, de Rham cohomology and algebraic geometry, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Arithmetic varieties and schemes; Arakelov theory; heights \(\varepsilon\)-constants and the Galois structure of de Rham cohomology. 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 paper under review deals with algebraic curves \(C\) of genus 2 over a field \(k\) of characteristic not 2 or 3, such that the subgroup \(J[3](k)\) of theirs Jacobians \(J\) contains a subgroup \(\Sigma(k)\) of order 9. The authors fully describe the genus 2 curves having a subgroup \(\Sigma(k)\subset J[3](k)\) of order 9 on which the Weil pairing is trivial, and phrase the question of classification of such curves in terms of partial level structure on principally polarized abelian surfaces. Furthermore, they determine a genus 2 curve \(C_{\mathrm{rst}}\) over \(k(r,s,t)\) such that a sufficiently general curve of the above type can be obtained by specializing \(r\), \(s\), \(t\). Moreover, various other interesting results about these curves are proved. Jacobians; Shafarevich-Tate group; class groups; isogenies; Kummer surfaces; genus 2 curves Bruin, Nils; Flynn, E. Victor; Testa, Damiano, Descent via \((3,3)\)-isogeny on Jacobians of genus 2 curves, Acta Arith., 165, 3, 201-223, (2014) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties Descent via \((3,3)\)-isogeny on Jacobians of genus 2 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 results of this paper constitute the author's Harvard doctoral thesis. The author constructs a compactification of the moduli space of pairs \((C,E)\) where \(C\) is a curve representing a point of \(M_g\) and \(E\) is a Mumford semistable vector bundle on \(C\) of rank \(r\) and degree \(e\). For \(g\geq 2\) let \(U_g(e,r)\) be the set of equivalence classes of pairs \((C,E)\) as above. The author constructs a projective variety \(\overline{U_g(e,r)}\) which parametrizes equivalence classes of slope-semistable torsion free sheaves on stable curves representing points of \(\overline{M_g}\).
\(U_g(e,r)\) corresponds functorially to an open dense subset of \(\overline{M_g}\) . There is a natural projection \(\eta\colon \overline{U_g(e,r)}\to\overline{M_g}\) satisfying the obvious properties to get a moduli space. In particular if \(C\in M_g\) then \(\eta^{-1}([C])\simeq U_C(e,r)/ \Aut(C)\) where \(U_C(e,r)\) is the well known compact moduli space of semistable vector bundles of degree \(e\) and rank \(r\) over \(C\).
The construction of \(\overline{U_g(e,r)}\) requires a lot of technical steps which are described with full details in the paper and we only sketch them here.
\textit{D. Gieseker} constructed in his book: ``Lectures on moduli of curves'' (1982; Zbl 0534.14012) a quasi-projective variety \(H_g\) as the closed subscheme of \(10\)-canonical stable curves of a suitable Hilbert scheme of curves in \(\mathbb{P}^{N+1}\) with the property that the G.I.T.-quotient (G.I.T. = geometric invariant theory) \(H_g/ SL_{N+1}\) is isomorphic to \(\overline{M_g}\).
Let \(U\) be the universal curve over \(H_g\). The author is inspired by the usual construction of \(U_g(e,r)\) over a fixed curve \(C\) and generalizes it in the relative setting, by replacing \(C\) with \(U\). He defines a relative Quot scheme parametrizing suitable quotients of \(\mathbb{C}^n\otimes{\mathcal O}_U\). The technical heart of the paper is the study of the action of \(SL_{N+1}\times SL_n\) over this Quot scheme according to the G.I.T. -- \(\overline{U_g(e,r)}\) is defined as the G.I.T. quotient of the Quot scheme above.
When \(r=1\), \(\overline{U_g(e,1)}\) is shown to be isomorphic to the compactification of the universal Picard variety constructed by \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589-660 (1994; Zbl 0827.14014)].
This paper is very well written and can be used also to learn a lot of techniques involving moduli problems and G.I.T. compactification of moduli space; slope-semistable vector bundles; Mumford semistable vector bundle Pandharipande, R., A compactification over \(\overline{M}_g\) of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc., 9, 2, 425-471, (1996) Algebraic moduli problems, moduli of vector bundles, Families, moduli, classification: algebraic theory, Geometric invariant theory A compactification over \(\bar M_ g\) of the universal moduli space of slope-semistable vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper was inspired by Kashiwara-Schapira's analogous theory over the field of complex numbers \(\mathbb{C}\) and various conjectures of Deligne. Using Belinson's singular supports, the author first constructs the characteristic cycles for constructible étale sheaves.
Let \(X\) be a smooth and projective variety of dimension \(d\) over a perfect field of characteristic \(p\), and \(\mathcal F\) a constructible étale sheaf on \(X\). The singular support \(SS(\mathcal{F})\) of \(\mathcal{F}\), defined by \textit{A. Beilinson} [Sel. Math., New Ser. 22, No. 4, 1797--1819 (2016; Zbl 1375.14058)], is a closed subscheme of \(T^*X\), the cotangent bundle of \(X\). All its irreducible components \(C_1, \dots, C_k\) are of dimension \(d\). The characteristic cycle \(CC(\mathcal{F}) \) of \(\mathcal{F}\) is a cycle supported on \(SS(\mathcal{F})\):
\[
CC(\mathcal {F}) = \sum_{i=1}^{n}n_iC_i
\]
where \(n_i\in \mathbb{Z}\). The coefficients \(n_i\) are defined using a generalization of Milnor's formula for the number of vanishing cycles.
Its main interest is that it can be used to compute the Euler-Poincaré characteristic of \(\mathcal{F}\) by the following index formula:
\[
\chi(X, F) = (CC(\mathcal{F}), T^*_XX)_{T^*X}
\]
where \(T^*_XX\) is the zero section. This provides a generalization to higher dimension of the Grothendieck-Ogg-Shafarevich formula.
The main ingredients of this paper are: Radon and Legendra transforms after Brylinski, geometric theory of Lefschetz pensils, ramifications theory for imperfect residue fields and Deligne's theory of vanishing cycles over general bases. characteristic cycles; singular supports; index formula; Euler-Poincaré characteristic; characteristic class Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Sheaves in algebraic geometry The characteristic cycle and the singular support of a constructible sheaf | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the Witt group of a general real projective curve is explicitly computed in terms of topological and geometrical invariants of the curve. The authors method is strongly inspired by \textit{R. Sujatha}'s [Math. Ann. 288, 89--101 (1990; Zbl 0692.14012)] computation of the Witt group of a smooth projective real surface and uses a comparison theorem between the graded Witt group and the étale cohomology groups. The second part of the paper concentrates on the torsion subgroup of the Picard group of a smooth geometrically connected (not necessarily complete) curve \(X\) over a real closed field \(R\). Let \(C\) be the algebraic closure of \(R\) and \(X_C: =X\times_{\text{Spec}\,R}\) Spec \(C\). We compute \(\text{Pic}_{\text{tors}}(X)\) and \(\text{Pic}_{\text{tors}} (X_C)\) using the Kummer exact sequence for étale cohomology. These computations depend on a new invariant \(\eta(X)\in\mathbb{N}\) (resp. \(\eta(X_C))\) which we introduce in this note. The author studies relations between \(\eta(X)\), \(\eta(X_C)\) and the level and Pythagoras number of curves using new results of \textit{J. Huisman} and \textit{L. Mahé} [J. Algebra 239, 647--674 (2001; Zbl 1049.14015)]. The last part is devoted to the study of smooth affine conics and hyperelliptic curves. For such curves we calculate the Witt group and the torsion Picard group determining the invariant \(\eta\). étale cohomology; Pythagoras number Monnier, J. -P.: Witt group and torsion Picard group of real curves. J. pure appl. Algebra 169, 267-293 (2002) Picard groups, Topology of real algebraic varieties, Algebraic theory of quadratic forms; Witt groups and rings, Real algebraic sets Witt group and torsion Picard group of real 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 celebrated theorem of Gross-Zagier [\textit{B.~Gross} and \textit{D.~Zagier}, ``Heegner points and derivatives of \(L\)-series'', Invent. Math. 84, 225--320 (1986; Zbl 0608.14019)] gives a formula relating certain linear combinations of Heegner points on integral models of the modular curve \(X_0(N)\) to Fourier coefficents of Eisenstein series that appear in the integral representation of an associated Rankin-Selberg \(L\)-function at its central value. A series of papers of \textit{S.~S.~Kudla}, e.g. [``Central derivatives of Eisenstein series and height pairings'', Ann. Math. (2) 146, No. 3, 545--646 (1997; Zbl 0990.11032)] and [``Integrals of Borcherds forms'', Compos. Math. 137, No. 3, 293--349 (2003; Zbl 1046.11027)], propose to extend such a formula to the general setting of Shimura varieties and Eisenstein series on higher rank groups. The authors give some compelling evidence for a special case of this conjecture, namely in the setting of Hilbert modular surfaces, where a moduli interpretation is available. In particular, by developing some techniques in deformation theory (e.g. the theory of displays due to \textit{T.~Zink} [``The display of a formal \(p\)-divisible group'', Astérisque 278, 127--248 (2002; Zbl 1008.14008)]), the authors are able to perform the calculations required to prove some special cases of Kudla's proposed conjecture, building on previous works of \textit{J.~H.~Bruinier} and second author ([``CM values of Hilbert modular functions'', Invent. Math. 163, No. 2, 229--288 (2006; Zbl 1093.11041)], [\textit{T.~Yang}, ``An arithmetic intersection formula on Hilbert modular surfaces'', Am. J. Math. 132, No. 5, 1275--1309 (2010; Zbl 1206.14049)], and [\textit{T.~Yang}, ``Arithmetic intersection on a Hilbert modular surface and the Faltings height'' (preprint)]).
Roughly, the authors prove the following type of result. Let \(F\) be a real quadratic field, with \(\sigma \in \text{Gal}(F/{\mathbb{Q}})\) the nontrivial automorphism of its Galois group over \({\mathbb{Q}}\). Associated to \(F\) is a Hilbert modular surface \(M\), which can be defined as the moduli space of all \(\mathcal{O}_F\)-polarized RM abelian surfaces \(\mathcal{A}\). Given a positive integer \(m\), Hirzebruch and Zagier constructed a divisor on \(M({\mathbb{C}})\), which was later given a moduli theoretic interpretation by Kudla-Rapoport [\textit{S.~S.~Kudla} and \textit{M.~Rapoport}, ``Arithmetic Hirzebruch-Zagier cycles'', J. Reine Angew. Math. 515, 155--244 (1999; Zbl 1048.11048)]. To describe this latter interpretation, let \(T(m)\) denote the moduli space of pairs \((\mathcal{A}, j)\), where \(\mathcal{A}\) is an \(\mathcal{O}_F\)-polarized RM abelian surface, and \(j\) is a Rosati fixed endomorphism satisfying \(j \circ j = m\) and \(j \circ x = x^{\sigma} \circ j\) for all \(x \in \mathcal{O}_F\). The ``forget-\(j\)'' morphism \(T(m) \rightarrow M\) has image given by a codimension one cycle whose complex fibre is equal to the divisor constructed by Hirzebruch and Zagier. The moduli space \(T(m)\) can then justifiably be referred to as a {\textit{Hirzebrich-Zagier divisor}}. The main aim of the work is to relate the intersection multiplicities of three such divisors to the Fourier coefficients of the central derivative of a certain Siegel Eisenstein series of genus three.
