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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 From the abstract: Let \(\mathcal M _m\) be the formal scheme which represents the functor of deformations of a one-dimensional formal module over \(\bar {\mathbb F}_p\) equipped with a level-\(m\)-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the \(\ell \)-adic étale cohomology of the generic fibre \(M_m\) of \(\mathcal M _m\) realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper the author shows without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces \(M_m\). local Langlands correspondence; Jacquet-Langlands correspondence; Lubin-Tate spaces; Drinfeld level structures; Lefschetz trace formula; rigid analytic geometry Strauch, M.: Deformation spaces of one-dimensional formal modules and their cohomology. Adv. Math. \textbf{217}(3), 889-951 (2008) Representations of Lie and linear algebraic groups over local fields, Linear algebraic groups over local fields and their integers, Modular and Shimura varieties, Representation-theoretic methods; automorphic representations over local and global fields Deformation spaces of one-dimensional formal modules and their cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians One perspective on the motive of an algebraic variety is to consider it as a collection of realizations, e.g. its Betti-, Hodge- and \(\ell\)-adic-realization. Equipped with the world of motives and étale cohomology is the formalism of Grothendieck's six functors for étale and \(\ell\)-adic sheaves. In his two groundbreaking œuvres [The Grothendieck six operations and the vanishing cycles formalism in the motivic world. I. Astérisque 314. Paris: Société Mathématique de France. (2007; Zbl 1146.14001)] and [Astérisque 315. Paris: Société Mathématique de France. (2007; Zbl 1153.14001)] the author has developed Grothendieck's six operations and the formalism of vanishing cycles in the stable motivic homotopy category of schemes. This leads to the natural question how this formalism behaves with respect to realizations in the world of stable motivic homotopy. The present paper provides a complete treatment of the case of Betti realization.
For a variety over a field of characteristic zero with an embedding into the field of complex numbers, the Betti realization is obtained by taking singular cohomology of the associated analytic space. The assignment of forming the associated analytic space can be extended to a functor from presheaves over smooth varieties to presheaves over smooth analytic spaces. Moreover it is possible to extend this functor to the stable motivic homotopy category. For this purpose, the author provides a thorough treatment of the homotopy category of presheaves over smooth analytic spaces in the first section. The Betti realization is then defined as the left derived functor of forming associated analytic spaces from the stable motivic homotopy category over a variety to the derived category of sheaves of abelian groups on the corresponding analytic objects. The main result states that Betti realization is compatible with Grothendieck's six operations and the nearby cycles functors. This is proven by constructing canonical isomorphisms between the respective composition of functors. This construction is given by providing general criteria for the compatibility of functors.
Despite the necessary level of abstract formalism the article is nicely written and provides detailed explanations with many interesting remarks at the sidelines. (The only obstacle for reading it is that the author assumes familiarity to a certain extent with the above cited fundamental articles and its notations and definitions. But this is just another impetus to study the work of the author.) motives; Betti realizations; Grothendieck's six operations; vanishing cycles formalism; model categories Ayoub, Joseph, Note sur les opérations de Grothendieck et la réalisation de Betti, J. Inst. Math. Jussieu, 9, 2, 225-263, (2010) Classical real and complex (co)homology in algebraic geometry, Motivic cohomology; motivic homotopy theory, Nonabelian homotopical algebra, Étale and other Grothendieck topologies and (co)homologies, Grothendieck topologies and Grothendieck topoi, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Resolutions; derived functors (category-theoretic aspects) Notes on Grothendieck's operations and Betti realizations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to generalise a classical argument for deducing algebraic models of Riemann surfaces from the
Riemann-Roch theorem. The method involves counting arguments based on equivariant resolutions. This paper starts by rehearsing a classical argument for the genus one surface with a set of divisors of the form \(nP\) for integer \(n\geq0\) to arrive at the cubic model [\textit{J. H. Silverman}, The arithmetic of elliptic curves. 2nd ed. New York, NY: Springer (2009; Zbl 1194.11005)]. The author then generalises this to the case of two-point divisors where a more complicated set of relations obtains, setting up an equivariant resolution of the coordinate ring in order to show that an explicit, finite set of quadratic and cubic relations generates all relations. The equivariant property in this case differs from that mentioned earlier in the introduction and corresponds to creation and annihilation of poles. Finally the author generalises the argument to two-point divisors on curves of the \((n, s)\) type for \(n\) and \(s\) coprime. equivariance; resolution; algebraic curve Jacobians, Prym varieties, Picard groups, Riemann surfaces; Weierstrass points; gap sequences, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Riemann-Roch theorems Equivariance and algebraic relations for 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 author in [``Nonabelian Jacobian of smooth projective varieties'', Science China, Vol. 56, No. 1, 1--42 (2013)] is a survey of the work done in [J. Differ. Geom. 74, No. 3, 425--505 (2006; Zbl 1106.14030)] as well as in [{I. Reider}, ``Nonabelian Jacobian of smooth projective surfaces and representation theory'', \url{arXiv:1103.4749}]. Central in this work is the construction of a nonabelian Jacobian \(J(X;L,d)\) of a smooth projective surface \(X\), a scheme over the Hilbert scheme \(X^{[d]}\) of subschemes of length \(d\) in \(X\), with a morphism to the stack of torsion free sheaves of rank 2 on \(X\) with determinant \(\mathcal{O}_X(L)\) and second Chern class \(d\). Among the various constructions covered in this survey, the existence of a sheaf of reductive Lie algebras on \(J(X;L,d)\) takes center stage. It originates from a well-chosen filtration on \(X^{[d]}\). This sheaf, which is the object of the second part of the paper, paves the way for the use of representation theoretic methods in the study of projective surfaces. There is some interesting work covered in the survey, though what makes it alluring are the possible applications. The seasoned algebraic geometer will no doubt appreciate the expansive coverage done in this work; worthy of further investigation are the connections with quantum gravity, homological mirror symmetry, the geometric Langlands program as well as quiver representations/Gromov-Witten invariants. Jacobian; Hilbert scheme; vector bundle; sheaf of reductive Lie algebras; Fano toric varieties; period maps; stratifications; Hodge-like structures; relative Higgs structures; perverse sheaves; Langlands program Reid, I, Nonabelian Jacobian of smooth projective surfaces -- a survey, Sci China Math, 56, 1-42, (2013) Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of orders, lattices, algebras over commutative rings, Variation of Hodge structures (algebro-geometric aspects), Geometric Langlands program (algebro-geometric aspects), Jacobians, Prym varieties Nonabelian Jacobian of smooth projective surfaces -- a survey | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal A_g\) be the reduction modulo a prime \(p\) of the moduli space of \(g\)-dimensional principally polarized abelian varieties. The author studies \(\ell\)-adic monodromies of subvarieties \(Z\) of \(\mathcal A_g\) which are stable under \(\ell\)-adic Hecke correspondences for any prime \(\ell\neq p\). He proves that if Hecke correspondences operate transitively on the set of irreducible components of \(Z\) and maximal points of \(Z\) are not supersingular, then the monodromy group is as large as possible and \(Z\) is irreducible. He also proves that the prime-to-\(p\) monodromy group is as large as possible under the same condition.
The proof uses the semisimplicity of the geometric monodromy group of a pure \(\mathbb Q_\ell\)-sheaf of a variety over a finite field due to Grothendieck and Deligne [\textit{P. Deligne}, Inst. Hautes Études Sci., Publ. Math. 52, 137--252 (1980; Zbl 0456.14014)]. The other part of proof is group-theoretic; he particularly shows that there are no subgroups of finite index in \(\text{Sp}_{2g}(\mathbb Q_\ell)\).
The results in this paper are useful to determine the irreducibility of subvarieties which are defined by certain properties of the underlying \(p\)-divisible groups and not contained in the supersingular locus. The method can be applied to some more general moduli spaces of PEL-type. Siegel moduli space; Hecke correspondence; monodromy C. Chai, ''Monodromy of Hecke-invariant subvarieties,'' Pure Appl. Math. Q., vol. 1, iss. 2, pp. 291-303, 2005. Complex multiplication and moduli of abelian varieties, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Abelian varieties of dimension \(> 1\), Algebraic moduli of abelian varieties, classification Monodromy of Hecke-invariant subvarieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this survey article the author shows how Serre's conjecture implies - via Frey's elliptic curves - Fermat's Last Theorem. (We remark that example 5.7 does not seem to be correct: The elliptic curve
\[
y^2 = x^3-6x^2+11x+3
\]
has rank 2 over \({\mathbb Q}\) with generators
\[
P_1 = (1,3)\quad \text{ and }\quad P_2 = (3,3).
\]
One obtains
\[
\begin{aligned} 2P_1 &= (37/9,-109/27),\\ 3P_1 &= (1177/196, 22833/2744),\\ 4P_1 &= (13381/106929,-72376055/1560896)\end{aligned}
\]
and
\[
P_1+P_2 = (2,-3).
\]
Since elliptic curves play an important role in this connection, an elementary and rather long introduction to the corresponding theory is given here.
A more detailed presentation of the theory of elliptic curves can be found in Silverman's first book [\textit{J. H. Silverman}, The arithmetic of elliptic curves. New York: Springer (1986; Zbl 0585.14026)] and for a detailed computational introduction into this field we refer to the book of \textit{S. Schmitt} and \textit{H. G. Zimmer} [Elliptic curves - a computational approach. de Gruyter Studies in Mathematics 31 (2003; Zbl 1195.11078)].
For Tate curves, besides \textit{P. Roquette}'s book [Analytic theory of elliptic functions over local fields. Göttingen: Vandenhoeck und Ruprecht (1970; Zbl 0194.52002)] the second book of \textit{J. H. Silverman} [Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151. New York: Springer (1994; Zbl 0911.14015)] appears to be a more suitable reference than the first one (see [43]).
Serre's conjecture asserts that any continuous irreducible odd representation
\[
\rho: G_{\mathbb Q} \to \text{ GL}_2({\mathbb F}_p)
\]
is modular, where \(G_{\mathbb Q}\) denotes the Galois group of the algebraic closure \(\overline{\mathbb Q}\) of the field \({\mathbb Q}\) of rationals, \({\mathbb F}_p\) is the finite field of prime characteristic \(p\), \(\overline{\mathbb F}_p\) its algebraic closure and \(\text{ GL}_2( \overline{\mathbb F}_p)\) the group of \(2\times2\)-matrices of determinant \(\not= 0\) over \(\overline{\mathbb F}_p\). We mention also that \(\rho\) is said to be \textit{odd} if \(\det(\rho(c)) = -1\), where \(c\) is the complex conjugation. Of course, Fermat's Last Theorem is the statement that the equation
\[
x^n + y^n = z^n,
\]
for positive integers \(n>2\), has no non-trivial solution.
For proofs the reader is referred to the literature (but it is occasionally not so easy to find them in textbooks). On the other hand it is impossible to include all proofs in an article of 32 pages especially since the not at all easy proof of Fermat's Last Theorem, which was finally carried out successfully only recently by Wiles and Taylor, is very long. elliptic curves; representations; modular forms; ramification; Ramanujan estimate Elliptic curves over global fields, Galois representations, Elliptic curves Elliptic curves, Serre's conjecture and Fermat's Last 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 Generalizing \textit{R. MacPherson} and \textit{K. Vilonen}'s method [Comment. Math. Helv. 63, No. 1, 89--102 (1988; Zbl 0638.32014)] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by \textit{O. Neto} and \textit{P. C. Silva} [C. R., Math., Acad. Sci. Paris 340, No. 2, 141--146 (2005; Zbl 1082.57005)]. The main result is an equivalence of categories between the category of perverse sheaves on \(\mathbf{C}^2\) stratified with respect to a singular plane curve and the category of \(n\)-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case. algebraic link; fundamental group; local systems; perverse sheaves; plane curve singularity Analytic sheaves and cohomology groups, (Co)homology theory in algebraic geometry, Curves in algebraic geometry Perverse sheaves on \(\mathbf{C}^2\) without vanishing cycles at the origin along a general plane curve with singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In an earlier paper Friedlander introduced and studied \(L_ rH_{2r+i}(X,\mathbb{Z}_ \ell)\), the ``Lawson homology'' groups associated to a projective variety \(X\) defined over an algebraically closed field of arbitrary characteristic \(p\neq\ell\), by using the étale homotopy in order to get algebraic invariants that are directly inspired from recent remarkable geometric-analytic ideas of Lawson arising from the study of algebraic cycles on \(X\) [see \textit{E. Friedlander}, Bull. Am. Math. Soc., New Ser. 20, No. 1, 49-53 (1989; Zbl 0696.14010) and \textit{H. B. Lawson} jun., Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006)]. In the paper under review, the author continues this program by generalizing Friedlander-Lawson's theory to quasi-projective varieties. Roughly speaking the idea is to use compactifications together with a relative version of Lawson homology. Lawson homology; étale homotopy; algebraic cycles; quasi-projective varieties Lima-Filho P.: Lawson homology for quasiprojective varieties. Compos. Math 84, 1--23 (1992) Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Algebraic cycles Lawson homology for quasiprojective 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 \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] proved the following decomposition theorem of the de~Rham complex:
Let \(X \to S\) be a smooth morphism of schemes of characteristic \(p > 0\). Assume that \(S\) lifts to a \(\mathbb{Z}/p^{2}\mathbb{Z}\)-scheme \(T\), and \(X\) admits a smooth lift over \(T\). Then \(\tau_{<p}F_{\ast}\Omega_{X/S}^{\bullet}\) is decomposable in the derived category of \(X'\).
Here, \(F\) is the relative Frobenius, \(X'\) is the Frobenius twist of \(X\), and \(\tau_{<p}\) is the ``canonical filtration''. We refer the reader to Professor Ekedahl's zbMATH review for more information.
The article under review generalizes the above theorem to formal schemes. The main theorem says that one can replace the every word ``scheme'' in Deligne-Illusie's theorem by ``locally noetherian formal scheme'', replace the de~Rham complex by its completed version, and the decomposition theorem is still valid, i.e., \(\tau_{<p}F_{\ast}\Omega_{X/S}^{\bullet}\) is decomposable.
To see that the authors' generalization is desirable, consider a singular projective variety \(Z\), embedded in \(\mathbb{P}^{r}_{k}\), \(k\) a perfect field. Then the formal completion of \(\mathbb{P}^{r}_{k}\) along \(Z\) is a smooth formal scheme over \(k\) which is liftable to \(W_{2}(k)\), and the authors' theorem can be applied.
``The strategy of the proof is similar to the classical method by Deligne and Illusie, but all the results of smoothness, deformation, and cohomology are needed in the setting of pseudo-finite maps of formal schemes.'' These foundational results have been previously explored by the authors.
The authors ``intend to apply the Decomposition Theorem to obtain vanishing theorems for formal schemes, with an eye towards the cohomology of singular varieties.'' formal scheme; de Rham complex de Rham cohomology and algebraic geometry, Formal neighborhoods in algebraic geometry, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry On the decomposition of the De Rham complex on formal 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 G and \(G'\) be two commutative algebraic groups, which are defined over the field \({\bar {\mathbb{Q}}}\) of algebraic numbers, and of positive dimension. Let \(\phi\) : G\({}'({\mathbb{C}})\to G({\mathbb{C}})\) be an analytic homomorphism, which is defined over \({\bar {\mathbb{Q}}}\) (this means that the induced map \(d\phi\) : \(T_{G'}({\mathbb{C}})\to T_ G({\mathbb{C}})\) on the tangent spaces is compatible with the \({\bar {\mathbb{Q}}}\) structure). If \(\phi (G')({\bar {\mathbb{Q}}})\neq 0\), then there exists an algebraic subgroup H of G, defined over \({\bar {\mathbb{Q}}}\) and of positive dimension, which is contained in \(\phi (G')\). This result extends earlier statements (Schneider, Lang, Baker, Masser,...), and has many important corollaries; some of these corollaries have been already published separately by the author: [J. Reine Angew. Math. 354, 164-174 (1984; Zbl 0543.10025); Invent. Math. 78, 381-391 (1984; Zbl 0584.10022)]. Further connected results are quoted in \textit{D. Bertrand} [Sémin. Bourbaki 1985/86, Exp. 652, Astérisque 145/146, 21-44 (1987; Zbl 0613.14001)]. Apart from Baker's method (which had already been developed on abelian varieties, mainly by Masser, Coates and Lang), the new element in the proof is a multiplicity estimate on group varieties (see the preceding review). analytic subgroups; algebraic points; commutative algebraic groups; multiplicity estimate; group varieties Wüstholz, G. : Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen , Annals of Math. 129 (1989), 501-517. Transcendence (general theory), Group varieties, Relevant commutative algebra Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. (Algebraic points on analytic subgroups of algebraic groups) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is concerned with quotients of algebraic groups and applications to moduli spaces of sheaves. The main goal is to study certain complexes of coherent sheaves over projective manifolds and their moduli spaces. A key problem is that in this setup the relevant automorphism group is usually not reductive. Hence it is necessary to study quotients by non-reductive groups. For this purpose a new notion of semistability is introduced as well as the notion of a ``quasi-good'' quotient. A ``quasi-good'' quotient has all the properties of a good quotient except that the projection morphism is not required to be affine. A notion of ``quasi-isomorphism'' is introduced such that, given two quasi-isomorphic \(G\)-varieties \(X\) and \(Y\), the \(G\)-variety \(Y\) admits a quasi-good quotient iff \(X\) does. This theory is then applied to certain complexes of coherent sheaves over projective manifolds. By a process called ''mutation'' one associates a new space of complexes of sheaves to a given one such that the new one is more readily accessible for quotient constructions. In this way the author proves the existence and projectivity of certain moduli spaces for complexes of coherent sheaves on a projective manifold. quotients of algebraic groups; quasi-good quotient; coherent sheaves; mutations; moduli space Jean-Marc Drézet, Quotients algébriques par des groupes non réductifs et variétés de modules de complexes, Internat. J. Math. 9 (1998), no. 7, 769 -- 819 (French). Algebraic moduli problems, moduli of vector bundles, Homogeneous spaces and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Algebraic quotients by non-reductive groups and moduli varieties of complexes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 ''La conjecture de Weil. II'' [Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)], \textit{P. Deligne} proves a fairly general fact regarding the non-vanishing of a large class of L-series on their line of convergence. Roughly speaking, he works with a locally compact group G which is an extension of a group \(\Gamma\) (either \({\mathbb{Z}}\) or \({\mathbb{R}})\) by a compact group, and an infinite collection \(\Sigma\) of conjugacy classes of G. A real quasi-character of \(\Gamma\) gives the notion of a norm of an element of \(\Sigma\), plus a family of one dimensional representations \(\omega_ s\) of G, indexed by the complex numbers. The author then looks at the Riemann surface of isomorphism classes of irreducible ''quasi-unitary'' representations of G (those of the form \(\omega_ s\) times a unitary representation). This space is the disjoint union of complex planes and possibly tubes, and comes equipped with a natural notion of ''real part''. He assumes that the ''Euler product'' corresponding to the norms of the elements of \(\Sigma\) converges for \(Re(s)>1\), which ensures the absolute convergence of the L- series associated to every quasi-unitary representation with real part strictly greater than 1. (The L-series is function on the space of representations, and is a natural infinite product of determinants taken over \(\Sigma\).) The theorem is that if the L-series continues meromorphically to \(Re(\tau)\geq 1\) and is holomorphic there except for possibly a simple pole at the original quasi-character, then the L series doesn't vanish on the line of representations with real part equal to 1, except for possibly one exceptional one dimensional order two representation. - Deligne's proof of this result is quite elegant (and goes back to an idea of Mertens), and essentially is a result about integer valued functions on representations of a compact group (here: orders of poles of the L-functions). It is extremely worthwhile in this context to look at the book by \textit{J.-P. Serre}, ''Abelian \(\ell\)-adic representations and elliptic curves'' (1968; Zbl 0186.257) to appreciate the origins and significance of some of Deligne's ideas, and to obtain an elementary calculus view of the Tate-Sato conjecture.
The author's paper, to my mind, doesn't help very much in understanding the above. In fact, I soon forgot about it and went to Deligne and Serre. However, it does have the virtue of publicizing Deligne's theorem, which, as the author says, should be brought to the attention of number theorists. unitary representations of compact groups; non-vanishing of class of L- series on their line of convergence; Gauss sums; Kloosterman sums Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic problems in algebraic geometry; Diophantine geometry, Jacobsthal and Brewer sums; other complete character sums, Unitary representations of locally compact groups The method of Hadamard and de la Vallée-Poussin (According to Pierre 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 The present book offers one more introduction to algebraic geometry, although during the past twenty years a rather great amount of textbooks on this subject has appeared. Thus one might ask the question of what is the specific character of this new textbook.
Generally speaking, algebraic geometric is amongst the oldest disciplines in mathematics. However, in the recent decades, it has undergone a radical change concerning its foundations, rigor, methods, interrelations with other branches in mathematics, and applications. In particular, algebraic geometry is intimately related to commutative algebra, algebraic number theory, complex analysis, differential geometry, topology, and --- nowadays --- also to special topics in mathematical physics. With regard to this widespread spectrum of interrelations and roots, foundational concepts and methods, purposes for application, and due to the rapid development that algebraic geometry is still undergoing, any introduction to this subject must be highly selective. And, in fact, it is precisely the selection of material, which makes the recent textbooks and standard references differ, apart from the level of the used methods and the completeness of the presented results.
The present textbook aims to introduce various aspects of algebraic geometry from the transcendental (i.e., complex-analytic and differential-geometric) point of view. This means, it does not intend to provide an introduction to ``abstract'' algebraic geometry from Grothendieck's scheme-theoretic viewpoint. Instead this textbook is developed around the algebraic theory of complex manifolds and varieties, with the main emphasis on the low-dimensional cases of curves and surfaces. In this regard, it is conceptually closely related to the much more advanced and comprehensive standard text ``Principles of algebraic geometry'' by \textit{Ph. Griffiths} and \textit{J. Harris} [New York 1978; Zbl 0408.14001)]. However, the present exposition is much more selective, and written at the second year graduate level. As the author himself points out, the book is basically designed as a guide to complex algebraic geometry for the nonexpert. --- With respect to this aim of his, the author has done an excellent job.
In chapter 1 he introduces preliminary materials from commutative algebra, projective algebraic geometry, and algebraic topology. This includes: affine and projective varieties, some ideal theory, analytic varieties, dimension theory, the degree of a variety in projective space, simplicial homology, intersection numbers, De Rham cohomology, Poincaré duality, and the Hodge decomposition theorem for complex manifolds. Here some proofs are carried out (e.g. Hilbert's basis theorem and Hilbert's Nullstellensatz); as for some other fundamental results, the author refers to the standard literature.
Chapter 2 provides the reader with various complex-analytical methods and techniques such as sheaves and their cohomology, complex vector bundles and Chern classes, line bundles, divisors, linear systems, Kähler manifolds and their Hodge theory, Hermitean vector bundles, and specific bundles over projective spaces. Special emphasis is put on computing Hodge numbers and Chern classes in several particular cases. --- The material here, although not given with complete proofs, is very well arranged and instructively presented. The standard reference for further reading is, again, the book of Griffiths and Harris.
Chapter 3 contains an exposition of algebraic curves and compact Riemann surfaces. Basically, this includes plane projective curves, the Plücker formulae, meromorphic functions and differential forms on Riemann surfaces (the Riemann-Hurwitz formula, the residue theorem, etc.), divisors, linear systems, projective embeddings of Riemann surfaces, elliptic curves, a discussion of Jacobians, Abel's theorem, Torelli's theorem, Weierstrass points, hyperelliptic curves, the Riemann-Roch theorem for curves, and some outlook to the Brill-Noether theory and special divisors.
Chapter 4 is devoted to the theory of compact algebraic surfaces and their Enriques classification. After a beautiful discussion of the intersection pairing on compact topological 4-manifolds, the author discusses the basic invariants of surfaces, (bi-)rational maps, the blow- up process, the notion of Kodaira dimension, ample divisors, ruled surfaces, rational surfaces, unirational surfaces, the Albanese variety of a surface, and the Enriques classification of surfaces of non-general type. Subsequently, a rather detailed treatment of \(K\)-3 surfaces and the Torelli theorem satisfied by them is given, and the Chern number geography for surfaces of general type is reviewed afterwards. The chapter ends with an outlook to complex spaces and singular surfaces. --- As for chapter 3 and chapter 4, which together form one main part of the book, proofs of the basic standard theorems are generally given. In addition, many more advanced topics are touched upon, outlined, or discussed with respect to their recent developments (e.g., moduli of \(K\)- 3 surfaces, the Miyaoka-Yau-Bogomolov inequality, etc.). It is, in fact, this combination of providing basic material with rigor and discussing more advanced (and recent) topics along this way, which gives the book its particular flavour and value. The interested reader is referred, as for further reading, to the existing more advanced references (or research articles), and he gets provided with quite a lot of motivation and profound basic knowledge for that.
The concluding chapter 5 deals with some fundamental techniques from Hermitean differential geometry and their applications in algebraic geometry. This reflects the author's special research interests and, due to this fact, offers some never-before-published material on the moving frame theoretic treatment of submanifolds in projective space. This chapter may be of particular interest for both algebraic geometers and differential geometers. The exposition starts with Grassmannians and their topological properties. This includes an account on metrics, Schubert calculus, unitary frames, local Hermitean geometry and curvature forms of Grassmannians, Chern numbers, and the universal bundle construction. The next topic concerns embedded curves in \(\mathbb{P}^ n\) and the Plücker formulae, whereas the following section is devoted to the theory of (higher) osculating maps of projective complex submanifolds. The author presents a proof of Weyl's formulae (and a generalization of them) relating Chern forms of osculating bundles to osculating Kähler forms. This may be thought of as a higher-dimensional analogue of the Plücker formulae given before, and represents really new material. The rest of the chapter discusses formulae of Gauss-Bonnet type for projective hypersurfaces (i.e., formulae which relate the total curvatures of hypersurfaces to their Chern numbers) in great detail and, as a nice application, surfaces in \(\mathbb{P}^ 3\) and \(\mathbb{P}^ 5\). --- Chapter 5 forms the other main part of the book and offers a lot of interesting material which can barely be found elsewhere.
The author has added, just for the convenience of the reader, two appendices. Appendix I (written by \textit{Robert Fisher}) gives some background material on complexification and complex differential forms, and appendix II discusses (with proofs) elliptic functions and complex elliptic curves.
Each chapter in the book starts with its own introduction, in which the aim, the organization of the material, and hints to further references are explained.
All in all, the present textbook is a highly welcome addition to the already existing ones. It is certainly particularly valuable for the analytically oriented beginner, who wants a profound introduction, without getting the whole material in full generality and completeness at the beginning. The text only assumes some familiarity with algebraic topology, complex function theory, and differential geometry. In this regard, the book is also useful for any working mathematician or physicist, who is a non-expert in algebraic geometry, but wants to get acquainted with algebro-geometric methods for further applications in his field of research. Finally, the book also provides an excellent text for teaching purposes at the graduate level in algebraic or complex-analytic geometry. transcendental algebraic geometry; Kaehler manifolds; Hodge theory; Hermitean vector bundles; algebraic curves; compact Riemann surfaces; compact algebraic surfaces; Torelli theorem; Chern number; moving frame; Schubert calculus; Grassmannians; complex-analytic geometry Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Curves in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), Riemann surfaces; Weierstrass points; gap sequences, Compact complex surfaces, Surfaces and higher-dimensional varieties Complex algebraic geometry. An introduction to curves and surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Suppose X is a scheme acted on by a reductive algebraic group G, and suppose a space X/G exists, together with an affine morphism \(\pi\) : \(X\to X/G\), and X/G can be covered by open affines such that over them \(\pi\) is given by the map \(Spec(A)\to Spec(A^ G)\). Then we call X/G the quotient of X by G. What we try to do here is to investigate the relation between the ordinary, complex topology of X and that of X/G.
The key results of part 1 of the paper are that there exists a closed subset \(C\subset Spec(A)\) such that
(a) The composite map \(C\to Spec(A)\to Spec(A^ G)\) is proper and surjective.
(b) C is a deformation retract of Spec(A), with a deformation retraction that commutes with \(\pi:\quad Spec(A)\to Spec(A^ G).\)
Part 2 concerns itself with the study of the Chern and Pontrjagin classes of bundles on the quotient: Sections 7, 8 and 9 are largely the technical background needed for section 10, where a method to show how to prove vanishing for Chern or Pontrjagin rings on quotient varieties is indicated. In sections 11 and 12 this technique, called the ``program'', is applied to one example. We obtain partial results on a conjecture of Ramanan about the vanishing of Pontrjagin classes on the moduli space of stable vector bundles of rank 2 and degree 1 over an algebraic curve.
In the sequel, there are outlined some rather wild conjetures. vanishing of Chern classes; topology of quotient varieties; vanishing of Pontrjagin classes; moduli space of stable vector bundles A. Neeman, \textit{The topology of quotient varieties}, Ann. of Math. \textbf{122} (1985), 419-459. Homogeneous spaces and generalizations, Topological properties in algebraic geometry, Group actions on varieties or schemes (quotients) The topology of quotient 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 A system of coordinates for the elements on the Jacobian variety of Picard curves is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an efficient algorithm for the reduction and addition of divisors. Exploiting the geometry of the Picard curves, a completely effective reduction algorithm is developed, which works for curves defined over any ground field \(k\), with \(\text{char}(k)\neq 3\).
In the generic case, the algorithm works recursively with the system of coordinates representing the divisors, instead of solving for points in their support. Hence, only one factorization is needed (at the end of the algorithm) and the processing of the system of coordinates involves only linear algebra and evaluation of polynomials in the definition field of the divisor \(D\) to be reduced. The complexity of this deterministic reduction algorithm is \(O(\text{deg}(D))\). The addition of divisors may be performed iterating the reduction algorithm.
See also the papers by the first two authors and \textit{J. A. Piñeiro Barceló}, Math. Nachr. 208, 149-166 (1999; Zbl 0960.14032) and the second author and \textit{J. A. Piñeiro Barceló}, Decoding of codes in Picrd curves, Preprint Nr. 96-30 Humboldt Univ. Berlin (1996). system of coordinates; Jacobian variety of Picard curves; efficient algorithm; reduction and addition of divisors; complexity E. Barreiro, J. Sarlabous, J. Cherdieu, Efficient reduction on the jacobian variety of Picard curves, Coding Theory, Cryptography and Related Areas, Springer, Berlin, 2000, pp. 13 -- 28. Jacobians, Prym varieties, Computational aspects of algebraic curves, Number-theoretic algorithms; complexity Efficient reduction on the Jacobian variety of 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 The main result of this paper is the following theorem: Let \(G\) be an almost simple, simply connected algebraic group over an algebraically closed field of positive characteristic \(p\). Let \(P\) be a proper parabolic subgroup of \(G\), put \(X:=G/P\) and let \(F\colon X\to X'\) be the Frobenius morphism on \(X\). If \(H^1(X,F^*T_{X'})=0\) (here \(T_{X'}\) denotes the tangent bundle on \(X')\), then \(X\cong {\mathbb P}^n\) for some \(n>0\). This is a generalization of a result of \textit{B. Haastert} [cf. Manuscr. Math. 58, 385-415 (1987; Zbl 0607.14010)]. The present authors use deformation theory and the ``complex geometry'' of \(X\) [cf. \textit{K. H. Paranjape} and \textit{V. Srinivas}, Invent. Math. 98, No. 2, 425-444 (1989; Zbl 0697.14037)] to get their result. algebraic group; non-vanishing theorems; flag manifolds; characteristic \(p\); parabolic subgroup; Frobenius; tangent bundle N. Lauritzen, V. Mehta, Differential operators and the Frobenius pull back of the tangent bundle on G/P, Manuscripta Math. 98 (1999), no. 4, 507--510. Homogeneous spaces and generalizations, Finite ground fields in algebraic geometry, Formal methods and deformations in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Differential operators and the Frobenius pull back of the tangent bundle on \(G/P\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((X,{\mathcal O}_X)\) be a noetherian formal scheme, and let \(A(X)\) be the category of \({\mathcal O}_X\)-modules, \(A_t(x)\), resp. \(A_{qc}(X)\) be the full subcategory of all torsion, resp. all quasi-coherent \({\mathcal O}_X\)-modules in \(A(X)\), \(A_{qct}(X)=A_t(X)\cap A_{qc}(X)\). Suppose that \(D(X)=D(A(X))\) is the derived category of \(A(X)\). Then \(D_{qct}(X)\) is the full subcategory of \(D(X)\) which consists of complexes with the homologies in \(A_{qct}(X)\).
The authors prove the generalization of \textit{A. Neeman}'s results [Topology 31, 519--532 (1992; Zbl 0793.18008)], namely they prove that there is a bijection between the set of rigid localizing subcategories of \(D_{qct}(X)\) and the subsets of \(X\). The methods proposed are similar to the methods developed by Neeman. Besides, the authors give the description of the acyclization functors associated with the subsets closed under specializations, namely the acyclization functors are the derived functors of sections with support. The authors also specify the relationship of the acyclization functors with Lipman's pairs. The localizing subcategories associated with such subsets by means of homological support are described, this provides the comparison of the obtained results with the Thomason's classification which is obtained in the case of a separated noetherian scheme.
The authors remind that the question of describing localizations for subsets that are neither stable for specialization nor generically stable still remains open. formal schemes; localizing subcategory; acyclization functors; rigid localizing subcategories; sheaves; stable for specialization subsets; cohomology Alonso Tarrío, Leovigildo; Jeremías López, Ana; Souto Salorio, María José, Bousfield localization on formal schemes, J. Algebra, 278, 2, 585-610, (2004) Localization of categories, calculus of fractions, Schemes and morphisms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Bousfield localization on formal 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 \(H_{d,g}\) \((H^0_{d,g}\) resp.) be the Hilbert scheme of locally Cohen-Macaulay (resp. smooth) curves \(C\) of degree \(d\) and genus \(g\), contained in the projective space \(\mathbb{P}^3_k\), where \(k\) is an algebraically closed field of characteristic zero. It is a classical result that if \(C\) is not contained in a plane then \(g\leq(d-2) (d-3)/2\). In the present paper the authors show that if \(d\geq 6\) and \(g\leq(d-3) (d-4)/2+1\), then \(H_{d,g}\) is not irreducible and not reduced. There exists an irreducible component which is not generically reduced and such that the underlying reduced scheme is smooth, of dimension \(3d(d-3)/2 +9-2g\). The points of this component correspond to curves \(C\) such that the values of the function \(n\to h^1(\mathbb{P}^3_k, {\mathcal I}_C(n))\) are maximal. The proof relies on a method developed by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017) and J. Reine Angew. Math. 439, 103-145 (1993; Zbl 0765.14017)]. The authors examine also the cases \(d<6\) and \(g>(d-3) (d-4)/2+1\). It is not known whether \(H_{d,g}\) is connected.
As a consequence, the authors exhibit examples of Hilbert schemes \(H^0_{d,g}\) which are not irreducible and not reduced. locally Cohen-Macaulay space curves; Rao module; Koszul module; Hilbert scheme Martin-Deschamps, M.; Perrin, D., Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit, Ann. Sci. École Norm. Sup. (4), 29, 757-785, (1996) Parametrization (Chow and Hilbert schemes), Plane and space curves The Hilbert scheme of locally Cohen-Macaulay space curves is (almost) never reduced | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f: X \rightarrow S\) be a smooth morphism of algebraic varieties. Then there exist the short exact sequence \(0 \rightarrow f^*\Omega_S \rightarrow \Omega_X \rightarrow \Omega_{X/S} \rightarrow 0\) and the spectral sequence \(E_r\) with \(E_1^{r,s} = H^{r+s}(X, f^*\Omega_S^r \otimes \Omega_{X/S}^{p-r} \otimes {\mathcal L})\) abuting to \(H^{r+s}(X, \Omega_X^p \otimes {\mathcal L})\) for an arbitrary line bundle \({\mathcal L}.\)
The purpose of the article is to construct a similar spectral sequence for a non-smooth morphism. In this case instead of sheaves \(f^*\Omega_S^\bullet\) and \(\Omega_X^\bullet\) the author considers the relative de Rham complexes which are objects of the derived category of the \({\mathcal O}_X\)-modules. When \(S\) is a smooth curve these complexes were constructed by \textit{S. J. Kovács} [J. Algebr. Geom. 5, No. 2, 369-385 (1996; Zbl 0861.14033)].