To be somewhat more precise, the authors establish the following main result (Theorem E). Let \(E\) be a quartic CM field containing \(F\) as a real quadratic subfield. Let us suppose that \(E\) is either Galois over \({\mathbb{Q}}\), with Galois group \(\text{Gal}(E/{\mathbb{Q}}) \cong {\mathbb{Z}}/4{\mathbb{Z}}\), or else that \(E\) is not Galois over \({\mathbb{Q}}\). Given a CM type \(\Sigma\) of \(E\), let us write \(E_{\Sigma}\) to denote its reflex field, with \(\mathcal{O}_{\Sigma}\) the ring of integers of \(E_{\Sigma}\). One can associate to \(E\) a {\textit{reflex algebra \(E^{\sharp}\)}} of \(E\), which is the \({\mathbb{Q}}\)-algebra defined up to isomorphism by the existence of an \(\text{Aut}({\mathbb{C}}/{\mathbb{Q}})\)-equivariant bijection
\[
\{ \text{CM types of \(E\)}\} \longrightarrow \text{Hom}(E^{\sharp}, {\mathbb{C}}), ~~~ \Sigma \longmapsto \phi_{\Sigma}
\]
for which \(\phi_{\Sigma}(E^{\sharp}) = E_{\Sigma}\). Let \(F^{\sharp}\) denote the maximal totally real subalgebra of \(E^{\sharp}\). In the setting described above, \(E^{\sharp}\) is a certain quartic CM field, with \(F^{\sharp}\) its maximal totally real subfield. Anyhow, one can associate to the pair \((E, \Sigma)\) an algebraic stack \(CM_{\Sigma}\), defined as the moduli space of principally polarized abelian varieties over \(\mathcal{O}_{\Sigma}\)-schemes with CM by \(\mathcal{O}_{\Sigma}\), satisfying a certain technical condition (the so-called Kottwitz condition of \(\S 3.2\)). The forgetful morphism \(CM_{\Sigma} \longrightarrow M_{\mathcal{O}_{\Sigma}}\) contains a codimension two cycle on \(M_{\mathcal{O}_{\Sigma}}\). To a given pair \(T(m)\) and \(CM_{\Sigma}\), one can associate a finite intersection pairing \(\langle T(m): CM_{\Sigma} \rangle_{\text{fin}}\), which is essentially the sum of lengths of all local rings in the intersection \(T(m) \bigcap CM_{\Sigma}\) (\(\S 5.4\)). Following the constructions of Kudla [loc. cit.] and Bruinier [\textit{J.~H.~Bruinier}, ``Borcherds products and Chern classes of Hirzebruch-Zagier divisors'', Invent. Math. 138, No. 1, 51--83 (1999; Zbl 1011.11027)], one can also define for any integer \(m\) an associated Green function \(G(m, v, \cdot)\) on \(M({\mathbb{C}})\), where \(v \in {\mathbb{R}}\) is some fixed parameter. Taking the (finite) sum over all complex points of \(CM_{\Sigma}\) then gives rise to a function \(G(m, v, CM_{\Sigma})\) on \(M({\mathbb{C}})\). One can then define the intersection of the formal pair \(\widehat{T}(m, v) := (T(m), G(m, v, \cdot))\) and \(CM_{\Sigma}\) to be the sum
\[
\langle \widehat{T}(m, v): CM_{\Sigma} \rangle_{\text{fin}} = \langle T(m): CM_{\Sigma} \rangle_{\text{fin}} + \frac{1}{2}G(m, v, CM_{\Sigma}).
\]
Now, turning to the automorphic side, the authors define an {\textit{\(\mathcal{O}_F\)-polarized CM module \({\mathbb{T}}\)}} to be a projective \(\mathcal{O}_E\)-module of rank one equipped with a perfect \({\mathbb{Z}}\)-valued symplectic form (satisfying some compatibility condition with respect to the \(\mathcal{O}_E\)-action). To such a module \({\mathbb{T}}\), there is an associated CM-type \(\Sigma\). The set \(X_{\Sigma}\) of isomorphism classes of such modules \({\mathbb{T}}\) with associated CM type \(\Sigma\) is finite. To each \({\mathbb{T}} \in X_{\Sigma}\), one can attach a quadratic space \(\mathcal{C}({\mathbb{T}})\) of rank two over the adele ring \({\mathbb{A}}_{F^{}\sharp}\), which is incoherent in the sense that it does not arise from any quadratic space over \(F^{\sharp}\). Using the Weil representation, one can associate to this quadratic space \(\mathcal{C}({\mathbb{T}})\) a Hilbert modular Eisenstein series \(E(\tau, s, {\mathbb{T}})\) of parallel weight one on \(\text{GL}_2({\mathbb{A}}_{F^{\sharp}})\). Here, \(\tau = u+iv\) denotes an element in the \(F^{\sharp}\) upper half space \(\mathfrak{H}_{F^{\sharp}}\), which can be identified with two copies on the usual upper-half space \(\mathfrak{H} = \{ z \in {\mathbb{C}}: \Im(z) > 0 \}\). One can then define
\[
E(s, \tau, \Sigma) = \sum_{ {\mathbb{T}} \in X_{\Sigma}} E(\tau, s, {\mathbb{T}})
\]
to denote the finite sum over modules \({\mathbb{T}} \in X_{\Sigma}\) of such Eisenstein series, which in the setting described above turns out not to depend on the choice of \(\Sigma\). The incoherence of the underlying quadratic space \(\mathcal{C}({\mathbb{T}})\) implies that \(E(\tau, s, \Sigma)\) vanishes at \(s=0\), whence the derivative at \(s=0\) is given by the sum
\[
E'(\tau, 0, \Sigma) = \sum_{\alpha \in F^{\sharp}} c_{\Sigma}(\alpha, v) q^{\alpha}
\]
for some parameter \(v \in {\mathbb{R}}\). Here, \(q^{\alpha} = e(\text{Tr}_{F^{\sharp}/{\mathbb{Q}}} (\alpha \tau))\), where \(e(x) = \exp(2\pi i x )\). Writing \(\iota: \mathfrak{H} \longrightarrow \mathfrak{H}_{F^{\sharp}}\) to denote the diagonal embedding, the pullback of \(E'(\tau, 0, \Sigma)\) to the complex upper-half space \(\mathfrak{H}\) is a non-holomorphic modular form of weight two, with Fourier series expansion
\[
E'(\iota(\tau), 0, \Sigma) = \sum_{m \in {\mathbb{Z}}} b_{\Sigma}(m, v) q^m,
\]
where the coefficients \(b_{\Sigma}(m, v)\) are given by the formula
\[
b_{\Sigma}(m,v) =
\sum _{\substack{ \alpha \in F^{\sharp} \\
\text{Tr}_{ F^\sharp/{\mathbb{Q}}^{(\alpha)}} =m }}
c_{\Sigma}(\alpha, v).
\]
The authors prove, under some mild technical hypothesis on the ramification of \(E\) (Hypothesis B), that for any parameter \(v \in {\mathbb{R}}\) and any nonzero integer \(m\),
\[
\langle \widehat{T}(m, v): CM_{\Sigma} \rangle = -\frac{1}{W_E} \cdot b_{\sigma}(m, v),
\]
where \(W_E\) denotes the number of roots of unity in \(E^{\times}\).
The first chapter of the monograph explains the main results. The second chapter introduces the notion of \(\mathfrak{c}\)-polarized RM and CM modules for \(\mathfrak{c} \supset \mathcal{O}_F\) a fractional \(\mathcal{O}_F\)-ideal, then describes how to construct quadratic forms from some associated spaces of endomorphisms. The third chapter defines \(\mathfrak{c}\)-polarized RM and CM abelian surfaces. The fourth chapter gives a construction of the Eisenstein series \(E(\tau, s, {\mathbb{T}})\). The fifth chapter describes the main technical results in more detail. The sixth and final chapter makes up the technical core of the work, developing techniques from the theory of Dieudonné modules, the Grothendieck-Messing deformation theory of \(p\)-divisible groups and Zink's theory of displays to perform various local calculations required for the proofs.
Though formidably technical, the reviewer recommends this beautiful monograph to anyone interested in the circle of conjecture proposed by Kudla et al., particularly from the point of view of arithmetic geometry. The work contains many useful references and intricate proofs that do not appear elsewhere, and is likely to be extremely useful to future progress in the area. Howard, Benjamin; Yang, Tonghai, Intersections of Hirzebruch-Zagier divisors and CM cycles, Lecture Notes in Mathematics 2041, viii+140 pp., (2012), Springer, Heidelberg Arithmetic aspects of modular and Shimura varieties, Research exposition (monographs, survey articles) pertaining to number theory, Complex multiplication and moduli of abelian varieties, Arithmetic varieties and schemes; Arakelov theory; heights Intersections of Hirzebruch--Zagier divisors and CM cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective variety defined over an algebraically closed field of characteristic \(p>0\). Let \(F\) denote the absolute Frobenius morphism on \(X\). A \textit{stratified bundle} on \(X\) is a sequence of locally free sheaves \(\{E_n\} _{n\in {\mathbb N}}\) together with isomorphisms \(\sigma _n: F^*E_{n+1}\to E_n\). \textit{D. Gieseker} [Ann. Sc. Norm. Super. Pisa 2, 1--31 (1975; Zbl 0322.14009)] conjectured that if the étale fundamental group of \(X\) vanishes then \(X\) has only trivial stratified bundles. The main aim of the paper under review is proof of this conjecture.
In characteristic zero the above conjecture is analogous to the fact that if étale fundamental group of \(X\) vanishes then a pro-algebraic completion of \(X\) also vanishes (this was known due to Grothendieck and, independently, Malcev).
Roughly speaking, the proof follows from application of \textit{E. Hrushovski}'s theorem on periodic points [\url{arXiv:math/0406514}]) to a Verschiebung morphism on the moduli space of semistable vector bundles on \(X\) (see the reviewer's paper in [Ann. Math. 159, 251--276 (2004; Zbl 1080.14014)]). fundamental group; stratified bundle; Verschiebung Esnault, H.; Mehta, V., Simply connected projective manifolds in characteristic \(p > 0\) have no nontrivial stratified bundles, Invent. Math., 181, 449-465, (2010) Positive characteristic ground fields in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Simply connected projective manifolds in characteristic \(p>0\) have no nontrivial stratified 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 An algebraic curve (smooth, projective, geometrically connected) is bielliptic if it may be presented as a double cover of some elliptic curve. It is an important problem to determine all those modular curves \(X\) (e.g., of shape \(X = X_0(N)\), \(X_1(N)\), \(X(N)\) with some natural number \(N\)) which are bielliptic or hyperelliptic, since that question is related to the question of whether \(X\) admits infinitely many points rational over a quadratic extension of \(\mathbb{Q}\). In the context just described, the problem has been solved for \(X = X_0(N)\) by \textit{F. Bars} [J. Number Theory 76, 154-165 (1999; Zbl 0964.11029)], following earlier work of \textit{A. P. Ogg} [Bull. Soc. Math. Fr. 102, 449-462 (1974; Zbl 0314.10018)]. In view of the familiar analogy between number fields and function fields, it is desirable to dispose of similar results for Drinfeld modular curves. In the important paper under consideration, the author manages, applying a variety of different strategies and ad hoc arguments, to describe all the bielliptic ones among the Drinfeld modular curves \(X_0(u)\) of Hecke type. These are defined over a rational function field \(K = \mathbb{F}_q(T)\), and \(u\) refers to a monic element of the polynomial ring \(A = \mathbb{F}_q[T]\). Theorem 4.6 states that, up to coordinate changes in \(T\), there are precisely 12 bielliptic curvs \(X_0(u)\), among which 6 are at the same time hyperelliptic. Together with his former results about the classification of hyperelliptic Drinfeld modular curves given in [Drinfeld modules, modular schemes and applications, Alden-Biesen 1996, World Scientific Publishing, 330-343 (1997; Zbl 0930.11039)], the author is able to list all the curves \(X_0(u)\) with infinitely many quadratic points over \(K\): see Theorem 5.3.
As an application, the following uniform boundedness theorem for Drinfeld \(A\)-modules is obtained, which sharpens a result of B. Poonen. Theorem 4: Given an irreducible \(\mathfrak p \in A = \mathbb{F}_q[T]\) and a finite field extension \(L\) of \(K\), there exists a uniform bound, depending only on \(q\), \(\mathfrak p\), and \(L\), for the size of the \(\mathfrak p\)-primary part of the \(L'\)-rational torsion of \(\phi\), where \(L'\) ranges over the quadratic extensions of \(L\) and \(\phi\) over the Drinfeld \(A\)-modules of rank two defined over \(L'\). Drinfeld modular curves; bielliptic curves; torsion of Drinfeld modules; uniform boundedness Schweizer, A.: Bielliptic Drinfeld modular curves. Asian J. Math. 5, 705--720 (2001) Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Rational points, Special algebraic curves and curves of low genus, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Modular and Shimura varieties, Global ground fields in algebraic geometry Bielliptic Drinfeld 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 \(F : \mathbb{C}^n \to \mathbb{C}^n\) be a morphism of complex affine spaces, i.e. in appropiate coordinate systems, \(F\) is given by \(n\) polynomials \(Y_i = F_i (X_1, \ldots, X_n) \in \mathbb{C}^n [X] = \mathbb{C}^n [X_1, \ldots X_n]\) of \(n\) indeterminates. Let \(F'(X) = \det J F (X)\), where \(JF(X)\) is the Jacobi matrix of \(F\).