As an application it is proved that the fibres of a smooth family \(Y \rightarrow S\) of projective varieties of general type with nef canonical bundle and semi-negative \(\Omega_S\) are birational. This is a common generalization of a paper by \textit{L. Migliorini} [J. Algebr. Geom. 4, No. 2, 353-361 (1995; Zbl 0834.14021)] and others. relative de Rham complex; hyperresolution; twisted cohomology; vanishing theorems; canonical singularity; non-smooth morphism; varieties of general type Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Birational geometry, Vanishing theorems in algebraic geometry, Families, moduli, classification: algebraic theory Relative de Rham complex for non-smooth morphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi :\mathfrak{X}\rightarrow S\) be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over \(\mathbb{C}\). On every such a family, suitable derivatives ``along the fibers'' [in the sense of \textit{R. F. Lax}, Math. Ann. 216, 35-42 (1975; Zbl 0291.32028)] of the relative Wronskian, as defined by \textit{D. Laksov} and \textit{A. Thorup} [in: Enumerative algebraic geometry, Proc. Zeuthen Symp., Copenhagen 1989, Contemp. Math. 123, 131-148 (1991; Zbl 0763.14013) and Ark. Mat. 32, No. 2, 393-422 (1994; Zbl 0839.14020)] are constructed. These are sections of suitable jets extensions of the \(g(g+1)/2\)-th tensor power of the relative canonical bundle of the family itself.
The geometrical meaning of such sections is discussed: The zero schemes of the \((k-1)\)-th derivative (\(k\geq 1\)) of a relative Wronskian correspond to families of Weierstrass points (WP's) having weight at least \(k\).
The locus in \(M_{g}\), the coarse moduli space of smooth projective curves of genus \(g\), of curves possessing a WP of weight at least \(k\), is denoted by \(\text{wt}(k)\). The fact that \(\text{wt}(2)\) has the expected dimension for all \(g\geq 2\) was implicitly known in the literature. The main result of this paper hence consists in showing that \(\text{wt}(3)\) has the expected dimension for all \(g\geq 4\). As an application we compute the codimension \(2\) Chow (\(\mathbb{Q}\)-)class of \(\text{wt}(3)\) for all \(g\geq 4\), the main ingredient being the definition of the \(k\)-th derivative of a relative Wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension \(2\) Chow (\(\mathbb{Q}\)-)classes in \(M_{4}\) (\(g\geq 4\)), corresponding to varieties of curves having a point \(P\) with a suitable prescribed Weierstrass gap sequence, relating to previous work of Lax. derivatives of relative Wronskians; families of Weierstrass points; coarse moduli spaces of curves; Chow classes Gatto L., Trans. Amer. Math. Soc. 351 pp 2233-- (1999) Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Fine and coarse moduli spaces Derivatives of Wronskians with applications to families of special 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 According to a theorem of \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323--448 (1962; Zbl 0201.35602)] it is possible to reconstruct, uniquely up to isomorphism, a noetherian scheme \(X\) from the category of coherent sheaves \(\text{Coh}(X)\) on \(X\). The purpose of this paper is to extend Gabriel's theorem to the category of twisted coherent sheaves on noetherian schemes. The author recalls definitions of Brauer groups of a scheme and of a coherent sheaf twisted by an element \(\alpha\) of Brauer-Grothendieck group. Denote by \(\text{Coh}(X,\alpha)\) the category of coherent \(\alpha\)-twisted sheaves over \(X\). Under the assumption that \(\alpha\) lies in the cohomological Brauer group of \(X\), the author gives a ringed space structure to the set \(E_{X,\alpha}\) of irreducible Serre subcategories of finite type of \(\text{Coh}(X,\alpha)\). The main theorem states that as a scheme \(E_{X,\alpha}\) is isomorphic to \(X\). In addition, it is proved that any equivalence between \(\text{Coh}(X,\alpha)\) and \(\text{Coh}(X,\beta)\) induces an isomorphism between \(X\) and \(Y\). Noetherian scheme; Brauer group; coherent twisted sheaves Perego, A.: A gabriel theorem for coherent twisted sheaves Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Brauer groups of schemes, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) A Gabriel theorem for coherent twisted sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a semisimple connected algebraic group over an algebraically closed field of characteristic zero and let \(\theta\) be an involution of \(G\) with set of fixed points \(G^\theta\). Let \(H\) be a subgroup of the normalizer of \(G^\theta\) which contains \(G^\theta\) and let \(X\) be a normal projective embedding of the symmetric variety \(G/H\). In the present paper, the author studies the effective cone and the big cone of \(X\), and a combinatorial criterion for the bigness of a nef divisor on \(X\) is given as well.
When \(X\) is \(\mathbb{Q}\)-factorial (i.e. when the rational Picard group equals the rational Weil divisor class group), an explicit description of the extremal rays of the effective cone is given, and an explicit description of the big cone is also deduced. The effective cone is described then in the non \(\mathbb{Q}\)-factorial case as the intersection of the real Picard group with an explicitly described polyhedral cone in the real Weil divisor class group.
Fix a maximal torus \(T\) of \(G\) and a Borel subgroup \(B\) containing \(T\). The last part of the paper is devoted to characterize combinatorially the bigness of a nef Cartier divisor \(D\) on \(X\), which up to linear equivalence may always be assumed \(B\)-stable: the criterion expresses the bigness of \(D\) in terms of the \(T\)-weights of the fibers of the associated line bundle \(\mathcal O(D)\) over the \(B\)-fixed points in \(X\). This is essentially done by reducing to the case where \(X\) is toroidal, i.e. when every \(B\)-stable prime divisor of \(X\) which contains a \(G\)-orbit is \(G\)-stable. Under this assumption, there exists a projective variety \(Z \subset X\) which is toric under the action of \(T\) and which uniquely determines \(X\) as a compactification of \(G/H\). A main step in the proof of the criterion in the toroidal case is the fact that the restriction to \(Z\) of a big divisor on \(X\) is still big. symmetric varieties; big divisors; effective divisors Ruzzi, A, A., \textit{effective and big divisors on a projective symmetric variety}, J. Algebra, 354, 20-35, (2012) Compactifications; symmetric and spherical varieties, Divisors, linear systems, invertible sheaves Effective and big divisors on a projective symmetric 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 It is well-known that the classical Weyl character formula for irreducible representations of a compact Lie group is a consequence of the classical Lefschetz fixed point formula applied to the corresponding generalized flag variety. In the context of Arakelov geometry, a fixed point formula of Lefschetz type has recently been formulated and proved by \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, 333-396 (2001; Zbl 0999.14002)]. Again by applying that formula to generalized flag varieties (now over Spec(\(\mathbb Z\))), the authors present, in the paper under review, a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over \(\text{Spec}(\mathbb Z)\) [see \textit{J. C. Jantzen}, ``Representations of algebraic groups'' (1987; Zbl 0654.20039)], except for the three exceptional cases \(G_2\), \(F_4\) and \(E_8\). The proof involves the computation of the equivariant Ray Singer analytic torsion associated with certain vector bundles on the corresponding complex generalized flag variety.
In the special case the flag variety is Hermitean symmetric, this computation has been carried out by \textit{K. Köhler} in a previous paper [J. Reine Angew. Math. 460, 93-116 (1995; Zbl 0811.53050)]. In the general case, the authors decompose the flag variety into Hermitean symmetric flag varieties by various fibrations and inductively apply a special case of a formula of \textit{X. Ma} [Ann. Inst. Fourier 50, 1539-1588 (2000; Zbl 0964.58025)], which relates the equivariant analytic torsion of the total space of a fibration to the equivariant analytic torsion of its base and its fibre. The authors in fact give a proof of this special case based on the arithmetic Lefschetz formula again. In the final chapter of the paper under review, the authors use the Jantzen sum formula to derive explicit formulae for the global height of ample line bundles on an arbitrary generalized flag variety. This way they recover formulas for projective spaces proved by \textit{H. Gillet} and \textit{C. Soulé} [Ann. Math. (2) 131, 163-203 (1990; Zbl 0715.14018) and 205-238 (1990; Zbl 0715.14006)], and for quadrics proved by \textit{J. Cassaigne} and \textit{V. Maillot} [J. Number Theory 83, 226-255 (2000; Zbl 1001.11027)]. integral representations of Chevalley schemes; Jantzen sum formula; Arakelov geometry; generalized flag variety; equivariant Ray-Singer torsion; Hermitean symmetric space; arithmetic Lefschetz formula Kai Köhler and Damian Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. I. Statement and proof, Invent. Math. 145 (2001), no. 2, 333 -- 396. , https://doi.org/10.1007/s002220100151 K. Köhler and D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. II. A residue formula, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 81 -- 103 (English, with English and French summaries). Christian Kaiser and Kai Köhler, A fixed point formula of Lefschetz type in Arakelov geometry. III. Representations of Chevalley schemes and heights of flag varieties, Invent. Math. 147 (2002), no. 3, 633 -- 669. Arithmetic varieties and schemes; Arakelov theory; heights, Grassmannians, Schubert varieties, flag manifolds, Determinants and determinant bundles, analytic torsion, Representation theory for linear algebraic groups, Heights A fixed point formula of Lefschetz type in Arakelov geometry. III: Representations of Chevalley schemes and heights of flag varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper the author proves two theorems, which are essentially generalizations of the results of \textit{M. Gros} and \textit{N. Suwa} [cf. Mém. Soc. Math. Fr., Nouv. Sér. 21, 87 p. (1985; Zbl 0615.14011), K-Theory 9, No. 3, 245--271 (1995; Zbl 0838.14014), Duke Math. J. 57, No. 2, 615--628 (1988; Zbl 0715.14011)] to regular schemes, which are not necessarily smooth over a perfect field. Both theorems concern the logarithmic Hodge-Witt cohomology of regular schemes. The first theorem is the purity of the logarithmic Hodge-Witt cohomology of an excellent regular pair \(Z\hookrightarrow X\) of characteristic \(p\). The second theorem is the Gersten type conjecture for the \(p\)-primary part of the Kato complex [cf. J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)] (the arithmetic Bloch-Ogus complex) of the spectrum of an excellent regular local ring of characteristic \(p\). This result is a generalization of the results of Jannsen-Saito-Sato [cf. \textit{U. Jannsen, S. Saito}, and \textit{K. Sato}, Étale duality for constructible sheaves on arithmetic schemes, preprint] and in fact the proof relies on reducing to their results. logarithmic Hodge-Witt cohomology; purity; Gersten -type conjecture Shiho, A.: On logarithmic Hodge--Witt cohomology of regular schemes. (2006)(preprint) \(p\)-adic cohomology, crystalline cohomology, Higher symbols, Milnor \(K\)-theory On logarithmic Hodge-Witt cohomology of regular schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is devoted to various special, largely technical results in the cohomology theory of schemes, as well as to generalizations of some known theorems in local cohomology, étale cohomology, \(\ell\)-adic cohomology, and the theory of perverse sheaves. Based upon the deep and extensive theories developed in \textit{A. Grothendieck}'s ``Séminaire de Géométrie Algébrique'' (SGA 2 -- SGA 7, 1962--1977) and by many authors thereafter, including the author himself, \textit{P. Deligne, O. Gabber, N. Katz, W. Messing, K. Fujiwara}, and others, two types of promoting results are established in this paper under review. The general context of this work is related to the so-called ``absolute purity conjecture'' (A.~Grothendieck) and its recent proof by \textit{O. Gabber} [cf. \textit{K. Fujiwara}, in: Algebraic Geometry 2000, Azumino, Proc. Symp. Nagano 2000, Adv. Stud. Pure Math. 36, 153--183 (2002; Zbl 1059.14026)].
More precisely, the author studies a scheme \(X\) of finite type over the spectrum \(S\) of a strictly Henselian discrete valuation ring with residue field of characteristic \(p > 0\) and a group \(\Lambda = \mathbb{Z}/\ell^{\nu}\cdot\mathbb{Z}\), where \(\ell\) is a prime number different from the characteristic \(p\) and \(\nu\) is a natural number. Then he derives an estimate for the étale depth of the constant sheaf \(\Lambda\) and proves a cohomological vanishing theorem for the fibres in \(X\). This leads to an affine Lefschetz-type theorem (à la SGA 4) which appears to be closely related to the absolute purity conjecture. In fact, the author provides here many details needed in Gabber's spectacular proof (still unpublished) of Grothendieck's old conjecture.
In the second part of this paper, the author studies the complex of vanishing cycles for the datum \((X,S,\Lambda )\), together with the so-called variation morphism of complexes established in SGA 7 I and SGA 7 II [Lect. Notes Math. 288 and 340 (1972/1973; see Zbl 0237.00013 and Zbl 0258.00005)]. This variation morphism depends on an element \(\sigma\) in the tame inertia group \(I_t\) of the given set-up, and the author shows under what conditions such a variation morphism is an isomorphism. Also this result is related to O. Gabber's proof of the absolute purity conjecture, as the author points out, and it might be seen as an (\(p\)-adic) analogue of a corresponding result in the transcendental theory of hypersurface singularities (à la J.~Milnor, 1968).
No doubt, the paper under review is a very important contribution towards a better understanding of O. Gabber's approach to Grothendieck's problem of absolute cohomological purity. cohomology of schemes; perverse sheaves; vanishing cycles; absolute purity conjecture; étale cohomology; \(\ell\)-adic cohomology; local cohomology; monodromy groups; Picard-Lefschetz theory Illusie, L, Perversité et variation, Manuscr. math., 112, 271-295, (2003) Étale and other Grothendieck topologies and (co)homologies, Structure of families (Picard-Lefschetz, monodromy, etc.), Fibrations, degenerations in algebraic geometry Perversity and variation | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 important paper completes the author's program of proving the Weil conjectures using purely \(p\)-adic techniques. It is a companion paper to the author's article [Duke Math. J. 134, No. 1, 15--97 (2006; Zbl 1133.14019)]. The main result of the paper under review (theorem 5.3.2, rigid Weil II over a point) is a weak analogue in rigid cohomology of \textit{P. Deligne}'s ''Weil II'' theorem on purity of higher direct images
[Publ. Math., Inst. Hautes Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)]. The proof follows \textit{G. Laumon}'s Fourier theoretic approach [Publ. Math., Inst. Hautes Étud. Sci. 65, 131--210 (1987; Zbl 0641.14009)], transposed into the setting of arithmetic \(\mathcal{D}\)-modules, and builds on the work of many people. We refer to the paper's informative introduction for more specific details and comments. Weil conjectures; rigid cohomology; \(\mathcal{D}\)-modules; F-isocrystals; Fourier transform Kedlaya, Kiran S., Fourier transforms and \(p\)-adic ``Weil II'', Compos. Math., 142, 6, 1426-1450, (2006) \(p\)-adic cohomology, crystalline cohomology, Varieties over finite and local fields, \(p\)-adic differential equations, Rigid analytic geometry Fourier transforms and \(p\)-adic `Weil II' | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let Y be a nonsingular irreducible projective curve of genus \(g\geq 2\) defined over an algebraically closed field of characteristic \(p>0.\) For an integer \(q\geq 1\), a basis for the space of holomorphic q-differentials on X gives a morphism \(\Phi_ q: X\to {\mathbb{P}}^{d-1}\), where \(d=g\) if \(q=1\) and \(d=(2q-1)(g-1)\) if \(q\geq 2\). This leads to the definition of q- Weierstrass points, which are basically those points on X which have non- generic intersection multiplicities with the hyperplanes of \({\mathbb{P}}^{d-1}.\)
The author uses a theorem of \textit{A. Neeman} [Invent. Math. 75, 359-376 (1984; Zbl 0555.14009)] to reprove a result of \textit{H. M. Farkas} and \textit{I. Kra} [``Riemann surfaces'', Graduate Texts Math. 71 (1980; Zbl 0475.30001)] that if an automorphism of X with order relatively prime to p has three or more fixed points then each of them is a q-Weierstrass point, for infinitely many values of q. He also gives some conditions which imply the existence of such an automorphism. fixed points of automorphism; q-Weierstrass points Garcia, A.--Remarks on fixed points of automorphisms and higher-order Weierstrass points in prime characteristic. Manuscripta Math. 69, 301--303 (1990) Riemann surfaces; Weierstrass points; gap sequences, Automorphisms of curves Remarks on fixed points of automorphisms and higher-order Weierstrass points in prime characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a hyperbolic curve over \(\overline{\mathbb{Q}_{p}}\). In [Publ. Res. Inst. Math. Sci. 40, No. 4, 1291--1336 (2004; Zbl 1078.14037)], \textit{A. Tamagawa} proved that for every closed point \(x\) of the stable reduction of \(X\), there exists a finite étale cover \(Y\) of \(X\) and an irreducible component of the stable reduction of \(Y\) that lies over \(x\).
In the paper under review, and under the assumption that \(X\) is a Mumford curve, the result is generalized by replacing the stable model of \(X\) by an arbitrary semistable model \(\mathcal{X}\) (and the stable model of \(Y\) by the minimal semistable model of \(Y\) over \(\mathcal{X}\)). This is realized by a clever construction of \(\mu_{p^n}\)-torsors on the Berkovich analytification \(X^{\mathrm{an}}\) of \(X\) (in order to construct points with positive genera on the corresponding covers, which necessarily correspond to generic points of any of their semistable reductions) using theta functions. As a corollary, the author deduces that every curve over \(\overline{\mathbb{Q}_{p}}\) admits a Zariski-dense open subset on which the same result holds.
In the sequel of the paper, the author studies tempered fundamental groups of Mumford curves. Let us recall that a tempered cover of a Berkovich space is a sort of a mix between a finite étale and a topological cover. The notion has been introduced by \textit{Y. André} [Period mappings and differential equations. From \({\mathbb C}\) to \({\mathbb C}_p\). Tôhoku-Hokkaidô lectures in arithmetic geometry. With appendices: A: Rapid course in \(p\)-adic analysis by F. Kato, B: An overview of the theory of \(p\)-adic unifomization by F. Kato, C: \(p\)-adic symmetric domains and Totaro's theorem by N. Tsuzuki. Tokyo: Mathematical Society of Japan (2003; Zbl 1029.14006)] and [Duke Math. J. 119, No. 1, 1--39 (2003; Zbl 1155.11356)]. The author proves that two Mumford curves over \(\overline{\mathbb{Q}_{p}}\) with isomorphic tempered fundamental groups have homeomorphic associated Berkovich spaces.
It is well-known that the Berkovich space associated to \(X\) may be reconstructed by the graphs of all the semistable reductions. The former result of resolution of nonsingularities then shows that it is actually enough to consider stable models of finite Galois covers up to action of the Galois group. The other essential ingredient is a result of Mochizuki that ensures that the tempered fundamental group is enough to recover the graph of the stable reduction (see [\textit{S. Mochizuki}, Publ. Res. Inst. Math. Sci. 42, No. 1, 221--322 (2006; Zbl 1113.14025)]).
Finally, The author handles the case of punctured Tate curves. Let \(q_{1}, q_{2} \in \overline{\mathbb{Q}_{p}}\) such that \(|q_{1}|, |q_{2}| <1\). If the tempered fundamental groups of \(\mathbb{G}_{m}/q_{1}^{\mathbb{Z}} - \{1\}\) and \(\mathbb{G}_{m}/q_{2}^{\mathbb{Z}} - \{1\}\) are isomorphic, then there exists \(\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\) such that \(q_{2} = \sigma(q_{1})\). tempered fundamental group; anabelian geometry; Berkovich spaces; nonsingularities; Mumford curves Rigid analytic geometry, Coverings of curves, fundamental group Resolution of nonsingularities for Mumford curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a foregoing paper [J. Reine Angew. Math. 575, 135--155 (2004; Zbl 1072.14053)], the authors had studied a smooth projective curve \(X\) with an action of a finite group \(G\) and they had proved that the Jacobian \(J(X)\) is isogenous to a product of particular abelian subvarieties in the form
\[
J(X)\sim B^{d_1}_1\times\cdots\times B^{d_r}_r,
\]
where the abelian subvarieties \(B_i\) correspond to the irreducible \(\mathbb Q\)-representations of the group \(G\), which also determine the numbers \(d_i\), \(1\leq i\leq r\). In the paper under review, the authors continue their study of this decomposition in terms of the curve and its group action itself. To this end, they introduce the notion of the Prym variety of a pair of coverings, which is roughly speaking the complement of the Prym variety of one morphism in the Prym variety of another morphism of curves. This concept, arising quite naturally in their framework, is first investigated under more general aspects, mainly by exhibiting the basic properties of Prym varieties of pairs of coverings. Then possible isogenies between ordinary Prym varieties and the newly defined Prym varieties of pairs of coverings are analyzed, and finally their occurrence is illustrated by the example of the isogeny decomposition of the Jacobian \(J(X)\) of a curve \(X\) acted on by the symmetric group \(S_5\). algebraic curves; Jacobians; isogeny; group actions; group representations Lange, H.; Recillas, S.: Prym varieties of pairs of coverings. Adv. geom. 4, 373-387 (2004) Jacobians, Prym varieties, Isogeny, Group actions on varieties or schemes (quotients) Prym varieties of pairs of coverings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective curve of genus \(g\geq 2\) and let \({\mathcal {SU}}_C(r)\) be the moduli space of semi-stable vector bundles of rank \(r\) on \(C\) with trivial determinant. Let \(J_{g-1}\) be the Jacobian of \(C\) parameterizing line bundles of degree \(g-1\) and \(\Theta \subset J_{g-1}\) be the canonical theta divisor. A general \(E\in {\mathcal {SU}}_C(r)\) defines a divisor \(\Theta_E\) in the linear system \(| r\Theta|\) and therefore there exists a rational map \(\theta : {\mathcal {SU}}_C(r) \dashrightarrow | r\Theta|\) which coincides to the morphism defines by the global sections of the determinant bundle of \({\mathcal {SU}}_C(r)\).
For \(r=2\), that \(\theta\) is an embedding if \(C\) is not hyperelliptic. In the case \(g=2\), for \(r=2\) \textit{M.S. Narasimhan} and \textit{S. Ramanan} [Ann. Math. (2) 89, 14--51 (1969; Zbl 0186.54902)] proved that \(\theta\) is an isomorphism, and for \(r=3\), \(\theta\) is a double covering ramified along a sextic hypersurface which is the dual of the ``Coble cubic'' [\textit{A. Ortega}, J. Algebr. Geom. 14, No.2, 327--356 (2005; Zbl 1075.14031)]. For higher rank and genus, very little is known.
The author proves in this paper that for genus 2 and any rank \(r\), the map \(\theta\) is generically finite, or equivalently, dominant because \(\dim {\mathcal {SU}}_C(r)=\dim | r\Theta|\). Moreover, \(\theta\) admits some fibers of dimension \(\geq [r/2]-1\). The method used to prove this result is to consider the fiber of \(\theta\) over a reducible element of \(| r\Theta|\) of a suitable form. The author affirms that this method is not, in principle, restricted to genus 2, but the geometry in higher genus becomes much more intricate.
For genus \(g=3\), the author proves that if \(r=3\), then \(\theta\) is a finite morphism. This result was known for a generic curve of genus 3 [\textit{M. Raynaud}, Bull. Soc. Math. Fr. 110, 103--125 (1982; Zbl 0505.14011)]; however the author believes that the method is more interesting than the result itself, because he translates the problem into a question of classical projective geometry which has been completely solved by \textit{U. Morin} [Atti Ist. Veneto 89, 907-926. (1930; JFM 56.1179.03)]. moduli of vector bundles; map associated to the determinant bundle Beauville A.: Vector bundles and theta functions on curves of genus 2 and 3. Am. J. Math. 128, 607--618 (2006) Vector bundles on curves and their moduli, Divisors, linear systems, invertible sheaves Vector bundles and theta functions on curves of genus 2 and 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 For schemes over the complex numbers, there are two comparison theorems between topological and algebraic cohomology. Firstly the étale cohomology of a constructible (torsion) sheaf is equal to its topological cohomology. Secondly for manifolds the algebraic de Rham topology coincides with the topological complex cohomology. Also for complex manifolds locally constant systems correspond to vector bundles with integrable connections (``Riemann-Hilbert correspondence'').
In the paper under review these are generalised to log-schemes and log-analytic spaces. The log-étale cohomology has been studied by one of the authors, and its log-analytic analogue uses a ``real blow-up'' of the underlying analytic space. The comparison for constructible sheaves holds in general, for the other two results one has to restrict the singularities of the spaces (fine and log smooth is sufficient, but the results are more general). Also the Riemann-Hilbert correspondence needs some hypothesis on the bundles (``unipotent monodromy at the boundary''). The strategy of the proof consists of reduction to the classical case via local computations. Riemann-Hilbert correspondence; log-schemes; log-analytic spaces; log-étale cohomology; comparison for constructible sheaves Kato, K.; Nakayama, C., Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C, Kodai Math. J., 22, 2, 161-186, (1999) Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over \(\mathbb{C}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(A_{g}\) be the moduli space of principally polarized abelian varieties over \({\mathbb C}\) of dimension \(g,\) and let \(M_{g}\) be the moduli space of smooth algebraically irreducible curves of genus \(g\) over \({\mathbb C}\). There exists a morphism of schemes \(J_{g} : M_{g} \rightarrow A_{g}\) called the Jacobi map which is defined by \(J_{g}(C) = (J(C), W_{g-1}),\) where \(J(C)\) is the Jacobian of the curve \(C\) and \(W_{g-1}\) is its theta divisor. One can define the Jacobian locus \(J_{g}\) in dimension \(g\) as the image of the Jacobi map, and the closed Jacobian locus \(\overline{J_{g}}\) as the closure of \(J_{g}\) in \(A_{g}.\) One has that \(\dim \overline{J_{g}} = 3g - 3\) for \(g > 1\), and \(\dim A_{g} = g(g+1)/2.\) From a general point of view could state the Schottky problem as the problem of characterize \(\overline{J_{g}}\) as a subscheme of \(A_{g}.\) There are several approaches to this problem in terms of geometric properties. Characterizing \(\overline{J_{g}}\) in terms of algebraic equations goes back to the original paper of Schottky-Jung and there have been proved partial results by R. Donagi and B. van Geemen. The problem of characterizing \(\overline{J_{g}}\) in terms of differential equation (the Novikov's conjecture) was solved by T. Shiota and M. Mulase. This approach is related to the geometric characterization Jacobian in terms of the existence of trisecant lines in its Kummer variety (R. Gunning. G. Welters and I. Krichever).
In this paper is to expose the approaches to the Schottky problem more directly related to the recent results proved by I. Krichever and S. Grushevsky. That is authors gave expose the approaches related with the existence of trisecants, the KP equation and the \(\Gamma_{00}\)-conjecture. moduli space of principally polarized abelian varieties; moduli space of smooth algebraically irreducible curves; Jacobi map; Jacobian of the curve; the Novikov's conjecture; trisecant lines; Kummer variety Theta functions and curves; Schottky problem, Theta functions and abelian varieties, Jacobians, Prym varieties Survey on the Schottky problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given an algebraic curve \( C \) of genus \( g \), it follows from the Riemann-Roch theorem that every non-special divisor on \( C \) of the form \(D-(\deg D)\infty\) is equivalent to \(\tilde{D}-g\infty\), where \(\tilde{D}=\sum_{k=1}^{g}P_k\), \(P_k\in C\), is \textit{a reduced divisor}. The first result in finding a reduced divisor was given in [\textit{D.G. Cantor}, Math. Comp. 48, 95--101 (1987; Zbl 0613.14022)]. Cantor's algorithm was inspired by reduction of quadratic forms. For low genera (\( g = 2, 3 \)) many authors worked on giving more explicit solutions to the reduction problem due to potential application in cryptography, see [\textit{P. Gaudry}, J. Math. Cryptol. 1, 243--265 (2007; Zbl 1145.11048)] and [\textit{A. V. Sutherland}, ``Fast Jacobian arithmetic for hyperelliptic curves of genus 3'', Preprint, \url{arXiv:1607.08602}] and the literature cited there.
The authors propose an effective algorithm that reduce an arbitrary degree non-special divisor on a hyperelliptic curve \(C\) to the equivalent one of degree \(g\). The advantage of the algorithm is that it involves only arithmetic operations on polynomials. The reduced divisor
is obtained in the form of solution to the Jacobi inversion problem. Explicit formulas defining the reduced divisor \( \tilde{D} \) are found, with the help of interpolation polynomials, for an initial divisor of the form \( D_{g+1}=\sum_{k=1}^{g+1}P_k \), \(D_{g+2}=\sum_{k=1}^{g+2}P_k \), \( D_{g+1}=2P_1+\sum_{k=2}^{g}P_k \) with all \( P_k\in C \) distinct, and for \(D_{g+1}=(g + 1)P \) with \( P\in C \). The polynomials defining \( \tilde{D} \) are expressed in terms of the points \( P_k \), their coefficients computed directly. An iterative reduction algorithm is given for \( D_{g+m}=\sum_{k=1}^{g+m}P_k\), \(m>2 \), with all \( P_k\in C \) distinct. It is proposed a cryptography application of the algorithm that is extended to divisors of arbitrary degrees comparing with the known ones in the theory of hyperelliptic functions.
The authors also solve an addition problem: given two non-special divisors \(D_1 \) and \( D_2 \) of degrees \( g + m_1 \) and \( g + m_2 \),
\( m_1, m_2 > 0 \), find a reduced divisor \( \tilde{D} \) of degree \( g \) such that \( D_1+D_2-(2g+m_1+m_2)\infty\sim\tilde{D}-g\infty\). The addition algorithm is also applicable to special divisors of degree \( m < g \) provided that their sum is a non-special divisor of degree greater than \( g \). reduced divisor; inverse divisor; non-special divisor; generalised Jacobi inversion problem Jacobians, Prym varieties, Special algebraic curves and curves of low genus, Uniformization of complex manifolds, Applications to coding theory and cryptography of arithmetic geometry Addition of divisors on hyperelliptic curves via interpolation polynomials | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R\) be a curve singularity, i.e. the complete local ring of some point of a reduced algebraic curve, always over an algebraically closed field \(K\) with \(\text{char} (K) \neq 2\). \textit{G.-M. Greuel} and \textit{H. Knörrer} showed that the finite Cohen-Macaulay representation type is connected with the behaviour under deformations: \(R\) is of finite type if and only if \(R\) dominates some of the simple plane curve singularities \(A_ n, D_ n, E_ 6, E_ 7, E_ 8\) [Math. Ann. 270, 417-425 (1985; Zbl 0553.14011)]. In this paper, the authors consider curve singularities of infinite type; these are either tame or wild by an earlier result of the authors [Math. Ann. 294, No. 3, 387-394 (1992; Zbl 0760.16005)]. The main result (theorem 1) states that \(R\) is of tame type if and only if it dominates one of the plane curve singularities \(T_{pq} = K[[X,Y]] / (X^ p+\lambda X^ 2Y^ 2+Y^ q)\), \(\lambda \neq 0,1\) a parameter. In order to prove the tameness of \(T_{pq}\), the authors first check (in section 3) the tameness of the nonplane singularity \(K[[X,Y,Z]] / (XY,X^ p+Y^ q+Z^ 2)=P_{pq}\), combining explicit calculations with results of an unpublished preprint of \textit{V. M. Bondarenko} (1988); then, in section 4, they demonstrate that \(T_{pq}\), as a deformation of \(P_{pq}\), must be of tame type, too. The key-result for proving the converse (in section 7) is a useful description of tameness (for curve singularities of infinite type) in terms of conditions for certain overrings near to the normalization of \(R\) (theorem 3 in section 6). tame Cohen-Macaulay type; deformations; curve singularities of infinite type Y. \textsc{Drozd}~\textsc{and} G.-M. \textsc{Greuel}, Cohen-Macaulay module type, Compos. Math. \textbf{89} (1993), 315-338. Singularities of curves, local rings, Formal methods and deformations in algebraic geometry Cohen-Macaulay module 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 \(l\) be a fixed prime number, and \(K\) a field of characteristic \(\operatorname{char}(K) \neq l\). The first key step in most strategies towards anabelian geometry is to develop a local theory allowing to recover inertia and/or decomposition groups of ``points'' using the given Galois-theoretic information. In the context of algebraic curves, one should eventually detect inertia/decomposition groups of the given curve within its étale fundamental group. On the other hand, in the birational setting, this corresponds to detecting inertia/ decomposition groups of arithmetically and/or geometrically meaningful places of the considered function field within its absolute Galois group. The earliest example of such a local theory is Neukirch's group-theoretic characterization of decomposition groups of finite places of global fields.{
} Two non-arithmetically based methods were proposed at about the same time for recovering inertial and decomposition groups of valuations using Galois groups. The first one relies on the theory of rigid elements introduced by \textit{R. Ware} [Can. J. Math. 33, 1338--1355 (1981; Zbl 0514.10015)] and developed by Arason-Elman-Jacob [\textit{J. Kr. Arason} et al., J. Algebra 110, 449--467 (1987; Zbl 0629.10016)], \textit{A. J. Engler} and \textit{J. B. Nogueira} [J. Algebra 166, No. 3, 481--505 (1994; Zbl 0809.12004)], \textit{J. Koenigsmann} (e.g., [J. Reine Angew. Math. 465, 165--182 (1995; Zbl 0824.12006)]) and \textit{I. Efrat} (e.g., [Pac. J. Math. 226, No. 2, 259--275 (2006; Zbl 1161.19002)]), among others. The second method is the theory of commuting-liftable pairs (CL-pairs) in Galois groups introduced by Bogomolov and developed by himself and Tschinkel. Its input is the much smaller pro-\(l\) abelian-by-central Galois group, but it requires that the base field contain an algebraically closed subfield. Motivation for the development of this theory primarily comes from the Bogomolov-Pop conjecture providing a precise formulation of Bogomolov's program for ``reconstructing'' higher-dimensional function fields over an algebraically closed field from their pro-\(l\) abelian-by-central Galois groups (see [\textit{F. A. Bogomolov}, Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 1, 32--67 (1991; Zbl 0736.12004)], and [\textit{F. Pop}, Invent. Math. 187, No. 3, 511--533 (2012; Zbl 1239.14025)]).{
}Until now, the two non-arithmetically based methods remained almost fully separate (although a connection between them has been suggested in the \(l^n\)-abelian-by-central situation by several authors). The paper under review provides an approach that unifies the two methods. Let \(n > 0\) be an integer or equal to infinity. Then the author shows that, for each sufficiently large integer \(N\) and any field \(K\) containing a primitive root of unity of degree \(l^{2l^{N}}\), there is a group-theoretical recipe which recovers (minimized) inertia and decomposition subgroups in the maximal \(l^n\)-elementary abelian Galois group of \(K\). This is obtained by using the structure of the \(l^{N}\)-abelian-by-central Galois group of \(K\). Moreover, if \(n\) is finite, then \(N\) is determined explicitly as well; when \(n = 1\), one may take \(N = 1\).{
}The paper under review obtains analogous results to those in the theory of commuting-liftable pairs, for the \(l^n\)-abelian-by-central and the pro-\(l\) abelian-by-central situations. For this purpose, the author applies the theory of rigid elements, while working under far less restrictive assumptions than needed by the Bogomolov and Tschinkel approach. Thus, Theorems 1 and 2 of the present paper yield a non-trivial generalization of results of [\textit{F. Bogomolov} and \textit{Y. Tschinkel}, Int. Press Lect. Ser. 3, No. I, 75--120 (2002; Zbl 1048.11090)]. As an application, the paper exhibits several important classes of function fields of zero characteristic whose absolute Galois groups are not realizable as absolute Galois groups of fields of nonzero characteristic. inertia and decomposition groups; commuting-liftable pairs; \(C\)-pairs; pro-\(l\) abelian-by-central Galois group; rigid elements of a field; Milnor \(K\)-theory; valuations; commuting-liftable subgroups of Galois groups Separable extensions, Galois theory, Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Commuting-liftable subgroups of Galois groups. 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 subject of this paper is the study of the moduli space \(M_{H}(r,c_1 ,c_2)\) of stable sheaves of rank \(r\) and Chern classes \(c_1\), \(c_2\) on a smooth projective surface \(X\) over \({\mathbb{C}}\) with respect to an ample divisor \textit{H}. The case considered here is that \(X\) is ruled over a non-rational curve \(C\) of genus \(g\) and \(r\geq 3\) (the case \(r=2\) has been studied by many authors).
The paper uses and generalizes methods of B. Qin (chamber structure), gives conditions for the existence of stable sheaves and computes the Picard group of the Gieseker-Maruyama compactification of \(M_{H}(r,c_1,c_2)\).
As an application of the chamber structure method, a way of computing Betti numbers of moduli spaces on \({\mathbb{P}}^2\) is given. polarization; moduli space; chamber structure; Chern classes; stable sheaves; ruled surface over a non-rational curve; Betti number K. Yoshioka, Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface, Internat. J. Math. 7 (1996), 411-431. Families, moduli, classification: algebraic theory, Rational and ruled surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Chamber structure of polarization and the moduli of stable sheaves on a ruled surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book is the first of three volumes of an introductory text on the theory of algebraic schemes. These three volumes, the original Japanese versions of which have gained a widespread acceptance as a standard introduction to this subject in Japan, are mainly designed for non-specialists in the field of algebraic geometry, and are aimed to provide the interested reader with the basic ideas and techniques of Grothendieck's theory of schemes, the most general, natural and rigorously elaborated form of algebraic geometry.
There is certainly no lack of good texts on algebraic geometry in its scheme-theoretic setting, but most of them are written for students and mathematicians who already have a fundamental, profound knowledge of (and experience in) both the classical theory of algebraic varieties and the allied commutative, homological, and categorical algebra. In contrast to this situation, the author of the present introduction to algebraic geometry has attempted to develop the subject in the modern scheme-theoretic language, basically so from the very beginning on, and to keep the prerequisites from algebra and classical algebraic geometry to an absolute minimum, just in order to provide a nearly self-contained introduction to scheme theory and its applications as a primer for the very beginner in the field, without leaving out the most crucial theorems and their complete proofs.