Jacobian conjecture: JC(n) Let \(F : \mathbb{C}^n \to \mathbb{C}^n\) be a morphism such that \(F'(x) \neq 0\) for all \(x \in \mathbb{C}^n\) (equivalently, \(F'(X) \in \mathbb{C}^*\)). Then \(F\) is an isomorphism, i.e. \(F\) has an inverse which is given also by polynomials.
A morphism \(N = (N_1, \ldots, N_n) : \mathbb{C}^n \to \mathbb{C}^n\) is called {nilpotent}, if its Jacobi matrix \(JN(X)\) is a nilpotent matrix in \(\mathcal{M}(n, \mathbb{C}[X])\).
Nilpotence conjecture: JN(n) A nilpotent morphism \(N : \mathbb{C}^n \to \mathbb{C}^n\) has at most one fixed point.
In this paper, the author proves the following interesting result:
Theorem 2. The Jacobian conjecture JC\((n)\) and the nilpotence conjecture JN\((n)\) are equivalent, i.e. JC\((n)\) is true for all \(n\) iff JN\((n)\) is true for all \(n\). Finally, some particular cases for JN\((n)\) are presented. affine spaces; automorphisms; Jacobian conjecture; nilpotence conjecture V.S. Kulikov, The Jacobian conjecture and nilpotent mappings, in \(Complex Analysis in Modern Mathematics\) (FAZIS, Moscow, 2001 in Russian), pp. 167-179. Eng. Trans. J. Math. Sci. 106, 3312-3319 (2001) Jacobian problem, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables The Jacobian conjecture and nilpotent mappings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 following result is proven in this paper. If the generic point of an irreducible subvariety of the moduli space of complex algebraic curves of genus \(g \geq 2\) classifies a curve whose Jacobian possesses nontrivial endomorphisms, then this subvariety has codimension \(\geq g-1\).
Moreover, one obtains the same bound on the codimension when one restricts to moduli of hyperelliptic curves. The bound turns out to be sharp, and one can explicitly describe the situations where equality arises.
The results given here compare quite nicely to Shimura's analogous theorems on moduli of principally polarized abelian varieties: 30 years ago he proved the same bounds on the codimension in that context and moreover found compatible descriptions of all cases where the bound is attained.
The proof is based on a description of the universal deformation of a pair \((C,e)\), in which \(C\) is a curve and \(e\) is an injection of a given field \(\neq \mathbb Q\) into the endomorphism algebra of the Jacobian of \(C\). bound on codimension; moduli of hyperelliptic curves; universal deformation; endomorphism algebra of the Jacobian Ciliberto, C., van der Geer, G., Teixidor i Bigas, M.: On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms. J. Algebr. Geom. 1, 215--229 (1992) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Low codimension problems in algebraic geometry, Rational and birational maps, Algebraic moduli of abelian varieties, classification On the number of parameters of curves whose Jacobians possess nontrivial 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 Let \(S\) be a Noetherian scheme. The qfh topology for \(S\) is generated by the finite families of finite morphisms such that their sum is a universal topological epimorphism. This paper compares the qhf and étale torsors over \(S\) for certain group schemes. \textit{V. Voevodsky} [Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)] showed that for each \(n\geq 0\) there are isomorphisms \(H_{et}^{n}( S,A) \cong H_{qfh}^{n}( S,A) \) for abelian schemes \(A\) which are locally constant for the étale topology.
In the work under review, a non-abelian version of this result is obtained in the case \(n=1\). Specifically, it is shown that for \(G\) a finite étale group scheme over \(S\) that the canonical map \(H_{et}^{1}( S,G) \to H_{qfh}^{1}( S,G) \) is a bijection. The main result is equivalent to the statement that the qfh stack completion map \( BG\to BSt_{qfh}G\) is a local weak equivalence in the étale topology. First it is established that finite constant simplicial sheaves with finitely many simplices of each degree satisfy a qfh descent condition on strict local Hensel rings and that the inverse image of a qfh sheaf over such a ring can be computed using filtered colimits. By restricting to the case where \(G\) is constant and finite (which can be done since the problem is a local one in the étale topology) the author quickly proves the main result. finite group schemes; torsors; qfh topology; étale topology J.F. Jardine, Finite group torsors for the qfh topology, preprint, 2000. Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Group schemes Finite group torsors for the qfh 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 Let C be a smooth curve over \({\mathbb{C}}\). We say that C is f-gonal if there exists a surjective morphism \(\phi:C\to {\mathbb{P}}^ 1\) of degree f. We assume C is of genus \(g\geq 2\) and that a base point \(P_ 0\in C\) is fixed. For a positive integer d a morphism I(d) from \(C^{(d)}\) to the Jacobian variety J(C) is defined by \(D\mapsto the\quad linear\quad equivalence\quad class\quad [D-P_ 0].\) For \(r\geq 0\), we put \(W^ r_ d=\{x\in J(C)| \dim(I(d)^{-1}(x))\geq r\},\) which is a Zariski closed subset of J(C). It is easy to see that C is d-gonal if and only if there exists a point in \(W^ 1_ d\) which is not contained in \(W^ 1_{d-1}+W^ 0_ 1\). Now we assume \(f\geq 3\) and that there exists no covering \(\phi:C\to \tilde C\) such that \(1<\deg(\phi)<f.\) Under this assumption the author gives sufficient conditions, concerning \(\dim(W^ r_ d)\) for some r, d, for C to be f-gonal, and he also proves that C is d-gonal for each \(d\geq g-f+2.\) These are the main results in the paper. In the last part of the paper some refinements of the above are tried in some examples. f-gonal smooth curve; Jacobian variety Coppens, M.R.M.: Some sufficient conditions for the gonality of a smooth curve. J. Pure Appl. Alg.30, 5--21 (1983) Jacobians, Prym varieties Some sufficient conditions for the gonality of a smooth curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0628.00007.]
From the introduction: \textit{M. Cornalba} and \textit{P. Griffiths} [Invent. Math. 28, 1-106 (1975; Zbl 0293.32026)] posed the following problem: Let \(X\subset {\mathbb{C}}^ 3\) be a smooth algebraic curve. Do there exist two holomorphic functions f, g of finite order on \({\mathbb{C}}^ 3\) such that X is the complete intersection of the surfaces \(\{f=0\}\) and \(\{g=0\}\), i.e., f and g generate the ideal of X? A holomorphic function \(f:\quad {\mathbb{C}}^ n\to {\mathbb{C}}\) is said to be of finite order, if there exists a polynomial P with real coefficients such that \(| f(z)| \leq e^{P(\| z\|)}\) for all \(z\in {\mathbb{C}}^ n\). By J. P. Serre one knows that it is not always possible to find two polynomials with this property. In fact, it follows from the solution of the Serre conjecture (given by D. Quillen and A. A. Suslin) that X is an algebraic complete intersection if and only if the canonical bundle of X is algebraically trivial (the same is true more generally for two-codimensional algebraic submanifolds of \({\mathbb{C}}^ n)\). On the other hand, one knows that any smooth analytic curve X in a Stein manifold M of dimension \(\leq 3\) is analytically a complete intersection.
The purpose of the present paper is to solve the problem of Cornalba- Griffiths. In fact we prove a more general theorem: Let \(X\subset {\mathbb{C}}^ n\) be an algebraic submanifold of pure codimension two such that the canonical bundle of X is topologically trivial. Then the ideal of X is generated by two entire functions of finite order. Note that the condition on the canonical bundle is necessary, since the normal bundle of every complete intersection is trivial. If X is a curve, this condition is automatically fulfilled, since on an open Riemann surface every holomorphic vector bundle is analytically (a fortiori topologically) trivial.
The proof uses analytic and algebraic methods. As an analytic tool we prove an extension and division theorem with growth conditions. - With the help of this theorem the problem is algebraically reduced to the application of a theorem of Quillen-Suslin on projective modules over a polynomial ring B[T] and in this application B is the ring of certain functions of finite order. topologically trivial canonical bundle; complete holomorphic functions of finite order; complete intersection; Serre conjecture Forster, Otto; Ohsawa, Takeo, Complete intersections with growth conditions.Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, 91-104, (1987), North-Holland, Amsterdam Complete intersections, Special algebraic curves and curves of low genus, Holomorphic functions of several complex variables Complete intersections with growth conditions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This work is a wonderful survey that reveals the author's genius in dealing with such relevant topics and still capable of establishing, in a natural and attractive way, a historical connection, both from a topological and geometric point of view, in the correlation of these topics with the Intersection Homology, that have been obtained in the context of singular varieties. More specifically, this article recalls the the first geometric definition as well as the theoretical sheaf definition allowing to describe the main properties of the intersection homology, going from the classical results in the manifold case to singular varieties. Furthermore, in the course of the work, the main tools in the frameworks of sheaf theory and derived categories are very well explained, in particular, to the definition of intersection homology, the local calculus eventually leads to ``sheafify'' the original geometric approach, thus obtaining the intersection sheaf complexes. In this context, the author highlights that the Deligne sheaf complex is of fundamental importance, the Deligne construction uses two tools: the ``pushing'' attaching property and the ``truncating'' operation. Also, in this article it is shown how several important concepts and results carry over from the usual (co-) homology of manifolds to intersection homology of singular varieties. The author reserved the last section to provide some applications of intersection homology, for example concerning toric varieties or asymptotic sets, since, according to the author, the main application and source, itself, of innumerable applications is the fascinating and fruitful topic of perverse sheaves, which unfortunately it is not possible to develop in such a survey. intersection homology; Poincaré duality; derived category Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Intersection homology and cohomology in algebraic topology, Singularities in algebraic geometry, Poincaré duality spaces, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Algebraic and analytic properties of mappings on manifolds, Derived categories, triangulated categories Intersection homology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We investigate local to global properties for commensurability in Mordell-Weil groups of abelian varieties and tori via reduction maps.
In more detail, local to global properties for detecting linear relations in Mordell-Weil groups
of abelian varieties and tori have been investigated by numerous authors. Commensurability
questions in the Mordell-Weil groups have not yet been investigated in relation to reduction maps. In this paper we establish the relations between local to global detecting properties and local to global commensurability properties. We apply these results to Mordell-Weil groups of abelian varieties and tori. The structure of the paper is as follows. At the end of this introduction we define local to global commensurability properties. We also define notion of strong commensurability
in abelian groups with finite torsion. Then we define local to global properties for strong commensurability. In section 2 we investigate relations between local to global commensurability properties and local to global detecting properties. In section 3 we give examples of classes of abelian varieties and tori where the local to global strong commensurability property holds. In both cases we show examples of classes of abelian varieties and tori where the criterion fails.
As a corollary we obtain, in each case, four different Deligne 1-motives over a ring of integers, which become all equal to a torsion 1-motive, after base change and application of reduction map for almost all residue fields. In section 4 we give examples where one can check the strong commensurability in Mordell-Weil groups of abelian varieties and tori by finite number of reductions. commensurability; abelian variety; torus; reduction map; Mordell-Weil group Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties Commensurability in Mordell-Weil groups of abelian varieties and tori | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Brill-Noether theory governs the behavior of linear series on algebraic curves. In this paper the authors study Brill-Noether theory for Du Val curves of genus \(g\) which are suitably general singular plane curves of degree \(3g\) having multiplicity \(g\) at eight points in \(\mathbb P^2\) and multiplicity \(g-1\) at a further ninth point. The main result shows that a general pointed Du Val curve satisfies the pointed Brill-Noether dimension theorem. Moreover, the authors show that a general pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. As a consequence, such Du Val curves provide explicit examples of smooth pointed Brill-Noether general curves of arbitrary genus defined over \(\mathbb Q\). The authors also establish a similar result for \(2\)-pointed curves by using curves on elliptic ruled surfaces. Brill-Noether theory; Du Val curves; rational and ruled surfaces 10.1007/s00029-017-0329-3 Special divisors on curves (gonality, Brill-Noether theory), Rational and ruled surfaces Du Val curves and the pointed Brill-Noether 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 Over \(\mathcal{M}_{g,n}\), the moduli space of genus \(g\) curves with \(n\) marked points, let \(\mathcal{J}^d_{g,n}\) denote the universal Jacobian of degree \(d\) line bundles. In the article under review, the authors study the question of extending the moduli space \(\mathcal{J}^d_{g,n}\) to a family over \(\overline{\mathcal{M}}_{g,n}\), the Deligne-Mumford moduli space of stable \(n\)-pointed genus \(g\) curves.
In addressing this question, the authors extend earlier work of \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1--90 (1979; Zbl 0418.14019)]. The idea is to first construct an affine space of stability conditions for the universal stable pointed curve.
Among other results, the authors determine the structure of such stability spaces \(V^d_{g,n}\). For example, they prove that if \(g \geq 2\), \(n \geq 1\) and \(N = N(g,n)\) is the number of boundary divisors of \(\overline{\mathcal{M}}_{g,n}\), then the stability space \(V^{g-1}_{g,n}\) is isomorphic to \(\mathbb{R}^{N-1}\times \mathbb{R}^n\) as an affine space. They also prove that, under this isomorphism, the decomposition of \(V^{g-1}_{g,n}\) into stability polytopes is the product decomposition of \(\mathbb{R}^{N-1}\) and \(\mathbb{R}^n\) into integer translates of finitely many hyperplanes.
In terms of the extent to which the compactified Jacobians \(\mathcal{J}_{g,n}(\phi)\) depend on the given stability parameter \(\phi\), the authors prove that if \(\overline{\mathcal{M}}_{g,n}\) is of general type, then there exist nondegenerate stability parameters \(\phi_1\) and \(\phi_2\) for which the compactified Jacobians \(\mathcal{J}_{g,n}(\phi_1)\) and \(\mathcal{J}_{g,n}(\phi_2)\) are non-isomorphic as Deligne-Mumford stacks.
Finally, given a nondegenerate stability parameter \(\phi \in V^d_{g,n}\), together with an integer vector \((k;d_1,\dots,d_n)\) which satisfies the condition that \[ k(2-2g)+d_1+\dots+d_n = d, \] the authors describe the locus of indeterminacy of the corresponding rational map \[ \sigma_{k,\mathbf{d}} : \overline{\mathcal{M}}_{g,n} \dashrightarrow \overline{\mathcal{J}}_{g,n}(\phi) \text{.} \] In doing so, they address a question which was raised by \textit{S. Grushevsky} and \textit{D. Zakharov} [Duke Math. J. 163, No. 5, 953--982 (2014; Zbl 1302.14039)]. compactified Jacobian; universal Jacobian; stability polytopes Families, moduli of curves (algebraic), Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification, Picard groups The stability space of compactified universal 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 Denote by \(M_{g,n}\) the set of isomorphism classes of tuples \((X, x_1, \dots, x_n)\), where \(X\) is a compact Riemann surface of genus \(g \geq 0\), and \(x_1, \dots, x_n\) are \(n \geq 1\) distinct ordered points in \(X\). It is well-known, mainly from the classical work of L. Bers, that \(M_{g,n}\) is the set of \(\mathbb{C}\)-rational points of a quasi-projective variety defined over \(\mathbb{Q}\). Analytically, \(M_{g,n}\) is the quotient of a holomorphically convex bounded domain \(T_{g,n}\) in \(\mathbb{C}^{3g - 3 + n}\) (provided that \(3g - 3 + n > 0)\) modulo a certain group \(\Gamma_{g,n}\) of holomorphic automorphisms of \(T_{g,n}\), which acts properly discontinuously. The space \(T_{g,n}\) is called the generalized Teichmüller space, and \(\Gamma_{g,n}\) is usually called the generalized Teichmüller modular group.
In the present paper, the author gives an algebraic description of the (analytic) Teichmüller spaces \(T_{g,n}\) in the critical case when \(2g - 2 + n > 0\) (and \(\Gamma_{g,n}\) is an infinite group). Historically, the real-algebraic approach to the study of the moduli of compact Riemann surfaces goes back to R. Fricke, who (about 100 years ago) used characters of Fuchsian groups in order to classify Riemann surfaces from the viewpoint of uniformization theory. These classifying parameters, sometimes called the Fricke moduli, have been intensively studied over the past decades by numerous authors. The present paper provides some interesting and important refinements in the theory of Fricke moduli. The author describes the variety of representations of a group \(\Gamma\) in \(SL_2 (\mathbb{R})\), and its quotient variety by the adjoint action of \(PGL_2\), as schemes over \(\mathbb{Z}\), and he associates to them a certain abstract ring \(R (\Gamma)\) which is called the universal character ring of \(\Gamma\) over \(\mathbb{Z}\). In the case of \(\Gamma = \Gamma_{g,n}\) (the generalized Teichmüller modular group), a certain factor ring of the universal character ring \(R (\Gamma_{g,n})\) turns out to be closely related to the Teichmüller space \(T_{g,n}\). More precisely, \(T_{g,n}\) is described as a component of the real-algebraic variety of \(\mathbb{R}\)-valued points in the affine scheme of this factor ring \(R_{g,n}\), and the local ring of \(T_{g,n}\) at a point is explicitly established as a concrete subring of \(\mathbb{R}\), representable as a homomorphic image of \(R_{g,n}\). This beautiful real-algebraic description of \(T_{g,n}\) is carried out in full detail. Some related results can be found in the author's recent papers [cf. ``Moduli space for Fuchsian groups'' in: Algebraic analysis, Vol. 2, 735-787 (1989; Zbl 0674.30034), ``The Teichmüller space and a certain modular function from a view point of group representations'' in: Algebraic Geom. rel. top., Proc. Int. Symp., Inchoen, Korea 1992, Conf. Proc. Lect. Notes Algebr. Geom. 1, 41-88 (1993), and ``Representation variety of a finitely generated group into \(SL_2\) or \(GL_2\)'' Preprint, Res. Inst. Math. Sci. Tokyo, RIMS-985 (1993; Zbl 0837.32010)]. semi-algebraic sets; real algebraic varieties; rational points; generalized Teichmüller space; moduli of compact Riemann surfaces; Fricke moduli Kyoji Saito: Algebraic Representation of the Teichmuller Spaces, The Grothendieck Theor of Dessins dnfants, Edited by L. Schneps, London Math. Soc. Lee. Note Ser. 200, Cambridge Univ. Press, 1994. Families, moduli of curves (analytic), Compact Riemann surfaces and uniformization, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Algebraic representation of the Teichmüller 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 Let \(X\) be a smooth geometrically connected curve of genus \(g\geq2\) over a field \(k\). For a divisor \(e\) of degree \(1\) on \(X\), \textit{B. H. Gross} and \textit{C. Schoen} [Ann. Inst. Fourier 45, No. 3, 649--679 (1995; Zbl 0822.14015)] constructed a cohomologically trivial cycle \(\Delta_e\) of codimension \(2\) on the triple product \(X^3\). \textit{S.-W. Zhang} [Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031)] derived an explicit formula for \(\langle\Delta_e,\Delta_e\rangle\), which led to some important new results.
The present paper obtains a variant of Zhang's formula, based on showing that the contributions from archimedean places occur as the norms at infinity of a certain canonical isomorphism of line bundles.
In more detail, let \(S\) be a smooth quasiprojective variety over \(\mathbb C\). Let \(\pi:\mathcal X\to S\) be a smooth projective family of curves of genus \(g\geq2\) over \(S\), let \(e\) be a flat divisor of relative degree \(1\) on \(\mathcal X/S\), let \(\omega\) be the relative dualizing sheaf of \(\mathcal X/S\), let \(x_e\) be the divisor class (of relative degree \(0\)) given by \((2g-2)e-c_1(\omega)\) (note that this differs from Zhang's notation by a factor of \(2g-2\)), and let \(\Delta_e\) be the relative Gross-Schoen cycle on the triple fiber self-product of \(\mathcal X/S\). Let \(\langle\omega,\omega\rangle\) and \(\langle x_e,x_e\rangle\) be the Deligne self-pairings of \(\omega\) and \(x_e\), respectively; they are line bundles on \(S\). Also let \(\langle\Delta_e,\Delta_e\rangle\) be the Bloch pairing; it too is a line bundle on \(X\).
The variant of Zhang's result proved in this paper, then, is that there is an isomorphism
\[
\langle\Delta_e,\Delta_e\rangle^{\otimes(2g-2)} \overset\sim{} \langle\omega,\omega\rangle^{\otimes(2g+1)} \otimes\langle x_e,x_e\rangle^{\otimes(-3)}
\]
of line bundles on \(S\), canonical up to sign, and whose norms at archimedean places are related to the contributions \(\varphi(X_v)\) at those places in Zhang's formula.
In addition, if \(g\geq3\) then the paper calculates the value of \(\langle\Delta_e,\Delta_e\rangle\) in the Picard group \(\text{Pic}(\mathcal M_{g,1}^c)\).
The basic tools used in the paper are normal functions and biextensions associated to the cohomology of the universal Jacobian. Gross-Schoen cycle; Beilinson-Bloch height Arithmetic varieties and schemes; Arakelov theory; heights, Algebraic cycles, Fibrations, degenerations in algebraic geometry Normal functions and the height of Gross-Schoen cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(T=\mathbb{G}_{m}^{n}\) be an \(n\)-dimensional algebraic torus over \(\overline{\mathbb{Q}}\), where \(n\geq 2\). For \(0\leq r\leq n-1\), let \(\mathcal{H}_{\,r}\) denote the union of all \(r\)-dimensional algebraic subgroups of \(T\). Set \(\mathcal{H}=\bigcup_{\,r=0}^{\,n-1}\mathcal{H}_{\,r}\).
In [[1]: \textit{E. Bombieri, D. Masser} and \textit{U. Zannier}, ``Intersecting a curve with algebraic subgroups of multiplicative groups'', Int. Math. Res. Not. 1999, No. 20, 1119--1140 (1999; Zbl 0938.11031)], the authors studied the intersection of a given closed and irreducible curve \(C\subset T\) with \(\mathcal{H}\). In particular, they showed that if \(C\) is not contained in a translate of a proper subtorus of \(T\), then the set \(C\cap\mathcal{H}(\overline{\mathbb{Q}})\) has bounded (Weil) height. The paper under review is concerned with a generalization of this result to higher-dimensional subvarieties \(X\subset T\). To explain the sort of results actually proved in this paper, we go back to the works [[2]: \textit{E. Bombieri} and \textit{U. Zannier}, ``Algebraic points on subvarieties of \(\mathbb{G}_{m}^{n}\)'', Int. Math. Res. Not. 1995, No. 7, 333--347 (1995; Zbl 0848.11030)] and [[3]: \textit{U. Zannier}, ``Appendix'', in ``Polynomials with special regard to reducibility. With an Appendix by Umberto Zannier'' by A. Schinzel. Encyclopedia of Mathematics and its Applications 77. Cambridge University Press (2000; Zbl 0956.12001), pp. 517--539] by the first and third authors.