Writing such a textbook, with that ambitious methodological program as the basic framework, is a very delicate task, which the author seems to have surpassed in a brilliant manner. Already the English translation of the first volume, which is being reviewed here, indicates the outstanding character of the entire text. -- The major part of this first volume is devoted to carefully preparing the concept of a scheme in the sense of Grothendieck. The first chapter of the book discusses the basics from the classical theory of algebraic varieties over an algebraically closed field, and that just to the extent that is needed to motivate and understand the concept of a scheme treated in the subsequent second chapter. More precisely, chapter 1 treats, in a brief but sufficiently detailed exposition, the following fundamental elementary topics: algebraic sets, Hilbert's Nullstellensatz, affine algebraic varieties, multiplicities and local intersection multiplicities, projective varieties, and an outlook to the ideal-theoretic and geometric aspects of polynomial equations over an arbitrary ring of coefficients.
This last section leads over to chapter 2, entitled ``Schemes'', which explains the prime spectrum of a commutative ring, together with its Zariski topology and sheaf-theoretic aspects, the algebraic principle of localization, the necessary toolkit from general sheaf theory, the concept of a general ringed space and, concludingly, arbitrary schemes, subschemes, and morphisms of schemes.
Chapter 3, the final chapter of this first volume, deals with the indispensible categorical interpretation of scheme theory. Along with a brief introduction to categories, functors, representable functors, and fibre products in categories, scheme-valued points are discussed, the existence of fibre products in the category of schemes is fully proved, and separated morphisms are analyzed.
This first volume ends with the last-mentioned topic, and the author points out in the preface that the subsequent volumes ``Algebraic Geometry. 2'' and ``Algebraic Geometry. 3'' will provide the reader with a unified understanding of algebraic geometry from the scheme-theoretic viewpoint, including sheaf cohomology and its applications to the theory of algebraic curves and surfaces as well as the complex-analytic aspects of algebraic geometry.
In this first volume under review, each chapter comes with a suggestive summary of the main results covered by it, and with a set of complementary exercises, the solutions of which are compiled at the end of the book. There is no bibliography, but perhaps there will be one at the end of the concluding third volume, as all three volumes are to be understood as an integrated treatise. Certainly, some hints to the existing standard texts for further or parallel reading would not have been displaced, but in regard of the author's goal to provide the non-expert reader, and originally the Japanese reader, with a coherent, self-contained textbook on the subject, the missing bibliography is not a serious deficiency whatsoever. This masterly written text, now also available to the international mathematical audience, tells its own tale and represents a highly welcome addition to the great standard textbooks on algebraic geometry. The author's hope that his book in three volumes could find the same benevolent acceptance by the international mathematical community as it did in Japan seems to be absolutely justified and realistic.
We may be looking forward to the English translation of the subsequent two volumes with great pleasure, students just as well as teachers in the field of algebraic geometry. theory of schemes Ueno, K.: Algebraic Geometry 1. American Mathematical Society, Providence (1999) Schemes and morphisms, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Algebraic geometry. 1. From algebraic varieties to schemes. Transl. from the Japanese by Goro Kato | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article gives, in succinct yet very readable form, a beautiful survey of recent developments in geometric Galois theory. The reviewer feels he should refrain from long-winded paraphrases and just try to give some idea of the principal topics, strongly recommending that everyone interested in Galois module theory look at Erez's paper directly.
The starting point is Fröhlich's theory of the \(\mathbb{Z} [G]\)-module structure of \(O_N\) with \(N/K\) an at most tamely ramified \(G\)-Galois extension. As a first classification, later developments either belong to ``tame geometric theory'' continuing classical tame theory into the multidimensional realm, or to the ``theory of Omega invariants'', extending the study of Chinburg's \(\Omega(3,N/K)\) for non-tame Galois extensions \(N/K\) to objects coming from K-theory or from motives. The main emphasis of this survey is on tame geometric theory. A common trait of most results is that they equate, or relate, an algebraic object with an invariant coming from \(L\)-functions, which is in the classical tame case the so-called root number class.
In the tame geometric theory, one considers a projective scheme \(X\) either smooth over a finite field (first subcase) or regular and flat over \(\text{Spec}(\mathbb{Z})\) (second subcase), with an action of a finite group \(G\) fulfilling an appropriate tameness condition: approximately this means that the inertia group of every point has order coprime to the residual characteristic. Let \(f: X\to Y=X/G\) be the quotient scheme. One has a ``refined Euler characteristic'' \(\chi\) which is a group homomorphism between two Grothendieck groups: at the source one takes coherent \(G\)-sheaves on \(X\), at the target cohomologically trivial \(R[G]\)-modules. Here \(R\) is the finite base field (first subcase) or \(\mathbb{Z}\) (second subcase). The existence of this refined characteristic hinges on the fact that for any coherent \(G\)-sheaf \(F\), the stalks of \(f_*F\) are \(G\)-cohomologically trivial, and this is closely related to Noether's theorem concerning the projectivity of \(O_N\) over \(\mathbb{Z}[G]\) for tame extensions \(N/K\).
The definition of the algebraic object is slightly different in the two subcases; we only say that the homomorphism \(\chi\) which we just discussed, and sheaves of differential forms on \(X\) are the main ingredients. (If in the second subcase \(X\) has dimension 1, that is, relative dimension 0 over \(\mathbb{Z} \), then only the zeroth sheaf of differential intervenes, and this is just the ring of functions of \(X\).) We do not go into the description of the analytic class whose equality with the algebraically defined class one wants to prove. It is a sum of two terms, the root number class and the ramification class. The theorems in the first subcase are due to \textit{T. Chinburg} [see for instance Ann. Math. (2) 139, 443-490 (1994; Zbl 0828.14007)], and in the second subcase (``generalized Fröhlich conjecture'') to \textit{T. Chinburg, B. Erez, G. Pappas} and \textit{M. J. Taylor} [see for instance Am. J. Math. 119, 503-522 (1997; Zbl 0927.14013)].
The article also discusses generalized \(\Omega\)-invariants \(\Omega_n(N/K)\) for \(n\geq 0\). They are an outgrowth of Chinburg's \(\Omega(3,N/K)\) and involve certain 2-extensions arising in algebraic \(K\)-theory. Their construction is due to \textit{T. Chinburg, M. Kolster, G. Pappas} and \textit{V. P. Snaith} [C. R. Acad. Sci., Paris, Sér. I 320, 1435-1440 (1995; Zbl 0840.11048)]. A very general motivic construction of \textit{D. Burns} and \textit{M. Flach} [Math. Ann. 305, 65-102 (1996; Zbl 0867.11081)] produces \(\Omega\)-invariants for various motives including Tate motives, and Burns and Flach (preprint) have shown how to recover the \(\Omega_n(N/K)\) from their own construction.
In a short section, Erez presents the nice and very general result of \textit{G. Pappas} [Duke Math. J. 91, 215-224 (1998)] concerning normal bases in elliptic division orders. Previous results [\textit{A. Srivastav} and \textit{M. J. Taylor}, Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)] had to rely heavily on elliptic functions. This rapid overview does not even mention all the topics covered in the article. A lot of deep mathematics goes into all of them. Another attractive feature of this article is that it includes proof sketches for selected special cases; we should also mention the extensive bibliography. Omega invariants; \(L\)-functions; tame coverings; algebraic \(K\)-theory; survey; geometric Galois theory; Galois module theory; tame geometric theory; refined Euler characteristic B. Erez , Geometric trends in Galois module theory ., 116 - 145 , in Galois representations and arithmetic algebraic geometry , eds A.J. Scholl and R.L. Taylor, LMS Lecture Notes 254 , Cambridge University Press , Cambridge , 1998 . MR 1696473 | Zbl 0931.11047 Integral representations related to algebraic numbers; Galois module structure of rings of integers, Finite ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Global ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Group actions on varieties or schemes (quotients), Étale and other Grothendieck topologies and (co)homologies, de Rham cohomology and algebraic geometry Geometric trends in Galois module 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 Let \(\text{Hilb}^{p(t)} (\mathbb P^n)\) be the Grothendieck Hilbert scheme that parametrizes closed subschemes \(X \subset \mathbb P^n\) with Hilbert polynomial \(p(t)\). \textit{A. Reeves} and \textit{M. Stillman} showed that the point in \(\text{Hilb}^{p(t)} (\mathbb P^n)\) corresponding to the lexicographic ideal is smooth [J. Algebr. Geom. 6, 235--246 (1997; Zbl 0924.14004)] and determines a canonical \textit{lexicographic component} of \(\text{Hilb}^{p(t)} (\mathbb P^n)\). \textit{I. Peeva} and \textit{M. Stillman} gave a version of this theorem for toric Hilbert schemes [Duke Math. J. 111, 419--449 (2002; Zbl 1067.14005)]. With \textit{M. Haiman} and \textit{B. Sturmfels} extending the Grothendieck Hilbert schemes to standard graded Hilbert schemes \(\mathcal H^{\mathfrak h} (R)\) parametrizing homogeneous ideals \(I\) with fixed Hilbert function \(\mathfrak h\) in a graded ring \(R\) [J. Algebr. Geom. 13, 725--769 (2004; Zbl 1072.14007)] it becomes natural to ask what the lexicographic points look like in that setting. \textit{D. Maclagan} and \textit{G. G. Smith} gave an analog to the Reeves-Stillman for standard multigraded Hilbert schemes in two variables [Adv. Math. 223, 1608--1631 (2010; Zbl 1191.14007)].
The authors give examples showing that the lexicographic point does not behave so nicely in \(\mathcal H^{\mathfrak h} (R)\) for more variables. They show that for \(S = k[x,y,z]\) and \(\mathfrak h = (1,3,4,4,3,3,3, \dots)\), the Hilbert scheme \(\mathcal H^{\mathfrak h} (S)\) is the union of two irreducible components of dimension \(8\) containing the lexicographic point in their intersection, so the lexicographic point is singular and does not correspond to a canonical component. They also give an example where the lexicographic point is not even Cohen-Macaulay. They also show for the exterior algebra \(E = \bigwedge k^5\) and \(\mathfrak h = (1,5,7,2)\) that the standard graded Hilbert scheme \(\mathcal H^{\mathfrak h} (E)\) is the union of two irreducible components of dimensions \(14\) and \(15\) which contain the lexicographic point in their intersection. standard graded Hilbert scheme; lexicographic component; lexicographic ideal; reducible scheme; exterior algebra Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Exterior algebra, Grassmann algebras On the smoothness of lexicographic points on 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 Let \(X\) be a nonsingular complex projective curve of genus \(g \geq 2\). Define \({\mathcal{F}}(X)\) as the collection of equivalence classes of morphisms \(f: X \rightarrow Y\) onto curves \(Y\) of genus \(g^{\prime}\geq 2\), up to isomorphisms \(Y \rightarrow Y^{\prime}\). This is a finite set, according to the De Franchis-Severi theorem.
\(Z_f\) in \(X \times X\) denotes an effective divisor associated with a morphism \(f: X \rightarrow Y\), the locally principal divisor being defined by \(f(x)=f(y)\). This is a reduced divisor. It only depends on the equivalence class \([f]\) in \({\mathcal{F}}(X)\), and the correspondence \([f] \mapsto Z_f\) is injective.
The main result about this is \textit{E. Kani}'s rigidity theorem [Invent. Math. 85, 185.-198, (1986; Zbl 0615.12017)] which states that, taking the homology class of the divisor, the map
\[
{\mathcal{F}}(X) \rightarrow H_2(X \times X, \mathbb Z), \;\;[f] \mapsto [Z_f]
\]
is still injective. The aim of the paper under review is to describe this map.
Define \({\mathcal{F}}_n(X)\) as the collection of equivalence classes of morphisms of degree \(n\). The author describes its image as a subset of an intersection of two loci: the locus of integral points \(V_n( \mathbb Z)=V_n \cap H_2(X \times X, \mathbb Z)\), where \(V_n\) is an algebraic set in \(H_2(X \times X, \mathbb C)\), and the locus of effective homology classes in the product. Here, \(V_n(\mathbb Z)\) is a finite set. The author writes explicit algebraic equations for \(V_n\) in a remarkably simple form. In the case of genus \(g=3\), certain computations are presented, which lead to a few explicit solutions in \(V_n \cap H_2(X \times X, \mathbb Q)\).
Different but related approaches to the subject have been developed by Martens, Tanabe, and Naranjo-Pirola. Genus of a curve; locally principal divisors; morphisms; homology Curves in algebraic geometry, (Co)homology theory in algebraic geometry Morphisms on an algebraic curve and divisor classes in the self product | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let K be an algebraically closed field and C be a curve in \({\mathbb{P}}^ 3(K)\), i.e. a 1-dimensional closed subscheme of \({\mathbb{P}}^ 3(K)\) without point componts (embedded or not). For a function f: \({\mathbb{Z}}\to {\mathbb{Z}}\) one defines the first difference \(\partial f\) of f by \(\partial f(n)=f(n)-f(n-1)\) (higher differences are defined in a similar way). In the present paper the authors associate to the curve C the couple \((\gamma_ C,M_ C)\), where \(\gamma_ C(n)=\partial^ 3(h^ 0({\mathcal J}_ C(n))-h^ 0({\mathcal O}_{{\mathbb{P}}^ 3}(n)))\), which is called ``caractère de postulation'' of C, \(M_ C\) the Rao module, and they determine the image of the map \(C\mapsto (\gamma_ C,M_ C)\). - Moreover they relate this problem to the study of the minimal curves in the bilinkage classes. postulation; Rao module; minimal curves in the bilinkage classes Martin-Deschamps, M.; Perrin, D.: Sur la classification des courbes gauches. Astérisque 184 (1990) Plane and space curves, Linkage, Projective techniques in algebraic geometry Sur la classification des courbes gauches. I. (Classification of space curves. I) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be an algebraically closed field of positive characteristic \(p\). Let \(\text{Aut}(\mathcal{X})\) be the automorphism group of a projective non-singular geometrically irreducible algebraic curve \(\mathcal{X}\) of genus \(g\geq2\). \textit{H. W. Henn} proved in [J. Reine Angew. Math. 302, 96--115 (1978; Zbl 0378.12011)] that \(\mathcal{X}\) is birationally equivalent to a Hermitian curve if \(|\text{Aut}(\mathcal{X})|>16g^3+24g^2+g\). In this paper, the authors prove this result for plane curves, with a better lower bound \(|\text{Aut}(\mathcal{X})|>3(g^2+g)\left(\sqrt{8g+1}+3\right)\). The ingredients of the paper are Hilbert's ramification theory, Stöhr-Voloch theory on Weierstrass points with respect to a base-point-free linear series and some results on finite groups, such as the Kantor-O'Nan-Seitz theorem. plane curve; automorphism group; hermitian curve Automorphisms of curves, Arithmetic algebraic geometry (Diophantine geometry) On the size of the automorphism group of a plane algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we can get the details of the paper reviewed above [\textit{D. Abramovich} and \textit{J. Wang}, Math. Res. Lett. 4, No. 2--3, 427--433 (1997; Zbl 0906.14005)]. The authors provide a new proof of Hironaka's well-known theorem on resolution of singularities as follows.
Theorem: Let \(X\) be a variety of finite type over an algebraically closed field \(k\) of characteristic 0, let \(Z\subset X\) be a proper closed subset. There exists a modification \(f:X_1\to X\), such that \(X_1\) is a quasi-projective nonsingular variety and \(Z_1= f^{-1} (Z)_{\text{red}}\) is a strict divisor of normal crossings.
Structure of the proof: (1) We choose a projection \(X\to P\) of relative dimension 1 and apply semistable reduction to obtain a model \(X'\to P'\) over a suitable Galois base change \(P'\to P\) with Galois group \(G\).
(2) We apply induction on the dimension to \(P\). We may assume that \(P\) is smooth, and the discriminant locus of \(X'/G\to P\) is a strict divisor of normal crossings.
(3) A few auxiliary blow-ups make the quotient \(X'/G\) toroidal.
(4) A theorem of \textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat} [``Toroidal embeddings'', Lect. Notes Math. 339 (1973; Zbl 0271.14017)] about toroidal resolutions finishes the argument. resolution of singularities; toroidal embedding; divisor of normal crossings [AJ]Abramovich, D. \&Jong, A.J. ed, Smoothness, semistability and toroidal geometry.J. Algebraic Geom., 6 (1997), 789--801. Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Smoothness, semistability, and toroidal geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give an algebraic proof of and generalise a result of [\textit{A. R. Magid}, Proc. Am. Math. Soc. 71, 164--168 (1978; Zbl 0393.20029)] from characteristic zero originally proven using the Lefschetz principle and comparison of étale and singular cohomology.
The main theorem (Theorem 2.11) is: ``Let \(F\) be a field and \(n \geq 2\) an integer which is not divisible by \(\mathrm{char}\,F\). Let further \(X\) be a regular and geometrically integral \(F\)-scheme of finite type, and \(T\) an \(F\)-scheme, such that either \(T_s\) is isomorphic as an \(F_s\)-scheme to a torus, or \(T\) is the ``trivial'' torus \(\mathrm{Spec}\,F\). We assume that (i) \(\mathrm{Pic}\,X_s = 0\), (ii) \(\mathrm{H}^0(X_s, \mathrm{K}^M_i/n) = 0\) for \(i = 1,2\), and (iii) \(F_s[X_s]^\times = F_s^\times\). Then the natural homomorphism \(p_T: {}_n\mathrm{Br}(T) \to {}_n\mathrm{H}^2_{\mathrm{et}}(X \times_F T,\mathbb{G}_m)\) along the projection \(p_T: X \times_F T \to T\) is an isomorphism.''
In section 3, the authors draw several corollaries from this theorem, for example Theorem 3.2: Let \(T\) and \(n\) be as in Theorem 2.11. ``Let \(F\) be a field of characteristic \(\neq 2\), and \(X_q \hookrightarrow \mathbb{P}^m_F\) be the projective quadric defined by the equation \(q = \sum_{i=0}^ma_ix_i^2\) with \(m \geq 4\) and \(a_i \in F^\times\) for all \(i = 0, 1, \ldots, m\). Let further \(X_{q,\mathrm{aff}} \subset X_q\) be the open affine quadric defined by \(x_0 \neq 0\). Then the pull-back homomorphisms \({}_n\mathrm{Br}(T) \to {}_n\mathrm{Br}(X_{q,\mathrm{aff}} \times_F T) \quad\text{and}\quad {}_n\mathrm{Br}(T) \to {}_n\mathrm{H}^2_{\mathrm{et}}(X_q \times_F T,\mathbb{G}_m)\) along the respective projections to \(T\) are isomorphisms.'' A further Corollary 3.14: ``Let \(H\) be a connected reductive group over a field \(F\), such that \([H,H]\) is of adjoint type. Assume that the characteristic of \(F\) does not divide the order of the fundamental group of the commutator subgroup of \(H\). Then \({}_n\mathrm{Br}(\mathrm{Rad}(H)) \cong {}_n\mathrm{Br}(H)\) for all integers \(n\) which are not divisible by \(\mathrm{char}\,F\) and coprime to the order of the fundamental group of the commutator subgroup \([H,H]\).''