Let \(X^{\circ}\) denote the complement in \(X\) of the union of all subvarieties of \(X\) which are translates of nontrivial subtori of \(T\). Then \(X^{\circ}\) is a Zariski-open subset of \(X\) [2]. Further, if \(X\) is defined over \(\overline{\mathbb{Q}}\), then \(X^{\circ}\cap\mathcal H_{1}\) is a set of bounded height [3]. In the paper under review the authors introduce a new set \(X^{{\circ}a}\), analogous to \(X^{\circ}\) and contained in it if \(X\neq T\), and show that it is (Zariski) open in \(X\) (in fact, a sharper ``structure theorem'' for \(X^{{\circ}a}\) is obtained. See Theorem 1.4 of the paper). The set \(X^{{\circ}a}\) is defined as the complement in \(X\) of the union of all ``anomalous'' subvarieties of \(X\). A positive-dimensional irreducible subvariety \(Y\) of \(X\) is \textit{anomalous} if it lies in a translate \(K\) of an algebraic subgroup of \(T\) and its dimension is strictly larger than \(\text{dim}\, X+\text{dim}\,K-n\) (so \(X\) and \(K\) do \textit{not} meet properly since \(\text{dim}\,(X\cap K)\geq \text{dim}\, Y>\text{dim}\,X+\text{dim}\,K-n\)).
The authors also state the following \textit{Bounded Height Conjecture}. Let \(X\) be an irreducible subvariety of \(T\) of dimension \(r\). Then \(X^{{\circ}a}\cap\mathcal H_{\,n-r}\) is a set of bounded height. When \(X\) is a curve \(C\) not contained in a translate of dimension \(n-1\), then \(X^{{\circ}a}=C\) and the conjecture is true by the result on curves quoted above. On the other hand, if \(r=n-1\) (i.e., if \(X\) is a hypersurface in \(T\)) then \(X^{{\circ}a}=X^{\circ}\) and the conjecture is true by the result from [3] cited above. No other instances where the conjecture is true are known, but the authors have promised to settle the case of planes in \(T\) in a subsequent publication.
The second main result of the paper establishes the existence of a finite collection \(\Psi\) of translations \(S\) of tori by torsion points, satisfying \(\text{dim}(X\cap S)\geq \text{dim}\,S-1\), such that \(X\cap\mathcal H_{\,1}=\bigcup_{S\in\Psi}(X\cap S)\cap \mathcal H_{\,1}\). The significance of this result is that it reduces the problem of describing \(X\cap\mathcal H_{\,1}\) for general \(X\) to the hypesurface case since \(X\cap S\) may be regarded as a hypersurface in \(S\) and \(S\) is essentially \(\mathbb G_{m}^{d}\) for some \(d\). No analogous description of \(X\cap\,\mathcal H_{\,2}\) is known at present. The paper also contains a result (Theorem 1.6) on lacunary polynomials with algebraic coefficients which has implications for irreducibility questions. This result (which, in the interest of brevity, we do not state here) extends work of Schinzel and of the first and third authors in [3].
The paper also discusses a set \(X^{ta}\) which is obtained by removing from \(X\) all ``torsion-anomalous'' subvarieties of \(X\) (to define a torsion-anomalous subvariety, simply repeat the definition of ``anomalous'' above specializing \(K\) to a translate of the form \(gH\), where \(g\) is a torsion element of \(T\) and \(H\) is an algebraic subgroup of \(T\)). The following conjectures are discussed: (a) Let \(X\) be an irreducible subvariety of \(T\) defined over \(\mathbb C\). Then \(X^{ta}\) is Zariski-open in \(X\), and (b) if \(X\) (as in (a)) has dimension \(r\), then \(X^{ta}\cap\mathcal H_{n-r-1}\) is a finite set. The authors also discuss generalizations of the above conjectures to the case of semi-abelian varieties. Finally, the third author corrects an inaccuracy which appears in the proof of Theorem 2 in [3]. heights; tori; lacunary polynomials Bombieri, E., Masser, D., Zannier, U.: Anomalous subvarieties--structure theorems and applications. Int. Math. Res. Not. \textbf{2007}: Article ID rnm057 (2007) Heights, Arithmetic varieties and schemes; Arakelov theory; heights Anomalous subvarieties -- structure theorems 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 Consider a curve of genus \(g\geq 4\) that is a (g-1) or fewer sheeted cover of \({\mathbb{P}}^ 1({\mathbb{C}})\) with only one point of (total) ramification and otherwise generic. The ramification point on the curve is then an exceptional Weierstrass point and the corollary to the last theorem of the paper is that, on the contrary, all the other Weierstrass points have normal gap sequence \(1,...,g-1,g+1\). More results on Weierstrass points and on the dimensions of subvarieties W of the moduli space of curves of genus g, representing curves with Weierstrass points of certain types, are obtained.
The method is by making up a deformation theory for a system consisting of a curve C, a finite set of its points and a line bundle on the curve. The obstruction to extending a section of the line bundle supported by the marked points to a deformation is indicated by a ''Kodaira-Spencer'' cohomology class in \(H^ 1(C,{\mathcal O})\). Thus the tangent spaces to the subvarieties W can be calculated. ramification point; exceptional Weierstrass point; dimensions of subvarieties; deformation theory Diaz, S., Tangent spaces in moduli via deformations with applications to Weierstrass points, Duke Math. J., 51, 905-922, (1984) Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Formal methods and deformations in algebraic geometry Tangent spaces in moduli via deformations with applications to 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 In the paper under review, the authors study the inverse Jacobian problem for the case of Picard curves over \({\mathbb C.}\) More precisely, the authors provide an algorithm (see Section 3) accepting input and returning output as follows:
``Input: A period matrix \(\Omega\in \mathbf{H}_3\) of the Jacobian of a Picard curve \(C\), and the transposed rational representation \(N\in {\mathbb Z}^{6\times 6}\) of the automorphism of the Jacobian \(\rho_*\) induced by the curve automorphism \(\rho(x,y)=(x,z_3y).\)
Output: The complex values \(\lambda\) and \(\mu\) in a Legendre-Rosenhain equation \(y^3=x(x-1)(x-\lambda)(x-\mu)\) for the Picard curve \(C\),'' where \(\mathbf{H}_3\) is the Siegel upper half-space, and \(z_3\) is a primitive third root of unity.
The results follow in a great part from earlier work of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)], but these have been presented to refine, correct or generalize the results of Koike and Weng.
One of the key reasons that the inverse Jacobian problem can be done for Picard curves is the following proposition in Section 3 (cf. Lemma 1 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)]).
\textbf{Proposition 1.} Let \(X\) be a simple principally polarized abelian variety of dimension \(3\) defined over an algebraically closed field \(k\). If \(X\) has automorphism \(\phi\) of order \(3\), then we have that \(X\) is the Jacobian of a Picard curve. Furthermore, let \(\rho\) be the curve automorphism \(\rho(x,y)=(x,z_3y),\) and let \(\rho_*\) be the automorphism of the Jacobian that it induces. Then we get \(\langle\phi\rangle=\langle\rho_*\rangle.\)
The authors of the paper under review mentioned that their proof followed the idea of Koike and Weng, but they fixed a gap by their reference to a result of Estrada. Furthermore, they remarked that the result can be seen by applying the classification of plane quartics and genus-3 hyperelliptic curves by their automorphism group [\textit{D. Lombardo} et al., ``Decomposing Jacobians via Galois covers'', Preprint, \url{arXiv:2003.07774}].
The algorithm mentioned above was made possible by establishing the following two theorems which are refined statements or generalizations of the results in [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)].
\textbf{Theorem 3.} Let \(C\) be a Picard curve defined over \({\mathbb C}\) given by \[y^3=x(x-1)(x-\lambda)(x-\mu),\] and consider the branch points \(P_0=(0,0),P_1=(1,0),P_{\lambda}=(\lambda,0),P_{\mu}=(\mu,0),\) and \(P_{\infty}\) at infinity. Let \(J(C)\) be the Jacobian of \(C\) with period matrix \(\Omega\), let \(\alpha\) be the Abel-Jacobi map with base point \(P_{\infty},\) and let \(\Delta\in J(C)\) be the associated Riemann constant. Then, for \(\eta\in {\lambda,\mu},\) we have \[\eta=\varepsilon_{\eta}\left(\frac{\theta[\widetilde{P_1}+2\widetilde{P_{\eta}}-\widetilde{P_0}-\widetilde{\Delta}](\Omega)} {\theta[2\widetilde{P_1}+\widetilde{P_{\eta}}-\widetilde{P_0}-\widetilde{\Delta}](\Omega)}\right)^3,\] where \(\varepsilon_{\eta}=\exp(6\pi i((\widetilde{P_{\eta}}-\widetilde{P_1})_1(\widetilde{P_0})_2+\widetilde{\Delta_1}(3\widetilde{P_1}+3\widetilde{P_{\eta}}-2\widetilde{\Delta})_2)).\)
The proof of the above theorem (see Section 2) uses Riemann's Vanishing Theorem, a result of Siegel, and a transition step (see Corollary 1 for details and for other undefined notations) which converts a product of Riemann theta constants as a product of theta constants with characteristics. For \(P\in C\), \(\widetilde{P}\) is a shorthand notation for the image under Abel-Jacobi map with identification of \(J(C)\) with \({\mathbb R}^{2g}/{\mathbb Z}^{2g}\) and reduction (so \(\widetilde{P}\in [0,1)^{2g},\) see (6) for details). The formula for \(\eta\) (i.e. \(\lambda\) or \(\mu\)) is to be compared with that of Corollary 11 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)], where the correction factor \(\varepsilon_{\eta}=1,\) because the authors of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)] took a four-element subset \(\{D_1,\cdots,D_4\}\) (of which \(\widetilde{P_{\eta}},\widetilde{P_1}\), etc, are analogues) of a set analogous to \(\Theta_3\) in Theorem 4 below with \(\sum_i D_i=\mathbf{0}.\) The authors of the paper under review clarified the existence of such four-element subset and its relation to the image of branch points under the Abel-Jacobi map. Accordingly Theorem 4 below (see Section 3) gives a refinement, clarification, or generalization of Corollary 11 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)].
\textbf{Theorem 4.} Let \(J(C)\) be the Jacobian of a Picard curve \(C,\) let \(\rho_*\) be the automorphism of \(J(C)\) induced by the curve automorphism \(\rho(x,y)=(x,z_3y).\) Let \(\mathcal{B}\) be the set of affine branch points of \(C,\) let \(\alpha\) be the Abel-Jacobi map with base point \(P_{\infty}=(0:1:0),\) let \(\Delta\) be the Riemann constant with respect to \(\alpha\) and define \[\Theta_3:=\{x\in J(C)[1-\rho_*]:\theta[\underline{x}+\underline{\Delta}](\Omega)=0\}.\] Then \(\alpha(\mathcal{B})\) and \(-\alpha(\mathcal{B})\) are the only subsets \(\mathcal{T}\subset J(C)\) of four elements such that:
\noindent (i) the sum \(\sum_{x\in \mathcal{T}}x\) is zero,
\noindent (ii) \(\mathcal{T}\) is a set of generators of \(J(C)[1-\rho_*],\) and
\noindent (iii) the set \(\mathcal{O}(\mathcal{T}):=\{\sum_{x\in \mathcal{T}}a_xx:a\in {\mathbb Z}^4_{\geq 0},\sum_{x\in \mathcal{T}}a_x\leq 2\}\) satisfies \[\mathcal{O}(\mathcal{T})=\Theta_3.\]
In Section 4, the authors of the paper under review applied their revised algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most \(4\).
In the Appendix, the third named author of the paper under review applied the tools used in Section 2 to correct a sign in the generalization of Takase's formula [\textit{K. Takase}, Proc. Japan Acad., Ser. A 72, No. 7, 162--165 (1996; Zbl 0924.14016)] for the inverse Jacobian problem for hyperelliptic curves, given in [\textit{J. S. Balakrishnan} et al., LMS J. Comput. Math. 19A, 283--300 (2016; Zbl 1404.11085)]. Picard curve; hyperelliptic curves; genus 3; inverse Jacobian; explicit algorithm Arithmetic ground fields for curves, Complex multiplication and moduli of abelian varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Special algebraic curves and curves of low genus, Theta functions and abelian varieties, Computational aspects of algebraic curves An inverse Jacobian algorithm for Picard 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 G be a group and X be a G-scheme. The q-th equivariant K-group \(K_ q(G,X)\) of X is defined to be the q-th K-group associated with the exact category of G-vector-bundles on X. The aim of this paper is to prove the following Adams-Riemann-Roch formula for elements \(x\in K(G,X):=\oplus_{q\geq 0}K_ q(G,X):\) Let \(f:X\to Y\) be a projective G- morphism of complete intersection and let \(\psi^ j\) be the j-th Adams operation (a certain polynomial in exterior power operations). Then \(f_*(\psi^ j(\theta^ j(\check T_ f)^{-1}\cdot x))=\psi^ j(f_*(x))\). Here \(\theta^ j(\check T_ f)^{-1}\in K_ 0(G,X)\) denotes the Adams multiplier (only depending on j and f) and \(f_*: K(G,X)\to K(G,Y)\) denotes the so-called Lefschetz trace (a generalization of the Euler characteristic).
In the nonequivariant case this theorem is the central part in the proof of the general Grothendieck-Riemann-Roch theorem (see, e.g., \textit{C. Soulé} [Can. J. Math. 37, 488-550 (1985; Zbl 0575.14015)]). In order to define \(\psi^ j\), a natural \(\lambda\)-structure on K(G,X) is constructed. In the affine case this is done by generalizing \textit{H. L. Hiller}'s constructions [J. Pure Appl. Algebra 20, 241-266 (1981; Zbl 0471.18007)] to the equivariant situation and in the general case by means of a generalized Jouanolou trick [\textit{J. P. Jouanolou}, Lect. Notes Math. 341, 293-316 (1973; Zbl 0291.14006)]. The proof of the theorem is based on an equivariant version of the deformation to the normal cone and on an equivariant excess intersection formula. G-scheme; equivariant K-group; exact category of G-vector-bundles; Adams- Riemann-Roch formula; Adams operation; Adams multiplier; Lefschetz trace; deformation to the normal cone; equivariant excess intersection formula B. Köck, Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten \(K\)-Theorie , J. Reine Angew. Math. 421 (1991), 189-217. \(K\)-theory of schemes, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Das Adams-Riemann-Roch-Theorem. (The Adams-Riemann-Roch 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 Suppose \(D\) is an effective \(\mathbb Q\)-Cartier divisor on a variety \(M\). If the section ring of \(D\) is finitely generated, the space \(M(D):= \text{Proj}\bigoplus_{m\geq 0} H^0(M, \mathcal O(mD))\) is called the model of of \(M\) associated to \(D\). When \(M\) is a moduli space, \(M(D)\) often has a moduli-theoretic meaning as well. Analyzing \(M(D)\) as above for all effective divisors \(D\) on a moduli space \(M\) is called Mori's program or the log minimal model program (LMMP) for \(M\). For example, the LMMP has been studied intensively for the Deligne-Mumford moduli space \(\overline{\mathcal M}_g\) of stable curves (called the Hassett-Keel program), which provides abundant information about the geometry of \(\overline{\mathcal M}_g\) and new compactifications of \({\mathcal M}_g\). In the paper under review the author carries out Mori's program completely for symmetric divisors on the moduli space \(\overline{M}_{0,6}\) of stable six-pointed rational curves, where a divisor is symmetric in the sense that it is invariant under the permutations of the marked points, see Theorem 1.1 for more details. As an application, the author gives an alternative proof of Mori's program for \(\overline{\mathcal M}_2\), which was first completed by \textit{B. Hassett} [Prog. Math. 235, 169--192 (2005; Zbl 1094.14017)], see Theorem 1.2 for more details. moduli space; birational geometry; Mori's program; Hassett-Keel program Moon, H-B, Mori's program for \(\overline{\mathrm M}_{0,6}\) with symmetric divisors, Math. Nachr., 288, 824-836, (2015) Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic) Mori's program for \(\overline{M}_{0,6}\) with symmetric divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In Ann. Math., II. Ser. 136, No. 2, 361-428 (1992; Zbl 0788.14014), \textit{E. M. Friedlander} and \textit{H. B. Lawson} jun. introduced the notion of an effective algebraic cocycle on an algebraic variety \(X\) with values in a variety \(Y\), and developed a ``bivariant morphic cohomology theory'' based on such objects. The fundamental objects of the theory are simply families of algebraic cycles on \(Y\) parameterized by \(X\). More precisely they are defined as morphisms from \(X\) to the Chow varieties of \(r\)-cycles on \(Y\) and can be represented as cycles on the product \(X\times Y\) which are equidimensional over \(X\). Such cocycles form a topological abelian monoid, denoted \({\mathfrak Mor} (X, {\mathcal C}_r (Y))\), and the morphic cohomology groups are defined to be the homotopy groups of its group completion \({\mathfrak Mor} (X, {\mathcal Z}_r (Y))\). This stands in analogy with (and, in fact, recovers by letting \(X=\) a point) the homology groups introduced and studied by \textit{E. M. Friedlander} [Compos. Math. 77, No. 1, 55-93 (1991; Zbl 0754.14011)] and [\textit{H. B. Lawson jun}; Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006)] and elsewhere. When \(Y= \mathbb{A}^n\), the theory is of strict cohomology type. It has a natural cup product given by the pointwise join of cycles, and a natural transformation (of ring functors) to \(H^*(X; \mathbb{Z})\). The main point of this paper is to establish a duality theorem between algebraic cycles and algebraic cocycles. The fundamental result (theorem 5.3) states that if \(X\) and \(Y\) are smooth and projective, then the graphing map \({\mathfrak Mor} (X, {\mathcal Z}_r(Y)) \hookrightarrow {\mathcal Z}_{m+r} (X\times Y)\) which sends \(Y\)-valued cocycles on \(X\) to cycles on \(X\times Y\) is a homotopy equivalence. The proof makes use of a new Chow moving lemma for families [\textit{E. M. Friedlander} and \textit{H. B. Lawson}, ``Moving algebraic cycles of bounded degree'' (to appear)]. If \(X\) is a smooth projective variety of dimension \(n\), our duality map induces isomorphisms \(L^sH^k (X) \to L_{n-s} H_{2n-k} (X)\) for \(2s\leq k\) which carry over via natural transformations to the Poincaré duality isomorphism \(H^k(X; \mathbb{Z}) \to H_{2n-k} (X;\mathbb{Z})\). More generally, for smooth projective varieties \(X\) and \(Y\) the natural graphing homomorphism sending algebraic cocycles on \(X\) with values in \(Y\) to algebraic cycles on the product \(X\times Y\) is a weak homotopy equivalence. The main results have a wide variety of applications. Among these are the determination of the homotopy type of certain algebraic mapping complexes and a computation of the group of algebraic \(s\)-cocycles modulo algebraic equivalence on a smooth projective variety. morphic cohomology groups; duality theorem; homotopy equivalence Friedlander, E., Lawson, H.B.: Duality relating spaces of algebraic cocycles and cycles. Topology 36, 533--565 (1997) Algebraic cycles, Homotopy theory and fundamental groups in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Generalized (extraordinary) homology and cohomology theories in algebraic topology Duality relating spaces of algebraic cocycles and cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M_g\) be the coarse moduli space of smooth complex projective curves of genus \(g\geq 2\) and denote by \(A_g\) the moduli space of principally polarized complex abelian varieties of dimension \(g\). Torelli's theorem states that the map \(J: M_g\to A_g\) assigning to each curve its Jacobian variety is injective. If \(J_g:= J(M_g)\) denotes the so-called Jacobian locus in \(A_g\), then the classical, very long-standing Schottky problem (posed by \textit{F. Schottky} in 1888; JFM 20.0488.02; 1888; JFM 20.0489.01) asks for characterizing \(J_g\) inside \(A_g\) in algebraic, analytic, or purely geometric terms, i.e., for distinguishing Jacobians among principally polarized abelian varieties of dimension \(g\).
During the past century, several approaches to solving the Schottky problem have been proposed and studied [cf. \textit{O. Debarre}, The Schottky problem: an update, in: Current topics in complex algebraic geometry. New York, NY: Cambridge University Press. Math. Sci. Res. Inst. Publ. 28, 57--64 (1995; Zbl 0847.14015)], with particularly spectacular progress having been achieved since the early 1980s.
Recently, motivated by some earlier work of \textit{P. Buser} and \textit{P. Sarnak} [Invent. Math. 117, No. 1, 27--56 (1994; Zbl 0814.14033)], B. Farb suggested to study the Schottky problem from the viewpoint of ``large scale geometry'' [cf.: \textit{B. Farb}, in: Problems on mapping class groups and related topics. Providence, RI: American Mathematical Society (AMS). Proceedings of Symposia in Pure Mathematics 74, 11--55 (2006; Zbl 1191.57015)]. This approach is based on the fact that the moduli space \(A_g\) admits a so-called ``asymptotic cone'' denoted by \(\text{Cone}_\infty(A_g)\), which is defined as a certain Gromov-Hausdorff limit of rescaled pointed spaces [cf. \textit{M. Gromov}, in: Progr. Math. 152. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)], and Farb's ``coarse Schottky problem'' can be stated as follows:
Describe, as a subset of a \(g\)-dimensional Euclidean cone, the subset of \(\text{Cone}_\infty(A_g)\) determined by the Jacobian locus \(J_g\) in \(A_g\).
In the paper under review, the authors provide a solution to the coarse Schottky problem by proving (Theorem 1.1) that \(J_g\) is coarsely dense in \(A_g\). This implies that the subset of \(\text{Cone}_\infty(A_g)\) determined by the Jacobian locus \(J_g\) actually coincides with \(\text{Cone}_\infty(A_g)\).
Their proof also shows that the Jacobian locus of hyperelliptic curves is coarsely dense in \(A_g\) as well (Theorem 1.2.).
Finally the authors study the boundary of the Jacobian locus \(J_g\) in the Baily-Borel compactification \(\overline A^{BB}_g\) and in the Borel-Serre compactification \(\overline A^{BS}_g\) of \(A_g\). Their precise description of these boundaries (Theorem 1.3. and Corollary 1.4.) is supplemented by a result identifying the boundary of the topological closure \(J_g\) in \(A_g\) itself as a product space defined by reducible Jacobians of dimension \(g\).
In the course of these subtle investigations, the authors (re-)prove and use the following extension property (Proposition 1.5.) of the Torelli map: The map \(J: M_g\to A_g\) extends to an algebraic map from the Deligne-Mumford compactification \(\overline M^{DM}_g\) of \(M_g\) to the Baily-Borel compactification \(\overline A^{BB}_g\) of \(A_g\). Schottky problem; moduli of algebraic curves; moduli of abelian varieties; Torelli map; coarse geometry; asymptotic cones; compactified moduli spaces Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Discrete subgroups of Lie groups, Differential geometry of symmetric spaces, Algebraic moduli of abelian varieties, classification, Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces The asymptotic 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 \(X\) be a (smooth, connected, projective) algebraic curve of genus \(g\) over the finite field \({\mathbb F}_q\) with \(q\) elements. The Weil bound for the number of \({\mathbb F}_q\)-rational points of \(X\) is
\[
\#X({\mathbb F}_q) \leq q+1 + 2gq^{1/2}.