In section 4, the authors study the morphisms \(\mathrm{Br}(R) \to \mathrm{Br}(R[T_1,\ldots,T_d])\) for rings \(R\) of positive characteristic. In particular, they prove Theorem 4.5: ``Let \(F\) be a field of characteristic \(p > 0\) and \(d \geq 1\) an integer. Then we have \(\mathrm{Br}(F[T_1,\ldots,T_d]) \cong \mathrm{Br}(F) \oplus \mathbb{Z}(p^\infty)^{(I)}\), where the set \(I\) is non-empty if (and only if) \(F\) is not perfect or \(d \geq 2\).'' Brauer groups of schemes; Applications of methods of algebraic \(K\)-theory Gille, S; Semenov, N, On the Brauer group of the product of a torus and a semisimple algebraic group, Isr. J. Math., 202, 375-403, (2014) Brauer groups of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the Brauer group of the product of a torus and a semisimple algebraic group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to relate Drinfeld modules with the theory of KP equations. For this we construct an immersion of the Drinfeld moduli schemes in an adelic infinite Grassmannian. In this way, Drinfeld moduli schemes will be locally closed subschemes of this Grassmannian.
This result is derived from two previous results: The first is that the moduli functor of vector bundles with certain level structures (defined below) over an algebraic curve, is representable in a natural way, by a closed subscheme of an infinite Grassmannian. It is obtained bearing in mind a relative Krichever morphism [see e.g. \textit{M. Mulase}, Proc. Symp. Pure Math. 49, Pt. 1, 39--50 (1989; Zbl 0676.58010) and \textit{I. Quandt}, Bayreuther Math. Schr. 52, 1--74 (1997; Zbl 0907.14004)]. The second is an equivalence between Drinfeld modules and diagrams of vector bundles called ``elliptic sheaves''. This equivalence was extended for level structures by \textit{G. W. Anderson} [Duke Math. J. 53, 457--502 (1986; Zbl 0679.14001)]. These results allow us to consider Drinfeld modules with level structures as diagrams similar to those of \textit{P. Deligne} and \textit{G. Lusztig} [Ann. of Math., II. Ser. 103, 103--161 (1976; Zbl 0336.20029)]. With the considerations presented here, we showed earlier [\textit{A. Álvarez Vazquez}, Int. J. Math. 11, No. 7, 949--968 (2000; Zbl 0982.14020)], more relations between elements of the theory of KP equations and Drinfeld modules: Baker's function, Schur pairs, etc. This issue was also studied by \textit{G. W. Anderson} [Duke Math. J. 73, No. 3, 491--542 (1994; Zbl 0807.11032)]. adelic infinite Grassmannian; Drinfeld moduli schemes; Krichever morphism; KP equations; Baker's function; Schur pairs A. Álvarez, Drinfeld moduli schemes and infinite Grassmannians, J. Algebra 225 (2000), no. 2, 822 -- 835. , https://doi.org/10.1006/jabr.1999.8158 A. Álvarez, Erratum: ''Drinfeld moduli schemes and infinite Grassmannians'', J. Algebra 229 (2000), no. 2, 794. , https://doi.org/10.1006/jabr.2000.8425 Erratum: ''Drinfeld moduli schemes and infinite Grassmannians'' [J. Algebra 225 (2000), no. 2, 822 -- 835; MR1741564 (2001e:11059a)] by A. Álvarez, J. Algebra 231 (2000), no. 2, 831. Drinfel'd modules; higher-dimensional motives, etc., Algebraic moduli problems, moduli of vector bundles, Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Group structures and generalizations on infinite-dimensional manifolds Drinfeld moduli schemes and infinite Grassmannians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is an enjoyable introduction to the study of the tautological ring of the moduli space of curves. It is aimed at a general public of mathematicians, even if some knowledge of geometry and topology is assumed. References for further reading on the subject are also included.
The article under review focuses on the combinatorial structure of the tautological ring, including a selection of the most important results and conjectures in that area. The approach to the subject is that of algebraic geometry.
The article begins by recalling the construction of the Grassmannian of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space. Following \textit{D. Mumford} [in: Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)], the Grassmannian is considered here as the prototypical example of a moduli space. Subsequently, the moduli space of \(n\)-pointed genus \(g\) smooth curves is presented, together with its Deligne-Mumford compactification. The basic properties of these spaces are explained, and examples presented. The tautological ring of the moduli space of curves is then introduced as the ring generated by ``all classes naturally coming from geometry''. Finally, some of the most important results and conjectures related to the properties of the tautological ring are explained, including (among others) Kontsevich's Theorem (Witten's Conjecture), the ELSV formula for Hurwitz numbers and Faber's Conjecture. R. Vakil, ''The moduli space of curves and its tautological ring,'' Notes Amer. Math. Soc., 50, No. 6, 647--658 (2003). Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry The moduli space of curves and its tautological ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let \(X=Spec(A)\), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime \({\mathfrak p}\) of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,\({\mathfrak p})\), correspond to the connected components of \(p^{-1}(Y)\), where \(p: X'\to X\) is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along \(U\cap Y\) and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,\({\mathfrak p})\), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X.
For closely related results proved with completely different topological techniques see the paper by \textit{G. Tedeschi}, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)]. analytic branches of an algebraic affine variety along a singular subvariety; henselian rings; geometric global branches Ramification problems in algebraic geometry, Henselian rings Global branches and parametrization | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 formal at meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper (part II, [\textit{K. S. Kedlaya}, J. Am. Math. Soc. 24, No. 1, 183--229 (2011; Zbl 1282.14037)]) we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the \textit{turning locus}, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by \textit{Y. André} [Invent. Math. 170, No. 1, 147--198 (2007; Zbl 1149.32017)] in the case of a smooth polar divisor. We also construct an \textit{irregularity sheaf} and its associated \(b\)-\textit{divisor}, which measure irregularity along divisors on blowups of the original space; this generalizes another result of André [loc. cit.] on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties. flat connection; irregularity; turning locus Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, Non-Archimedean analysis Good formal structures for flat meromorphic connections. III: Irregularity and turning loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [\textit{K. Thas}, Proc. Japan Acad., Ser. A 90, No. 1, 21--26 (2014; Zbl 1329.14009)] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, \(\mathbb{F}_1\)) to a so-called ``loose graph'' (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over \(\mathbb{F}_1\) such as combinatorial \(\mathbb{F}_1\)-projective and \(\mathbb{F}_1\)-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of [loc. cit.], and show that Deitmar schemes which are defined by finite trees (with possible end points) are ``defined over \(\mathbb{F}_1\)'' in Kurokawa's sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated \(\mathbb{F}_q\)-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara's zeta function (which is trivial in this case). Using a process called ``surgery,'' we show that one can determine the zeta function of a general loose graph and its associated {Deitmar, Grothendieck}-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the \textit{Grothendieck ring of}\(\mathbb{F}_1\)\textit{-schemes} along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara's zeta function for graphs in a number of examples. field with one element; Deitmar scheme; loose graph; zeta function; Ihara zeta function Mérida-Angulo, M.; Thas, K., Deitmar schemes, graphs and zeta functions, J. Geom. Phys., 117, 234-266, (2017) Schemes and morphisms, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Deitmar schemes, graphs and zeta functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0722.00001.]
\textit{G. Faltings} [``Diophantine approximation on abelian varieties'' (preprint, Princeton 1990); see also Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] proved the theorem 1 below, which generalizes the Mordell conjecture proved by himself [Invent. Math. 73, 349-366 (1983); erratum: ibid. 75, 381 (1984; Zbl 0588.14026)] too, six years before. The author presents an outline of the proof of Faltings' new theorem. This is a welcome guide line for each specialist of arithmetic algebraic geometry who wants to understand the proof of:
Theorem 1. Let \(A\) be an abelian variety over a number field \(K\), \(\overline\mathbb{Q}\) the field of all algebraic numbers and \(X\) a closed algebraic subvariety of \(A\) such that \(X\) does not contain over \(\overline\mathbb{Q}\) any translated abelian subvariety \(P+B\), \(P\in A(\overline\mathbb{Q})\), \(B\) abelian, \(\dim(B)>0\). Then the set \(X(K)\) of \(K\)- points on \(X\) is finite.
After presenting the main result the author introduces derivations and the index of bundle sections in the second section. Furthermore a splitting theorem is proved for certain subvarieties of \(\mathbb{P}^{n_ 1}\times\cdots\times\mathbb{P}^{n_ m}\). --- In the third section one finds a proof of ampleness of sheaves \(L(-\varepsilon,s_ 1,\ldots,s_ m)\) on \(A^ m\) depending on positive rational parameters, \(m\gg 0\), using numerical results of algebraic geometry, especially the theorem of Nakai- Moishezon-Kleiman. --- In the fourth section classical numerical results on lattice points due to Minkowski and Siegel are brought together with Arakelov theory of bundles in order to estimate norms of sections of the above sheaves. The assumption of the existence of infinitely many \(K\)- points on \(X\) leads to a contradiction via complicated estimations for norms of derived sections on this way. It should be mentioned that the paper involves a deep connection between methods of the theory of transcendental numbers and arithmetic algebraic geometry. arithmetic varieties; Arakelov theory; finiteness theorem; Diophantine approximation; transcendental numbers L. SZPIRO , Sur les solutions d'un système d'équations polynomiales sur une variété abélienne (d'après G. FALTINGS and P. VOJTA ) (Séminaire N. Bourbaki, No. 729, June 1990 ). Numdam | Zbl 0746.14010 Rational points, Abelian varieties of dimension \(> 1\), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Transcendence (general theory), Algebraic theory of abelian varieties On the solutions of a system of polynomial equations on an abelian variety [after G. Faltings and P. Vojta] | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal{X}\) be a regular scheme proper over \(\mathrm{Spec}\,\mathbb{Z}\). Constructing perfect Weil-étale complexes \(R\Gamma_{W,c}(\mathcal{X},\mathbb{Z}(n))\) of abelian groups for any \(n \in \mathbb{Z}\) and Weil-Arakelov complexes in the bounded derived category of locally compact abelian groups with an exact triangle
\[
R\Gamma_{\mathrm{ar},c}(\mathcal{X},\mathbb{Z}(n)) \to R\Gamma_{\mathrm{ar},c}(\mathcal{X},\tilde{\mathbb{R}}(n)) \to R\Gamma_{\mathrm{ar},c}(\mathcal{X},\tilde{\mathbb{R}}/\mathbb{Z}(n)) \to
\]
(without constructing a topos) under standard assumptions \(\mathbf{AV}(\overline{\mathcal{X}}_{\text{ét}},n), \mathbf{L}(\overline{\mathcal{X}}_{\text{ét}},n), \mathbf{L}(\overline{\mathcal{X}}_{\text{ét}},d-n)\) and \(\mathbf{B}(\mathcal{X},n)\), the authors give a conjectural description of the vanishing order \(\mathrm{ord}_{s=n}\zeta(\mathcal{X},s)\) and the leading Taylor coefficient \(\zeta^*(\mathcal{X},n) \in \mathbb{R}\) of the zeta function of \(\mathcal{X}\) building upon work of \textit{S. Lichtenbaum} [Compos. Math. 141, No. 3, 689--702 (2005; Zbl 1073.14024); Ann. Math. (2) 170, No. 2, 657--683 (2009; Zbl 1278.14029); in: Motives and algebraic cycles. A celebration in honour of Spencer J. Bloch. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences.. 249--255 (2009; Zbl 1245.14023)], \textit{T. Geisser} [Math. Ann. 330, No. 4, 665--692 (2004; Zbl 1069.14021); Duke Math. J. 133, No. 1, 27--57 (2006; Zbl 1104.14011)] and the second author [Duke Math. J. 163, No. 7, 1263--1336 (2014; Zbl 1408.14076)].
For \(\mathcal{X}\) smooth over the spectrum of the ring of integers of a number field, the authors prove compatibility with the Tamagawa number conjecture [\textit{J.-M. Fontaine} and \textit{B. Perrin-Riou}, Proc. Symp. Pure Math. 55, 599--706 (1994; Zbl 0821.14013); \textit{S. Bloch} and \textit{K. Kato}, Prog. Math. 86, 333--400 (1990; Zbl 0768.14001)], in particular with the analytic class number formula and the conjecture of Birch-Swinnerton-Dyer. Zeta functions; Zeta-values; Weil-étale cohomology; Arakelov theory Étale and other Grothendieck topologies and (co)homologies, Motivic cohomology; motivic homotopy theory, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Weil-étale cohomology and zeta-values of proper regular arithmetic 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 \(\pi \colon X\to S\) be a morphism of arithmetic varieties and \(\overline E\) a Hermitian vector bundle on \(X\). The arithmetic Riemann-Roch theorem of \textit{H. A. Gillet} and \textit{C. Soulé} [Invent. Math. 110, No. 3, 473--543 (1992; Zbl 0777.14008)], computes the arithmetic first Chern class of the determinant line bundle of \(\overline E\) with respect to \(\pi \), equipped with the Quillen metric. The special case of the trivial line bundle on the projective space over \(\text{Spec}\,\mathbb Z\) was proved by \textit{H. A. Gillet} and \textit{C. Soulé} [Topology 30, No. 1, 21--54 (1991; Zbl 0787.14005)] using a computation of the spectrum of the Laplacian. In the present paper another proof of this case is given. The author shows that it is a formal consequence of a Künneth type formula for the determinant bundles, together with the results concerning the behavior of Quillen metrics with respect to the holomorphic immersions, as obtained by \textit{J.-M. Bismut}, \textit{H. A. Gillet} and \textit{C. Soulé} [in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 249--331 (1990; Zbl 0744.14015)], and \textit{J.-M. Bismut} and \textit{G. Lebeau} [Publ. Math., Inst. Hautes Étud. Sci. 74, 1--297 (1992; Zbl 0784.32010)]. This new proof is interesting in that it greatly simplifies the original argument and sheds some light on the reason for the appearance of the \(R\)-genus of Gillet-Soulé in the arithmetic Riemann-Roch theorem. Bost, J.-B., Analytic torsion of projective spaces and compatibility with immersions of Quillen metrics, Int. Math. Res. Not., 8, 427-435, (1998) Arithmetic varieties and schemes; Arakelov theory; heights, Riemann-Roch theorems, Determinants and determinant bundles, analytic torsion Analytic torsion of projective spaces and compatibility with immersions of Quillen metrics. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 curve (not necessarily irreducible) on a surface \(S\) and \(\omega_D\) its dualizing sheaf. In this note we study some properties of both the adjoint system \(|K+ D|\) and \(\omega_D\). These properties can be useful in various situation where Reider's method [\textit{I. Reider}, Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0633.14010)] cannot be applied and can be used, for instance, to simplify the proofs of the birationality of \(\varphi_{3K}\) for minimal surfaces of general type with \(p_g=0\) of \textit{E. Bombieri} and \textit{F. Catanese} [in: C. P. Ramanujam. A tribute, Tata Inst. Fundam. Res., Stud. Math. 8, 279-290 (1978; Zbl 0423.14003)] and of \textit{Y. Miyaoka} [Invent. Math. 34, 99-111 (1976; Zbl 0337.14010)]. curve on a surface; dualizing sheaf; adjoint system; minimal surfaces of general type M. MENDES LOPES, Adjoint systems on surfaces, Bollettino U.M.I. (7), 10-A (1996), pp. 169-179. Zbl0861.14023 MR1386254 Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves Adjoint systems on surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a curve of genus g defined over an algebraically closed field K, and let \(D=\{D_ t\}_{t\in T}\) be an algebraic family of divisors on C which is parametrized by a curve T. In order to be able to decide whether such a family is ''contained in a linear series'', i.e. whether all the divisors \(D_ t\) are mutually linearly equivalent, \textit{G. Castelnuovo} [R. Accad. Lincei, Rend., Ser. V 15, 337-344 (1906)] introduced a remarkable invariant \((D)\geq 0\) which has the property that \(z(D)=0 \Leftrightarrow\) all the \(D_ t\) are linearly equivalent ( \(\Leftrightarrow D\equiv 0\), i.e. \(D\sim a\times C+T\times A,\) with \(a\in Div(T)\) and \(A\in Div(C));\) for this reason, \textit{C. Torelli} [Rend. Circ. Mat. Palermo 37, 27-46 (1914)] later called z(D) the ''equivalence defect'' of D. Castelnuovo's principal result is that this invariant may be calculated from the fundamental invariants attached to the family by the formula
\[
(*)\quad z(D)=frac{1}{2}\sigma(D,D),
\]
where \(\sigma(D,D)=2n(D)\nu(D)-\gamma(D)\) is the ''Severi-Weil metric'' on Di\(v(T\times C)\), and \(n(D)=(D\cdot t\times C)\) (for \(t\in T)\), \(\nu(D)=(D\cdot T\times P)\) (for \(P\in C)\), and \(\gamma(D)=(D.D).\) From (*) one obtains the inequality \(\sigma(D,D)\geq 0,\) which is often referred to as the ''Inequality of Castelnuovo-Severi'' [cf. \textit{A. Mattuck} and \textit{J. Tate}, Abh. Math. Semin. Univ. Hamburg 22, 295-299 (1958; Zbl 0081.376)].