\]
Similarly, if \(X\) is replaced by a (smooth, connected, not necessarily projective) \(d\)-dimensional variety, the following WD bound (WD = Weil-Deligne) holds:
\[
\#X({\mathbb F}_{q^n}) \leq \text{WD}(X,{\mathbb F}_{q^n}) \leq \sum_{i\geq 0} q^{in/2} h^i(X),
\]
with equality on the right hand side if \(X\) is projective. Here, \(n\) is any natural number, \({\mathbb F}_{q^n}\) is the extension of degree \(n\) of \({\mathbb F}_q\), \(h^i(X)\) the \(i\)-th (compact supports) Betti number of \(X\),
\[
\text{WD}(X,{\mathbb F}_{q^n}) = \sum_{i,j} \alpha_{i,j},
\]
where \(\alpha_{i,j}\) \((1 \leq j \leq h^i(X))\) are the eigenvalues of the geometric Frobenius element \(\text{Frob}_q\) acting on the \(\ell\)-adic cohomology module \(H^i(X,{\mathbb Q}_{\ell})\) with compact supports. Put \(h(X) := \sum_{i} h^i (X)\). According to \textit{S. Vladut} and \textit{V. Drinfeld} [Funct. Anal. Appl. 17, 53--54 (1983); translation from Funkts. Anal. Prilozh. 17, No. 1, 68--69 (1983; Zbl 0522.14011)],
\[
\limsup\frac{\#X({\mathbb F}_{q^n})}{h(X)} \leq \frac{q^{n/2}-1}{2},
\]
if \(X\) varies among curves and \(h(X)\) tends to infinity. This is sharper than the estimate derived from the WD bound, and is in fact an equality if \(n\) is fixed and even. Families of curves realizing the upper bound (``asymptotically optimal families'') may be constructed as families of Drinfeld modular curves with certain level structures: cf. \textit{S. Vladut} and \textit{Yu. Manin} [J. Sov. Math. 30, 2611--2643 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 209--257 (1984; Zbl 0629.94013), \textit{E.-U. Gekeler}, J. Number Theory 90, No. 1, 166--183 (2001; Zbl 0979.11036)]. The author proposes a conjectural generalization (Conjecture 1.4) of these facts, where now \(X\) has dimension \(n-1\) and runs through certain modular varieties of Drinfeld modules of rank \(n\). (For \(n = 2\), one finds the aforementioned asymptotically optimal families of curves.) He is able (Theorems 1.6 and 1.7) to derive the conjecture from another natural though unproved assertion (Conjecture 1.5; it states that the cohomology of such \(n-1\) dimensional modular varieties is essentially concentrated in the middle dimension \(n-1\)), which provides strong evidence for its validity. Anyone interested in that type of questions is advised to read the author's excellent 6-page-introduction (and, of course, the rest of the paper), where he gives precise definitions (omitted in the present review) and displays the relations and possible implications between known and conjectured statements. rational points; Drinfeld modular varieties; Bruhat-Tits buildings Papikian, The number of rational points on Drinfeld modular varieties over finite fields Not Article ID, Int Math Res pp 94356-- (2006) Varieties over finite and local fields, Drinfel'd modules; higher-dimensional motives, etc., Finite ground fields in algebraic geometry, Modular and Shimura varieties The number of rational points on Drinfeld modular varieties 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 Let \(f: X\to Y\) be a generically smooth semistable curve of genus \(g\geq 2\) over a smooth projective curve \(Y\) over an algebraically closed field. Let \(\overline{K}\) be the algebraic closure of the function field of \(Y\) and \(C\) the generic fibre of \(f\). Let \(j: C(\overline{K})\to \text{Jac} (C)(\overline{K})\) be the Abel-Jacobi embedding with respect to the canonical line bundle of \(C\). Then Bogomolov's conjecture over function fields says that, if \(f\) is non-isotrivial, then the image of \(j\) is discrete in terms of the semi-norm \(\|\;\|_{NT}\) given by the Néron-Tate height pairing on \(\text{Jac} (C)(\overline{K})\). In another paper [see Compos. Math. 105, No. 2, 125-140 (1997)], \textit{A. Moriwaki} proved this conjecture under the assumption that the stable model of \(f:X\to Y\) has only geometrically irreducible fibres. In the present paper the author proves an effective version of Bogomolov's conjecture in the general case. To be more precise:
For any \(P\in \text{Jac} (C)(\overline{K})\) and \(r\geq 0\) set \(B_C (P,r):= \{x\in C(\overline{K}): \| j(x)- P\|_{NT}\leq r\}\) and \(r_C(P):= -\infty\) if \(B_C(P,0)\) is infinite and \(r_C(P):= \sup\{r\geq 0: \#(B_C (P,r))< \infty\}\) otherwise. Then the main result of the paper is that there is an effectively calculated positive number \(r_0\) with \(\inf\{r_C (P)\}\geq r_0\) where the infimum is taken over all \(P\in \text{Jac} (C)(\overline{K})\). Jacobian; function field; Abel-Jacobi embedding; Bogomolov's conjecture A. Moriwaki, Bogomolov conjecture for curves of genus 2 over function fields, J. Math. Kyoto Univ. 36 (1996), 687-695. Algebraic functions and function fields in algebraic geometry, Jacobians, Prym varieties, Arithmetic ground fields for curves Bogomolov conjecture for curves of genus 2 over function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(B\) be a smooth projective curve of genus \(\gamma\) over the complex numbers and let \(f:A\to B\) be a non-isotrivial semi-abelian scheme over \(B\) with projective generic fiber of relative dimension \(g\). Let \(U\subset B\) be the locus above which the fibers are projective, and let \(S=B-U\) (a finite set). Thus \(f:A_U\to U\) is abelian, and \(f:A\to B\) is the connected component of its Néron model. Denote by \(g_0\) the dimension of the fixed part of \(A\) and \(s=|S|\). Let \(e:B\to A\) be the identity section, and let \(W:=e^* \Omega_{A/B}\). Various authors have dealt with upper and lower bounds for the degree of \(W\).
\textit{G. Faltings} [Invent. Math. 73, 337-347 (1983; Zbl 0588.14025)], for example, shows that \(\deg(W)\leq g(3\gamma +s+1)\) while \textit{L. Moret-Bailly} [``Pinceaux de variétés abéliennes'', Astérisque 129 (1985; Zbl 0595.14032)] shows that \(\deg(W)\leq (g-g_0)(\gamma-1)\) in the case where \(A/B\) is smooth. In this paper, we improve a bit on Faltings, in the general case:
Theorem. Let \(f:A\to B\) be a non-isotrivial semi-abelian scheme of relative dimension \(g\) with projective generic fiber. Then \(\deg(W)\leq{(g-g_0) \over 2}(2\gamma -2+s)\), where \(g_0\) is the dimension of the fixed part and \(s\) is the number of non-projective fibers.
The method of proof is an easy extension of Moret-Bailly's (loc. cit.) -- The reason writing it down in full is because of the recently emerging connection with the ABC conjectures. moduli space; ABC conjectures Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Global ground fields in algebraic geometry ABC inequalities for some moduli spaces of log-general type | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper studies some explicit examples of curves \(C\) of genus 2 with multiplication in a principal order of a real quadratic number field of discriminants 5, 8 and 12. The method is to compute correspondences inducing an endomorphism of the Jacobian of the curve implementing a method due to \textit{P. B. van Wamelen} [in: Discovering mathematics with Magma. Reducing the abstract to the concrete. Berlin: Springer. 117--135 (2006; Zbl 1146.14033)]. The correspondences are of bidegree \((2,d)\) with \(d = 2,3, 4\) or 5. They are explicitly given by curves \(Z\) admitting 2 coverings onto \(C\). One of the examples works over the universal family of Hilbert modular surfaces of discriminant 5. The induced action of the correspondences on the spaces of holomorphic one-forms is determined which certifies the equations for the correspondences. Finally, an application to the dynamics of the corresponding Riemann surfaces is given. correspondences; genus-2 curves Kumar, Abhinav; Mukamel, Ronen E., Real multiplication through explicit correspondences, LMS J. Comput. Math., 19, suppl. A, 29-42, (2016) Jacobians, Prym varieties, Modular and Shimura varieties, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Real multiplication through explicit correspondences | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For part I see \textit{B. H. Gross} and \textit{D. B. Zagier}, Invent. Math. 84, 225--320 (1986; Zbl 0608.14019).]
Let \(X_ 0(N)\) be the modular curve of level \(N\) and \(J_ N\) its Jacobian, and denote by \(J^*_ N\) the Jacobian of \(X_ 0(N)/w_ N\) where \(w_ N\) is the Fricke involution. For a fundamental discriminant \(D<0\) and \(r\in {\mathbb Z}\) with \(D\equiv r^ 2 (4N)\) we let \(Y_{D,r}\) be the corresponding Heegner divisor in \(J_ N\) and \(Y^*_{D,r}\) be its image in \(J^*_ N\). Let \(f\) be a normalized newform in \(S_ 2(\Gamma_ 0(N))\) and \(L(f,s)\) be its \(L\)-series. Suppose that the root number of \(L(f,s)\) is \(-1\).
The main result of the paper then states that the subspace of \(J^*_ N({\mathbb Q})\otimes {\mathbb R}\) generated by the \(f\)-eigencomponents of all Heegner divisors \((y^*_{D,r})_ f\) with \((D,2N)=1\) has dimension \(1\) if \(L'(f,1)\neq 0\).
More precisely, \((y^*_{D,r})_ f=c((r^ 2- D)/4N,r)y_ f\), where \(c(n,r)\) is the coefficient of \(e^{2\pi i(n\tau +rz)}\) in a Jacobi form \(\phi_ f\) of weight 2 and index \(N\) corresponding to \(f\) in the sense of \textit{N.-P. Skoruppa} and \textit{D. Zagier} [Invent. Math. 94, 113--146 (1988; Zbl 0651.10020)] and \(y_ f\in (J^*({\mathbb Q})\otimes {\mathbb R})_ f\) is independent of \(D\) and \(r\) with \(\langle y_ f,y_ f\rangle =L'(f,1)/4\pi \| \phi_ f\|^ 2\) \((\langle\cdot, \cdot\rangle\)=canonical height pairing).
This result is in accordance with the conjectures of Birch and Swinnerton-Dyer which in the above situation (i.e. under the assumption \(\text{ord}_{s=1}L(f,s)=1)\) would predict that \(\dim (J^*_ N({\mathbb Q})\otimes {\mathbb R})_ f=1\). derivatives of \(L\)-series; modular curve; Jacobian; Heegner divisor; conjectures of Birch and Swinnerton-Dyer Gross B., Kohnen W. and Zagier D., Heegner points and derivatives of \textit{L}-series. II, Math. Ann. 278 (1987), no. 1-4, 497-562. Arithmetic ground fields for curves, Jacobi forms, Arithmetic aspects of modular and Shimura varieties, Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Heegner points and derivatives of \(L\)-series. 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 author studies the ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining and Hilbert schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review the author proves various results on the Koszul cohomology (such as duality theorem, vanishing theorems, Lefschetz theorems etc.) thus developing algebraic techniques for describing the equations defining the image of a complex manifold under an embedding in a projective space. As an application, the author describes the structure of a minimal free resolution of the ideal of the embedding of a smooth curve C by a complete linear system of divisors of a sufficiently large degree. Another application concerns the so-called Arbarello-Sernesi module AS(X,L): if X is a compact complex manifold and L is an analytic line bundle over X, then \(AS(X,L)=\oplus_{q\in {\mathbb{Z}}}H^ 0(X,K_ X\otimes L^ q)\) viewed as a module over \(S(H^ 0(X,L))\) (where S denotes the symmetric algebra). The author shows that if \(| L|\) does not have fundamental points and maps X onto an n-dimensional variety \((n=\dim X)\), then, with a few exceptions, AS(X,L) is generated in degree \(\leq n-1\) and its relations are generated in degrees \(\leq n\) (this generalizes Petri's result for curves). Other applications include various local Torelli theorems. Koszul cohomology; duality theorem; vanishing theorems; Lefschetz theorems; image of a complex manifold; embedding; minimal free resolution of the ideal of the embedding of a smooth curve; Arbarello-Sernesi module; local Torelli theorems M. Green, Koszul cohomology and the cohomology of projective varieties, J. Differential Geom. 19 (1984), 125-171. (Co)homology theory in algebraic geometry, Analytic sheaves and cohomology groups, Complex manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Transcendental methods, Hodge theory (algebro-geometric aspects), Vanishing theorems Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline S\) be a curve, and \({\mathcal X}\to\overline S\) be a proper algebraic morphism smooth over \(S\subset\overline S\); then for each integer \(p\), \(S\) parametrizes the family of intermediate Jacobians
\[
J^p({\mathcal X}_s)=H^{2p-1} ({\mathcal X}_s,\mathbb{C})/(F^pH^{2p-1}({\mathcal X}_s) +H^{2p-1}({\mathcal X}_s, \mathbb{Z}))
\]
where \(F^\bullet\) is the Hodge filtration. This paper proposes an analytic compactification \(\overline{J^p}\to\overline S\) of the family \(J^p\to S\) having the meromorphic extension property: For any algebraic morphism \(Y@>\rho>> \overline S\) and any admissible (in the sense of variations of mixed Hodge structures) lifting \(\sigma:\rho^{-1}(S)\to J^p\) of \(\rho|_{\rho^{-1}(S)}\), there exists a projective birational morphism \(Y'@>\varphi>>Y\) such that \(\varphi\circ\sigma\) extends to a morphism from \(Y'\) to \(\overline{J^p}\).