The main purpose of the paper under review is to investigate the formula (*). As it turns out, if one uses Castelnuovo's original definition of z(D), then (*) is no longer valid if char(K)\(\neq 0\); one therefore has to redefine the invariant z(D) in a suitable way. Two such methods are known: (1) via intersection numbers involving the \(\theta\)-divisor on the Jacobian variety \(J_ C\) of C [due to \textit{G. Castelnuovo}, Atti R. Accad. Lincei, Rend., Cl. Sci. Fis. Mat. Nat. Ser. V 30, 50-55, 99-103, 195-200, 355-359 (1921)] and also \textit{A. Weil} [''Variétés abéliennes et courbes algébriques'' (Paris 1948; Zbl 0037.162)]; and (2) via intersection theory on symmetric products of curves and Schubert's formula (in char(K)\(\neq 0\), due to \textit{J. Igusa} [J. Math. Soc. Japan 1, 147-197 (1949; Zbl 0039.032)]). - In this paper another method is presented, one that does not leave the surface \(X=T\times C\); it is as follows. For any family \(D=\{D_ t\}_{t\in T}\) let \(\delta(D)=\min_{t} \ell(D_ t)\) denote the ''generic dimension'' of D (here, \(\ell(D_ t)=\dim H^ 0(C,{\mathcal L}(D_ t))),\) and let \(var(D)=\sum(\ell(D_ t)-\delta(D_ t))\) denote the ''total variation'' of the family. Then one has: Theorem: Let \(D\in Div(X/T)^+,\) i.e. let D be an effective divisor on \(X=T\times C\) which is flat over T. If \(\delta(D)=1\), then
\[
(**)\quad \nu(D)+var(D)\leq frac{1}{2}\delta(D,D).
\]
Moreover, if in addition we have \(n(D)=g\) and \(var(D)=0,\) then equality holds in (**); i.e. we have \(\nu(D)=frac{1}{2}\sigma(D,D).\) Thus, if we put \(z'(D)=\max(\nu(D')+var(D')),\) where the maximum extends over all divisors \(D'\in Div(X/T)^+\) with \(\delta(D')=1\) and \(D'\equiv D,\) then on the one hand it is easy to see that \(z'(D)=0 \Leftrightarrow\) \(D\equiv 0\) (of course one has \(z'(D)\geq 0),\) and on the other hand we have ''Castelnuovo's formula'': \(z'(D)=frac{1}{2}\sigma(D,D).\) (Also, it is easy to see that in characteristic 0 one has \(z'(D)=z(D)\).) Moreover, from the above theorem it is possible to deduce ''Castelnuovo's formula on Jacobians'', viz.
\[
(***)\quad(T\cdot \alpha_ D^*(\Theta))=frac{1}{2}\sigma(D,D),
\]
where \(\Theta\) denotes the theta- divisor on the Jacobian variety \(J_ C\) of C, \(\alpha_ D:J_ T\to J_ C\) denotes the homomorphism induced by D, and ( \(\cdot)\) denotes the intersection number of the Jacobian \(J_ T\) of T. By a more careful analysis of the above proof one also obtains a formula for the local intersection number \((T\cdot \alpha_ D^*(\Theta))_ x\) (provided that this intersection number is defined), which after summation over \(x\in J_ T\) yields the ''global formula'' (***). algebraic family of divisors on a curve of genus g; Castelnuovo-Severi inequality; finite characteristic; linear series; equivalence defect; theta-divisor on the Jacobian variety Kani E.: Castelnuovo's equivalence defect. J. Reine Angew. Math. \textbf{352}, 24-70 (1984). Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Theta functions and abelian varieties, Projective techniques in algebraic geometry On Castelnuovo's equivalence defect | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A determinantal representation of a projective plane curve F(x) of degree \(n\) is a matrix U(x) of order \(n\) whose entries are linear forms such that \(\det (U(x))=cF(x)\) for some constant c. Two determinantal representations \(U_ 1, U_ 2\) are called equivalent if there exist constant matrices M, N such that \(U_ 2=MU_ 1N\). If F(x) is a smooth curve, ker(U(x)) is a vector bundle of rank 1. Up to equivalence, the determinantal representations of F(x) are parametrized, via the class of divisors of the corresponding vector bundle, by the points of the Jacobian variety of F(x) not on the exceptional subvariety [see \textit{R. J. Cook} and \textit{A. D. Thomas}, Q. J. Math., Oxf. II. Ser. 30, 423-429 (1979; Zbl 0437.14004)]. Based on this fact, the author shows that one can build up all the non-equivalent determinantal representations of F(x) by applying a finite sequence of the so-called elementary transformations to F(x) whose number is bounded by the genus of F(x). determinantal representation of a projective plane curve; divisors; Jacobian variety Vinnikov V. (1990) Elementary transformations of determinantal representations of algebraic curves. Linear Algebra Applications 135: 1--18 Curves in algebraic geometry, Vector bundles on curves and their moduli, Determinantal varieties, Jacobians, Prym varieties Elementary transformations of determinantal representations 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 aim of this paper is to contribute to the study of Prym-Tyurin varieties. We recall that to a correspondence of a curve \(C\), satisfying the equation \(\varphi^ 2+(m-2)\varphi-(m-1)=0\), \(m\) a positive integer, it is associated an abelian subvariety of the Jacobian \(J(C)\).
The authors prove that, given a polarized abelian variety \(X\), to any of its abelian subvarieties can be associated an equation as above, this gives a geometric interpretation of the exponent \(m\). -- In fact let \((X,L)\) be a polarized abelian variety, \(\varepsilon\) an idempotent of \(\text{End}_ \mathbb{Q}(X)\) symmetric with respect to the Rosati involution, and \(n\) the minimal positive integer such that \(n\varepsilon\) belongs to \(\text{End}(X)\), we denote by \(X^ \varepsilon\) the image of the endomorphism, then the correspondence \(\varepsilon\to X^ \varepsilon\) is a bijection between the set of symmetric idempotents of \(\text{End}_ \mathbb{Q}(X)\) and the set of abelian subvarieties of \(X\). When \(X\) is principally polarized, \(n\) can be interpreted as the exponent of \(X^ \varepsilon\) and once we set \(\varphi=1-n\varepsilon\) we obtain a solution for the equation. Moreover some properties of the exponents are studied, in particular it is obtained, as a consequence, de Franchis' theorem on the finite number of morphisms onto curves of genus \(g\geq 2\). Prym-Tyurin varieties; correspondence of a curve; Jacobian; subvariety of abelian variety; de Franchis' theorem; number of morphisms onto curves \beginbarticle \bauthor\binitsC. \bsnmBirkenhake and \bauthor\binitsH. \bsnmLange, \batitleThe exponent of an Abelian subvariety, \bjtitleMath. Ann. \bvolume290 (\byear1991), page 801-\blpage814. \endbarticle \OrigBibText C. Birkenhake and H. Lange. The exponent of an abelian subvariety. Math. Ann. , 290:801-814, 1991. \endOrigBibText \bptokstructpyb \endbibitem Theta functions and abelian varieties, Jacobians, Prym varieties, Rational and birational maps The exponent of an abelian subvariety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0744.00029.]
Let \(C\) be a smooth connected complex curve of genus \(g\), and for all integers \(r\) and \(d\), let \(W^ r_ d(C)\) be the subscheme of \(Pic^ d(C)\) parametrizing isomorphism classes of line bundles of degree \(d\) with at least \(r\) independent sections. When \(C\) is general, the latter has dimension \(\rho = g - (r + 1)(g - d + r)\); when \(\rho \geq 2\), results of Fulton and Lazarsfeld imply that the Albanese variety of \(W^ r_ d(C)\) is isomorphic to \(\text{Pic}^ d(C)\). When \(\rho = 1\), the scheme \(W^ r_ d(C)\) is a smooth curve; it generates \(\text{Pic}^ d(C)\) hence induces a surjection \(\text{Jac}(W^ r_ d(C)) \to \text{Pic}^ d(C)\). The kernel \(K^ r_ d(C)\) is defined over the field \(\mathcal K\) of rational functions of the moduli space of pointed curves of genus \(g\), and is in general nontrivial. It is proved that if \(C\) is general of genus \(g\geq 3\) and \(\rho = 1\), the kernel \(K^ r_ d(C)\) is connected, and that the ring of rationally determined endomorphisms (i.e. defined over \(\mathcal K\)) of the abelian variety \(K^ 1_ d(C)\) is isomorphic to \(\mathbb{Z}\). An example of Pirola shows that the latter does not hold for \(r\geq 2\). The authors believe that the ring of all endomorphisms of \(K^ 1_ d(C)\) should still be isomorphic to \(\mathbb{Z}\), but they could not prove it. This theorem implies the extension to the case \(\rho = r = 1\) of a result of \textit{C. Ciliberto} [Duke Math. J. 55, 909-917 (1987; Zbl 0657.14013)] who proved it for \(\rho \geq 2\): the group of rationally determined line bundles on the curves of the universal family over any component of the Hurwitz scheme of coverings of \(\mathbb{P}^ 1\) of degree \(d\) and genus \(g \geq 3\) is generated by the relative canonical bundle and the bundle \({\mathcal O}_{\mathbb{P}^ 1}(1)\). The question remains whether this statement still holds for \(\rho = 1\) and \(r\geq 2\), or for \(\rho = 0\). The proof of the first result proceeds by degenerating the curve to a reducible tree-like curve, and by studying the limit of \(K^ 1_ d(C)\) using limit linear series. limit linear series; higher Picard group; higher Jacobian; complex curve; isomorphism classes of line bundles of degree \(d\); Albanese variety; moduli space of pointed curves Ciro Ciliberto, Joe Harris, and Montserrat Teixidor i Bigas, On the endomorphisms of \?\?\?(\?\textonesuperior _{\?}(\?)) when \?=1 and \? has general moduli, Classification of irregular varieties (Trento, 1990) Lecture Notes in Math., vol. 1515, Springer, Berlin, 1992, pp. 41 -- 67. Jacobians, Prym varieties, Families, moduli of curves (algebraic), Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves On the endomorphisms of \(\text{Jac}(W_ d^ 1(C))\) when \(\rho=1\) and \(C\) has general moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((A,\theta)\) be a principally polarized abelian variety. One defines \(\Gamma_{00}\) as the set of global sections of \({\mathcal O}_ A(2\Theta)\) which vanish with multiplicity at least 4 at the origin. The set \(V(\Gamma_{00})\) consists of the points in \(A\) on which every section in \(\Gamma_{00}\) vanishes.
A result of G. Welters states that if \(A\) is the Jacobian of a curve \(C\) then \(V(\Gamma_{00})\) consists of all divisor classes in \(\text{Jac}(C)=\text{Pic}^ 0(C)\) of the form \([a-b]\) for \(a,b\in C\). It is conjectured that this characterizes Jacobians: if \((A,\Theta)\) is \textit{not} a Jacobian, then \(V(\Gamma_{00})=\{0\}\).
The present note proves that \(V(\Gamma_{00})=\{0\}\) in case \((A,\Theta)\) is the intermediate Jacobian of a smooth cubic 3-fold in \(\mathbb P^ 4\), and in case \(A\) is the Prym variety associated with an étale double cover \(\tilde C\to C\), where \(C\) is a smooth plane curve and the covering corresponds by class field theory to a divisor \(D\) on \(C\) (with \(2D\sim 0)\) such that \(H^ 0(C,{\mathcal O}_ C(2)\otimes D)=(0)\).
The main ingredients in the proofs are Beauville's description of the singularities of the theta divisor of such an intermediate Jacobian, and an interesting generalization of the fact that if \(a\in \Theta\) is a singular point on the theta divisor, then \(V(\Gamma_{00})\subset (\Theta +a)\cup (\Theta-a)\). Schottky problem; principally polarized abelian variety; jacobian of a curve; intermediate jacobian of a smooth cubic 3-fold; Prym variety BEAUVILLE (A.) , DEBARRE (O.) , DONAGI (R.) , VAN DER GEER (G.) . - Sur les fonctions thêta d'ordre deux et les singularités du diviseur thêta , C.R. Acad. Sc. Paris, t. 307, 1988 , p. 481-484. MR 90b:14053 | Zbl 0699.14057 Theta functions and abelian varieties, Picard schemes, higher Jacobians, Jacobians, Prym varieties, Coverings of curves, fundamental group, \(3\)-folds On second order theta functions and the singularities of the theta divisor | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The notion of the generalized Jacobian of a complete integral curve \(X\) (over a field \(k\)) with respect to a non-trivial effective divisor \(D\) is introduced. This generalizes the concept of the generalized Jacobian such as given by Rosenlicht-Serre. --- This new generalized Jacobian has an interpretation as a rigidificator for the Jacobian. generalized Jacobian of a complete integral curve Jacobians, Prym varieties Generalized Jacobians and rigidificators | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 and let \(L\) be a line bundle on \(X\). Fix a point \(x\in X\). \textit{J.-P. Demailly} [in: Complex algebraic varieties, Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-104 (1992; Zbl 0784.32024)] introduced the real number \(\varepsilon (L,x)= \inf_{C\ni x} {L.C \over m_x(C)}\), which is called the Seshadri constant of \(L\) at \(x\). Here the infimum is taken over all irreducible curves \(C\) passing through \(x\) and \(m_x(C)\) is the multiplicity of \(C\) at \(x\). For example, if \(L\) is very ample then \(\varepsilon (L,x)\geq 1\).
Proposition 1. Let \(X\) be a surface with \(\rho(X)= \text{rk(NS}(X))=1\) and let \(L\) be an ample generator of \(\text{NS} (X)\). Let \(\alpha\) be an integer with \(\alpha^2\leq L^2\). If \(x\in X\) is a very general point, then \(\varepsilon (L,x)\geq \alpha\). In particular if \(\sqrt{L^2}\) is an integer, then \(\varepsilon (L,x)= \sqrt{L^2}\).
Proposition 2. Let \(X\) be the Jacobian of a hyperelliptic curve of genus \(g\geq 2\) with \(\rho(X)=1\). And let \(\Theta\) be the theta divisor on \(X\). Then \(\varepsilon (\Theta)= \varepsilon (\Theta,x) \leq{2g \over g+1}< \root g\of{g!} =\root g \of {\Theta^g}\). In particular if \(X\) is an irreducible principal polarized abelian surface, then \(\varepsilon (\Theta)= \varepsilon (\Theta,x)= {4\over 3}< \sqrt 2= \sqrt{\Theta^2}\).
Proposition 3. Let \(X\) be a general principal polarized abelian variety of dimension \(g\) with theta divisor \(\Theta\). Then
\[
\varepsilon (\Theta)= \varepsilon (\Theta,x) \leq{\root g-1\of {{g!\over 2^{g-3} (2^g-1)}}} <\root g\of{g!} =\root g\of {\Theta^g}.
\]
Proposition 4. In dimension \(n\), if the Seshadri constant is non-maximal, then it is a \(d\)-th root of a rational number for some \(1\leq d\leq n-1\). Néron Severi group; Seshadri constant; Jacobian of a hyperelliptic curve; principal polarized abelian variety; theta divisor Steffens A. (1998). Remarks on Seshadri constants. Math. Z. 227: 505--510 Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Theta functions and abelian varieties Remarks on Seshadri constants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an integral projective curve defined over the complex numbers and \(D_ 1,\dots,D_ n\) mutually disjoint effective Cartier divisors on \(X\). A parabolic structure on a torsion free coherent sheaf \(E\) on \(X\) consists on a filtration of the vector space \(H^ 0(E\mid D_ j)\) for each \(j=1,\dots,n\), and on some weights associated to these filtrations allowing one to define a parabolic sheaf and the notion of (semi-)stable generalized parabolic sheaf. This technical notion occurs naturally when one studies vector bundles on singular curves by pulling them back to the desingularization of the curve. The author proves existence and studies the moduli spaces of semi-stable generalized parabolic sheaves of a certain type (for which the filtrations have only 2 steps). As an application, she obtains partial desingularizations of moduli spaces of torsion free sheaves on a nodal curve. effective Cartier divisors on integral projective curve; vector bundles on singular curves; desingularization; moduli spaces of semi-stale generalized parabolic sheaves Vector bundles on curves and their moduli, Singularities of curves, local rings, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) Generalized parabolic sheaves on an integral projective curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be an algebraic number field, \(C\) a hyperelliptic curve of genus \(2\) defined by \(y^{2}=f(x)\), where
\[
f(x)=x^{5}+a_{1}x^{4}+ a_{2}x^{3}+a_{3}x^{2}+a_{4}x+a_{5}\in K[x],
\]
\(\mathcal B\) the set of finite Weierstrass points, \(J\) the Jacobian variety of \(C\), \(\Theta\) the theta divisor of \(J\). For a divisor \(D\) of degree \(0\) on \(C\), denote by \(\overline D\) its image in \(J=\text{ Pic}^{0}(C)\). For \(P\in C(\mathbb C)\), let \(u^{(P)}\in\mathbb C^2\) (which is defined modulo the period lattice \(\Lambda\)) be the hyperelliptic integral
\[
\left(\int_{\infty}^{P}{dx\over 2y},\;\int_{\infty}^{P}{xdx\over 2y}\right).
\]
For \(z\in\mathbb C^{2}\), let \(\tilde z\) denote the image of \(z\) in \(J=\mathbb C^{2}/\Lambda\). The map
\[
(P_{1},P_{2})\mapsto u^{(P_{1})}+u^{(P_{2})}
\]
gives a surjection from the symmetric \(2\)-product \(S^{2}(C)\) of \(C\) to \(J\). For two points \(P_{1}\) and \(P_{2}\) on \(C\) with coordinates \((x_{1},y_{1})\) and \((x_{2},y_{2})\), respectively, and for \(u=u^{(P_{1})}+u^{(P_{2})}\), define
\[
\Phi(u)={\mathfrak p}_{111}(u)- {\mathfrak p}_{12}(u){\mathfrak p}_{122}(u)+ {\mathfrak p}_{22}(u){\mathfrak p}_{112}(u),
\]
where
\[
{\mathfrak p}_{11}(u)={F(x_{1},x_{2})-2y_{1}y_{2}\over (x_{1}-x_{2})^{2}}, \qquad {\mathfrak p}_{12}(u)=-x_{1}x_{2}, \quad {\mathfrak p}_{22}(u)=x_{1}+x_{2},
\]
\[
{\mathfrak p}_{111}(u)= 2{y_{2}\psi(x_{1},x_{2})- y_{1}\psi(x_{2},x_{1})\over (x_{1}-x_{2})^{3}}, \qquad {\mathfrak p}_{112}(u)= 2{x_{2}^{2}y_{1}- x_{1}^{2}y_{2}\over x_{1}-x_{2}},
\]
\[
{\mathfrak p}_{122}(u)= -2{x_{2}y_{1}- x_{1}y_{2}\over x_{1}-x_{2}}, \qquad {\mathfrak p}_{222}(u)= 2{y_{1}- y_{2}\over x_{1}-x_{2}},
\]
where
\[
\psi(x_{1},x_{2})=x_{1}^{3}x_{2}( 3x_{1}+x_{2})+4a_{1}x_{1}^{3}x_{2} +a_{2}x_{1}^{2}(x_{1}+3x_{2})+2a_{3}x_{1} (x_{1}+x_{2})+ a_{4} (3x_{1}+x_{2})+ 4a_{5}
\]
and
\[
F(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}( x_{1}+x_{2})+2a_{1}x_{1}^{2}x_{2}^{2} +a_{2}x_{1}x_{2}(x_{1}+x_{2})+2a_{3} x_{1} x_{2}+ a_{4} (x_{1}+x_{2})+ 2a_{5}.