The following application is given: Let \(X\) be projective of dimension \(2p\), \(c\) a primitive Hodge class of degree \(2p\) on \(X\) and \(X\to\mathbb{P}^1\) a Lefschetz pencil; then if the value \(\nu_c(t)\in J^p(X_t)\) of the associated normal function belongs to the image of the Abel-Jacobi map for \(t\) general in \(\mathbb{P}^1\), there exists a codimension-\(p\) algebraic cycle in \(\widetilde X\), the restriction of which is homologous to zero in \(X_t\) and which has a multiple of \(\nu_c\) as associated normal function. (Here \(\widetilde X\) ist the blow-up of \(X\) along the intersection of two general hyperplane sections of \(X)\). intermediate Jacobians; analytic compactification; variations of mixed Hodge structures M. Saito, ''Admissible normal functions,'' J. Algebraic Geom., vol. 5, iss. 2, pp. 235-276, 1996. Picard schemes, higher Jacobians, Variation of Hodge structures (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations Admissible normal functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper deals with logarithmic geometry (or logarithmic spaces) in the sense of \textit{K. Kato} [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. Throughout the paper, in which the author assumes that the reader is familiar with the basics of this conceptual framework [\textit{L. Illusie}, in: Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspect. Math. 15, 183--204 (1994; Zbl 0832.14015)], a log structure on a scheme \(X\) means a log structure on the étale site \(X_{\text{ét}}\) of \(X\) in the sense of Kato.
In this context, the purpose of the present paper is to introduce a stack-theoretic approach to Kato's theory of logarithmic structures. More precisely, for any fine log scheme \(S\) with underlying scheme \(\mathring S\), the author constructs a fibred category \(\text{Log}_S\to (\mathring S\)-schemes). His main result consists in the proof of the fundamental theorem stating that \(\text{Log}_S\) is an algebraic stack locally of finite presentation over the underlying scheme \(\mathring S\). Moreover, it is shown that a morphism of fine log schemes \(f:X\to S\) defines tautologically a morphism \(\text{Log}(f) : \text{Log}_X\to \text{Log}_S\) of algebraic stacks, and that the association \(S\mapsto \text{Log}_S\) defines a 2-functor from the category of log schemes to the 2-category of algebraic stacks. It is then explained how this 2-functor can be used to reinterpret and study some original basic notions in Kato's logarithmic geometry.
The fine analysis carried out in this paper is enhanced by an appendix, in which the author compares the notions of log structure in the fppf, étale, and Zariski topology, respectively. Part of this comparison is used in the course of the main body of the paper, and the rest is included for the sake of completeness.
As the author points out, his main theorem on the structure of the stack \(\text{Log}_S\) has further applications which are not discussed in the present paper. Namely, one can develop the theory of log crystalline cohomology using a theory of crystalline cohomology over stacks [cf. \textit{M. C. Olsson}, Crystalline cohomology of schemes over algebraic stacks, Preprint 2002], and also the deformation theory of log schemes can be analyzed using the structure of \(\text{Log}_S\). In addition, the main theorem of the present paper has a natural place in the moduli theory of fine log schemes [cf. \textit{M. C. Olsson}, Tohoku Math. J., II. Ser. 55, No.~3, 397--438 (2003; Zbl 1069.14015)]. The author intends to discuss these subjects more thoroughly in forthcoming papers. Grothendieck topologies; étale topologies; sites; algebraic spaces; log structures; fibred categories Olsson, M. C., Logarithmic geometry and algebraic stacks, Ann. Sci. Éc. Norm. Supér. (4), 36, 5, 747-791, (2003) Algebraic moduli problems, moduli of vector bundles, Generalizations (algebraic spaces, stacks), Schemes and morphisms, Local structure of morphisms in algebraic geometry: étale, flat, etc., Fibered categories Logarithmic geometry and 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 Let~\(f: \mathbb{R}^n\rightarrow \mathbb{R}\) be a polynomial map such that its first-order partial derivatives generate a zero-dimensional ideal. In an earlier paper [\textit{N. Dutertre}, J. Pure Appl. Algebra 139, No. 1-3, 41-60 (1999)], the author established, among other things, a formula for the Euler characteristic of the fiber~\(f^{-1}(0)\) in terms of the signature of a quadratic form, if~\(f\) is a proper map (loc. cit., theorem~5.2). In the present paper, this formula is generalized to not necessarily proper polynomial maps. The main ingredients of the proof are the global residue [see \textit{E. Becker, J. P. Cardinal, M.-F. Roy} and \textit{Z. Szafraniec} in: Algorithms in algebraic geometry and applications. Proc. MEGA-94 Conf., Santander 1994, Prog. Math. 143, 79-104 (1996; Zbl 0873.13013)] and Morse theory for manifolds with boundary [\textit{H. A. Hamm} and \textit{Lê Dũng Tráng}, J. Reine Angew. Math. 389, 157-189 (1988; Zbl 0646.14012)]. Euler characteristic; fiber of polynomial map; isolated singularity Dutertre, N.: Sur la fibre d'un polynôme de rn à points critiques isolés. Manuscripta math. 100, 437-454 (1999) Topology of real algebraic varieties, Topological properties in algebraic geometry, Singularities in algebraic geometry On the fiber of a polynomial with isolated critical 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 \(\overline{M}_{g, n}({\mathbb{P}}^r, d)\) be the Kontsevich--Manin coarse moduli space of stable maps of degree \(d\) from \(n\)-pointed genus \(g\) curves into the \(r\)-dimensional projective space \({\mathbb{P}}^r\). The authors construct \(\overline{M}_{g, n}({\mathbb{P}}^r, d)\) via geometric invariant theory. The method follows the one used in the case \(n=0\) in [\textit{D. Gieseker}, Lectures on moduli of curves. Lectures on Mathematics and Physics. Mathematics, 69. Tata Institute of Fundamental Research, Bombay. Berlin-Heidelberg-New York: Springer (1982; Zbl 0534.14012)], but the proof that the semistable set is nonempty is entirely different. The construction is only valid over \(\text{Spec}\,{\mathbb{C}}\), but a special case, a GIT presentation of the moduli space of stable curves of genus \(g\) with \(n\) marked points, is valid over \(\text{Spec}\,{\mathbb{Z}}\). moduli space; geometric invariant theory; stable curve Elizabeth Baldwin and David Swinarski, A geometric invariant theory construction of moduli spaces of stable maps, Int. Math. Res. Pap. IMRP 1 (2008), Art. ID rp. 004, 104. Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) A geometric invariant theory construction of moduli spaces of stable maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f:X \to Y\) be a continuous map between two locally compact topological spaces. Assume that a topological group \(G\) acts on \(X\) and on \(Y\) in such a way that \(f\) is a \(G\)-map. Then one can define the categories \(Sh_ G (X)\) and \(Sh_ G (Y)\) of \(G\)-equivariant sheaves. In the case when the group \(G\) is trivial one can define the derived categories \(D(X)\) and \(D(Y)\) of the abelian categories \(Sh(X)\) and \(Sh(Y)\) of sheaves on \(X\) and \(Y\) respectively, together with natural functors \(f^*\), \(f_ *\), \(f^ !\), \(f_ !\), \(\Hom\), \(\otimes\), etc. The main aim of the book under review is to define and study the triangulated categories \(D_ G (X)\) and \(D_ G (Y)\) together with similar functors and with a forgetful functor For: \(D_ G \to D\), satisfying certain natural compatibility conditions. Simple examples show that, unless the group \(G\) is discrete, the derived category of \(Sh_ G\) is not the right candidate for the triangulated category \(D_ G\). Therefore the theory developed here is very subtle. After supplying the necessary details for the construction of \(D_ G\), the authors give an algebraic description of the triangulated category \(D_ G(pt)\) when \(G\) is a connected Lie group, in terms of \(DG\)-modules over a natural \(DG\)- algebra \(A_ G\). This description is particularly important for the main applications of the theory. In the last part of the book the authors illustrate their theory with a computation of the equivariant intersection cohomology (with compact supports) of toric varieties. equivariant sheaves; derived categories; triangulated categories; equivariant intersection cohomology; toric varieties J. Bernstein and V. Lunts, \textit{Equivariant sheaves and functors. }Lecture Notes in Mathematics, 1578. Springer-Verlag, Berlin, 1994. Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to category theory, Toric varieties, Newton polyhedra, Okounkov bodies Equivariant sheaves and functors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 discusses the Brauer-Manin obstruction for integral points on hyperbolic smooth curves \(X\) defined over number fields \(k\). Let \(S\) be a finite subset of places of \(k\). By definition, Brauer-Manin suffices for \(X\) if the diagonal image of \(S\)-integral points of \(X\) (which is finite by Siegel's theorem) coincides with the subset of the product of local integral points outside \(S\) that are orthogonal to \(S\)-locally constant elements of the Brauer group.
The authors conjecture that Brauer-Manin suffices for any hyperbolic curves \(X\) contained in \(\mathbb{P}^{1}\). The main result of the paper gives several equivalent formulations of the conjecture, which concern about the adelic intersection in the generalized Jacobian of \(X\). In particular, if \(X\) is \(\mathbb{P}^{1}\) minus three points, the conjecture implies an old conjecture on ``exponential Diophantine equation'' raised by \textit{T. Skolem} [Avh. Norske Vid. Akad. Oslo 1937, No. 12, 1--16 (1937; Zbl 0017.24606, JFM 63.0889.03)].
Besides the main result, the authors also consider higher genus cases, they show that:
- If \(X\) is an open subset of a projective curve of genus at least \(2\), then the fact that Brauer-Manin suffices for \(X\) follows from the conjecture raised by \textit{V. Scharaschkin} [Local-global problems and the Brauer-Manin obstruction. Ph.D. thesis. University of Michigan (1999)] and \textit{A. Skorobogatov} [Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015)] saying that the Brauer-Manin obstruction is the only obstruction to weak approximation for rational points on smooth projective curves.
If \(X\) is an open subset of an elliptic curve with finite Mordell-Weil group and finite Tate-Shafarevich group, then Brauer-Manin suffices for \(X\). Here the authors give an example (an elliptic curve over \(\mathbb{Q}\) minus the infinity point) to show that the condition on finiteness of the Mordell-Weil group is necessary. integral points; Brauer-Manin obstruction; hyperbolic curves David Harari & José Felipe Voloch, The Brauer-Manin obstruction for integral points on curves, Math. Proc. Camb. Philos. Soc.149 (2010), p. 413-421 Varieties over global fields, Rational points The Brauer-Manin obstruction for integral points on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve of genus \(g\geqslant 2\) over an algebraically closed field \(k\) of characteristic \(p>0\), and \(F:X\to X^{(1)}\) the relative Frobenius morphism. Let \(\mathfrak{M}_X^s (r,d)\) (resp. \(\mathfrak{M}_X^{ss} (r,d)\)) be the moduli space of (resp. semi-)stable vector bundles of rank \(r\) and degree \(d\) on \(X\). We show that the set-theoretic map \(S_{\mathrm{Frob}}^{ss} :\mathfrak{M}_X^{ss} (r,d) \to \mathfrak{M}_{X^{(1)}}^{ss}(rp,d+r(p -1)(g-1))\) induced by \([\mathcal{E} ]\mapsto [F_\ast(\mathcal{E})]\) is a proper morphism. Moreover, the induced morphism \(S_{\mathrm{Frob}}^s :\mathfrak{M}_X^s (r,d) \to \mathfrak{M}_{X^{(1)}}^s (rp,d + r(p - 1)(g - 1))\) is a closed immersion. As an application, we obtain that the locus of moduli space \(\mathfrak{M}_{X^{(1)} }^s (p,d)\) consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety \(\mathrm{Jac}X\) of \(X\). Frobenius morphism; stable vector bundle; moduli space; stratification Li, L., The morphism induced by Frobenius push-forward, Sci. China Math., 57, 1, 61-67, (2014) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure The morphism induced by Frobenius push-forward | 0 |
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