\]
Let \(v\) be an archimedean place of \(K\), \(\widehat{\lambda}_{v}\) the canonical local height on \(J-\Theta\) normalized by \(\widehat{\lambda}_{v}(2z)=4\widehat{\lambda}_{v}(z)+ v\bigl(\Phi(z)\bigr)\).
Denote by \(\langle a,b\rangle_{v}\) the Néron local pairing explicitly given by \(\langle a,b\rangle_{v}=g_{a}(b)\) where \(g_{a}\) is Green's function attached to \(a\). Assume \(P_{1}\) and \(P_{2}\) on \(C(K)\) are such that \(b=P_{1}-P_{2}\) satisfies \(\overline b\not\in\Theta\). Let \(z_{b}=u^{(P_{1})}-u^{(P_{2})}\in\mathbb C^{2}\). A basis of the tangent space at \(P_{i}\) is
\[
2y_{i}{\partial\over\partial x}= f'(x_{i}){\partial\over\partial y}.
\]
A uniformizer at \(P_{i}\) is
\[
{x-x_{i}\over 2y_{i}} \quad \text{if }P_{i}\not\in{\mathcal B} \quad\text{and } {y-y_{i}\over f'(x_{i})} \quad \text{if }P_{i}\in{\mathcal B}.
\]
The main result of this paper is that Néron's local pairing and the canonical height are related by
\[
\langle b,b \rangle_{v}= 2\widehat{\lambda}_{v}(\widetilde z_{b}).
\]
This enables the author to compute numerically the canonical local height at archimedean places. In particular he checks numerically the Birch and Swinnerton-Dyer conjecture in some cases. height functions; Jacobian surfaces; Jacobian variety of a hyperelliptic curve; Néron's local pairing; canonical local height at archimedean places; Birch and Swinnerton-Dyer conjecture Yoshitomi K.: On height functions on Jacobian surfaces. Manuscripta Math. 96, 37--66 (1998) Jacobians, Prym varieties, Theta functions and abelian varieties, Arithmetic varieties and schemes; Arakelov theory; heights, Abelian varieties of dimension \(> 1\), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Elliptic curves On height functions on Jacobian surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a continuation of [\textit{V. Brînzănescu}, Manuscr. Math. 79, No. 2, 187-195 (1993; Zbl 0847.32032) and ibid. 84, No. 3-4, 415-420 (1994; Zbl 0873.14033)]. Here one gets a theorem describing the Neron-Severi group \(NS(X)\) modulo torsion for a non-kähler elliptic surface \(X\to B\). This also gives a description of the torsion \(\text{Tors }NS(X)\). For elliptic bundle surfaces \(X\to B\) one also obtains a nice description of \(NS(X)/\text{Tors }NS(X)\) in the style of the Appel-Humbert theorem. nonalgebraic elliptic surface; vector bundle; Neron-Severi group; Jacobian variety of a curve; Chern classes Holomorphic bundles and generalizations, Elliptic surfaces, elliptic or Calabi-Yau fibrations Neron-Severi group for nonalgebraic elliptic surfaces. III | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal N}= {\mathcal N} (k,\vec d)\) be the moduli space of rank 2 semi-stable parabolic vector bundles with fixed trivial determinant over a smooth projective curve \(C\). We determine the Picard group of \({\mathcal N}\) and define Cartier divisors in the linear system \(| {\mathfrak L}^{\text{par}} |\), where \({\mathfrak L}^{\text{par}}\) is an ample determinant line bundle over \({\mathcal N}\). If \(k=2\) and \(d_i=1\), we show an isomorphism between the space of global sections \(H^0 ({\mathcal N}, {\mathfrak L}^{\text{par}})\) and a direct sum of spaces of theta functions of order 4 over the Jacobian of \(C\). moduli space of rank 2 semi-stable parabolic vector bundles; Picard group; Cartier divisors; determinant line bundle; theta functions; Jacobian Christian Pauly, Fibrés paraboliques de rang 2 et fonctions thêta généralisées, Math. Z. 228 (1998), no. 1, 31 -- 50 (French). Vector bundles on curves and their moduli, Theta functions and abelian varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Determinantal varieties Rank 2 parabolic bundles and generalized theta functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper answers affirmatively the question in the survey article by \textit{P. A. Griffiths} [``An introduction to the theory of special divisors on algebraic curves'', Regional Conf. Ser. Math. 44 (1980; Zbl 0446.14010)]: Does the variety \(W_d^r\) of linear systems on a general curve of genus \(g\) with degree \(d\) and dimension at least \(r\) have the Brill-Noether dimension \(g - (r +1)(g - d +r)\)? Moreover the authors determine the class of this variety in the cohomology of the Jacobian and show that it is without multiple components.
The proof is by a detailed geometrical analysis of a classical degeneration idea of Castelnuovo's. It is formalized as the Castelnuovo-Severi-Kleiman conjecture (CSK): the family of \(P^k\)'s in \(P^d\) meeting the chords of a rational normal curve in \(P^d\) has the same dimension as if the chords were lines in general position, and the family has no multiple components.
The paper has three parts: (I) The reduction to CSK -- except for the absence of multiple components to \(W_d^r\); (II) The proof of CSK; (III) Proofs of the absence of multiple components.
(I) has been previously achieved by \textit{S. Kleiman} [Adv. Math. 22, 1--31 (1976; Zbl 0342.14012)]. The proof in this paper is by a geometrical argument based on duality of special divisors. (II) uses a degeneration of the chords to span an osculating flag. (III) is again by degeneration. The degenerate case is chosen so that number of intersections of two varieties as a set equals the algebraic intersection number. The techniques are those of classical algebraic geometry and Schubert calculus. variety of linear systems on a general curve; cohomology of Jacobian; Castelnuovo-Severi-Kleiman conjecture; algebraic intersection number; Schubert calculus Griffiths, P. \& Harris, J.,On the variety of special linear systems on a general algebraic curves, Duke Math. J.,47(1980), 233--272. Jacobians, Prym varieties, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of special linear systems on a general algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((X,\theta)\) be a principally polarized abelian variety of dimension \(n\) and let \(\theta_2\) denote the integral cohomology class \(\theta^2/2\). The paper deals with the subvarieties of \(X\) whose class is an integral multiple of \(\theta_2\). Inspired by the situation occurring for the jacobian of a general curve, the author proves that for \((X,\theta)\) general and \(n\geq 7\) every subvariety of \(X\) of class \(d\theta_2\), with \(d\) odd, has a singular locus of dimension at least \(n/3-2 \log_2n\). On the other hand he also shows that for \(d\) even \(\leq 16\) and \(n\geq 11\) every subvariety of \(X\) of class \(d\theta_2\) is the zero locus of a section of a rank-2 vector bundle on \(X\) which is an extension of two line bundles. algebraic cycles; principally polarized abelian variety; jacobian of a general curve Theta functions and abelian varieties, Low codimension problems in algebraic geometry, Theta functions and curves; Schottky problem Subvarieties of codimension 2 of an abelian 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 We denote by \({\mathcal M}^ 0_ g\) the moduli space of smooth curves of genus \(g\) \((g\geq 3)\) without automorphisms, and by \({\mathcal C}_ g@>\pi>>{\mathcal M}^ 0_ g\) the universal curve over \({\mathcal M}^ 0_ g\). For any integer \(d\), we denote by \(\psi_ d:{\mathcal T}^ d_ g\to{\mathcal M}^ 0_ g\) the universal Picard (Jacobian) variety of degree \(d\); the fiber \(J^ d(C)\) over a point \([C]\) of \({\mathcal M}^ 0_ g\) parametrizes line bundles on \(C\) of degree \(d\), modulo isomorphism. We describe a group \({\mathcal N}({\mathcal T}^ d_ g)\) (which we call the relative Néron-Severi group of \({\mathcal T}^ d_ g)\) defined to be the group of line bundles on \({\mathcal T}^ d_ g\), modulo the relation that two line bundles are equivalent if their restrictions to the fibers of the map \(\psi_ d\) are algebraically equivalent.
Lemma: The Néron-Severi group of the Jacobian of a curve \(C\) with general moduli is generated by the class \(\theta\) of its theta divisor.
We can define an embedding of groups \(\varphi_ d:{\mathcal N}({\mathcal T}^ d_ g)\hookrightarrow\mathbb{Z}\). To describe the group \({\mathcal N}({\mathcal T}^ d_ g)\) is equivalent to finding the generator \(k^ d_ g\) of the image of the map \(\varphi_ d\).
Theorem: For \(d=0,\ldots,g-1\) the numbers \(k^ d_ g\) are given by the following formula: \(k^ d_ g=(2g-2)/\text{g.c.d}(2g-2,g+d-1)\). universal Picard variety; universal Jacobian variety; moduli space of smooth curves; relative Néron-Severi group Kouvidakis, A., The Picard group of the universal Picard varieties over the moduli space of curves, J. Differential Geom., 34, 3, 839-850, (1991) Picard groups, Jacobians, Prym varieties, Families, moduli of curves (algebraic) The Picard group of the universal Picard varieties over the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a non-hyperelliptic smooth curve over an algebraically closed field. Let \(\Theta\) be a theta divisor in the Jacobian \(J(C)\) of \(C\), and denote by \(\hbox{Sing}_ 2(\Theta)\) the locus of double points on \(\Theta\). A point \(p\in \hbox{Sing}_ 2(\Theta)\) is called a rank-4 double point on \(\Theta\) iff the quadric \(Q_ p\) defined (via the canonical embedding of \(C\)) by the quadratic tangent cone of \(\Theta\) at \(p\) has rank four. The famous ``rank-4 quadrics conjecture'' is then the assertion that, for \(g\geq 4\), the set
\[
\{Q_ p\mid p\in \hbox{Sing}_ 2(\Theta) \hbox{ a rank-4 double point}\}
\]
spans linearly the vector space \(I_ 2(C)\) of all quadrics containing the canonical image of \(C\). - This conjecture was proved over the groundfield \(\mathbb{C}\) of complex numbers, by Andreotti-Mayer (1967) for trigonal curves of genus \(g\geq 4\), by Arbarello-Harris (1981) for all canonical curves of genus \(g=6\), and finally by M. Green for all canonical curves and all genera \(g\geq 4\) [cf. \textit{M. L. Green}, Invent. Math. 75, 85-104 (1984; Zbl 0542.14018)].
In the present paper, the authors give a detailed proof of the rank-4 quadrics conjecture in the more general case of an arbitrary algebraically closed groundfield \(k\) of characteristic different from 2. Their (almost definite) solution of the rank-4 quadrics problem as well as the methods and proofs were already announced and outlined in two previous papers by the authors [cf. Trans. Am. Math. Soc. 307, No. 2, 647-674 (1988; Zbl 0674.14026) and in: Theta Functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, Part I, 571-579 (1989; Zbl 0702.14001)]. This approach is based upon the observation that for any non-hyperelliptic curve \(C\) of genus \(g\geq 5\) the rank-4 quadrics conjecture is essentially equivalent to the statement saying that the canonical image of \(C\) in \(\mathbb{P}^{g-1}\) is precisely the intersection of quadrics \(\{Q_ p\mid p\in \hbox{Sing}_ 2(\Theta)\}\). Then the strategy is to verify this kind of ``generic constructive Torelli theorem'' for curves separately and, consequently, to obtain a proof of the rank-4 quadrics conjecture along this way.
The proof of the constructive Torelli theorem for non-hyperelliptic curves of genus \(g\geq 5\) over an arbitrary algebraically closed field \(k\) with \(\hbox{char} (k)\neq 2\), on its part, is obtained by a very igenious and subtle analysis of the infinitesimal deformation theory for the singularities of theta divisors. Using Kempf's theorem on abstract first order deformations of theta divisors, Welters' algebraic analogue of the transcendental heat equation [cf. \textit{G. E. Welters}, Compos. Math. 49, 173-194 (1983; Zbl 0576.14042)], and an algebraic version of the classical infinitesimal Torelli theory, the authors develop a suitable (reduced) local Kodaira-Spencer theory for deformations of algebraic Jacobians and, with the aid of that, establish the constructive Torelli theorem for such curves by a dimension argument.
Thus the important innovation in this paper consists in relating the rank-4 quadrics problem to the deformation theory of singularities, and this method, because of its apparent generality, not only yields a generalization of the rank-4 quadrics problem to almost arbitrary groundfields, but also provides a general tool for tackling analogous problems for other classes of abelian varieties (e.g., Prym varieties, etc.)
The paper is very clearly and carefully written. The authors have spent great effort at motivating remarks and meticulous proofs, in order to make their strategy, methods and results as comprehensible as possible. Altogether, this is a very beautiful, important and inspiring treatise. theta divisor in the Jacobian of a non-hyperelliptic smooth curve; rank-4 double point; rank-4 quadrics conjecture; generic constructive Torelli theorem; infinitesimal deformation theory for the singularities of theta divisors SMITH (R.) , VARLEY (R.) . - Deformations of theta divisors and the rank 4 quadrics problem , Compositio Math., t. 76, 1990 , n^\circ 3, p. 367-398. Numdam | MR 92a:14025 | Zbl 0745.14012 Theta functions and curves; Schottky problem, Geometric invariant theory, Jacobians, Prym varieties, Theta functions and abelian varieties, Algebraic moduli of abelian varieties, classification, Singularities of curves, local rings, Local deformation theory, Artin approximation, etc., Singularities in algebraic geometry Deformations of theta divisors and the rank 4 quadrics 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 \(X_k\) be a proper, smooth and geomerically connected curve over a global field \(k\), and let \(A\) be the Jacobian variety of \(X_k\). Let \(X\) be a 2-dimensional proper, regular model of \(X_k\). After the work of A. Grothendieck, J. Milne gave a connection between the Tate-Shafarevich group of \(A\) and the Brauer group of \(X\) under the assumption that the index \(\delta_v\) of \(X_{k_v}\) equals 1 for all prime \(v\) of \(\pi\). In this paper, the author generalizes Milne's formula when the \(\delta_v\)'s are no longer equal to 1 and thereby answers partially a question posed by Grothendieck. For the proof, the author generalizes Milne's methods and employs the Albanese-Picard pairing of Poonen and Stoll instead of the Cassels-Tate pairing used by Milne. In an appendix, the compatibility of the Cassels-Tate pairing with the Albanese-Picard pairing of Poonen and Stoll is verified. Brauer groups; Tate-Shafarevich groups; Jacobian variety; index and period of a curve; Cassels-Tate pairing González-Avilés C.: Brauer groups and Tate-Shafarevich groups. J. Math. Sci. Univ. Tokyo 10, 391--419 (2003) Varieties over global fields, Arithmetic ground fields for abelian varieties, Global ground fields in algebraic geometry Brauer groups and Tate-Shafarevich groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Néron-Severi group of elliptic bundles over a curve is studied. An elliptic bundle \(X \to B\) is a principal fiber bundle over a complex, compact, connected smooth curve \(B\) whose typical fiber and structure group are an elliptic curve \(E\). Denote by \(E_B\) the sheaf of germs of local holomorphic maps from \(B\) to \(E\) and consider the cohomology class \(\xi\) in \(H^1 (E_B)\) which defines the elliptic bundle \(X \to B\). The author proves that, if \(c(\xi) \neq 0\), the group \(NS(X)/ \text{Tors} NS (X)\) is isomorphic to the group of morphisms of abelian varieties \(\Hom (J_B,E)\), where \(J_B\) denotes the Jacobian variety of \(B\). Moreover, also in case \(c(\xi) = 0\), the author determines the rank of the Néron-Severi group in terms of the genus and of the period matrix of the curve \(B\). Néron-Severi group; elliptic bundles over a curve; group of morphisms of abelian varieties; Jacobian variety Brînzănescu, V, Neron-Severi group for non-algebraic elliptic surfaces I: elliptic bundle case, Manuscr. Math., 79, 187-195, (1993) Compact complex surfaces, Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Elliptic curves Neron-Severi group for nonalgebraic elliptic surfaces. I: Elliptic bundle case | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author constructs compactifications of the generalized Jacobian variety of a semistable curve with smooth components from the viewpoint of logarithmic structures in the sense of Fontaine-Illusie. The above compactifications have a cohomological interpretation. compactifications; Jacobian variety of a semistable curve; logarithmic structures Takeshi Kajiwara, Logarithmic compactifications of the generalized Jacobian variety, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993), no. 2, 473 -- 502. Jacobians, Prym varieties, Picard schemes, higher Jacobians Logarithmic compactifications of the generalized Jacobian 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 Higher-dimensional algebraic geometry, that is the study of algebraic varieties of dimension greater than two, is a branch of algebraic geometry which is largely still in progress.
Opposed to the theories of algebraic curves and surfaces, which are basically classical and almost perfectly understood, in the meantime, the classification theory of higher-dimensional varieties is a comparatively young subdiscipline of algebraic geometry and, in its systematic framework, barely older than twenty-five years. However, an amazing quantity of knowledge in this new area has accumulated, in the course of the past twenty years, together with a vast spectrum of new, specific concepts, methods, techniques, and deep-going results. The whole subject has now reached a stage where it is rather hard for the beginner to enter this magic world built up by S. Mori, F. Campana, M. Reid, Y. Kawamata, J. Kollár, Y. Miyaoka, V. Shokurov, V. Iskovskikh, E. Viehweg, and many others.
As a result, higher-dimensional algebraic geometry, in its present state of art, somehow makes the impression of a private club of ultimate experts in the field, and has remained much more confidential than it should have. Certainly, this is particularly due to the fact that there has been no beginner-friendly textbook on this topic, so far, guiding the non-expert through the specific terminology, the considerable amount of technicalities, and the systematic strategy underlying the classification theory of higher-dimensional algebraic varieties.
The book under review represents the first attempt to provide an easily accessible introduction to this subject, and that in the form of a systematic textbook. Of course, there are already several excellent survey articles on topics in higher-dimensional algebraic geometry, and there are also a few special research monographs and high-level reference books, for example \textit{J. Kollár}'s book: ``Rational curves on algebraic varieties'' (1996; Zbl 0877.14012) but those can scarely be regarded as comprehensive or easy introductions to the field.
The author's textbook is based on notes from a class taught at Harvard University during the spring term 1999. Due to the introductory character of this text, the author has not tried to be exhaustive, nor to write another reference book. He has, on the contrary, selected suitable parts of the theory and tried to give basic definitions, essential proofs, and important examples with as many details as possible. In this vein, the material covered in this book falls roughly into three parts:
(1) preparatory and standard facts on Cartier divisors and parameter spaces for morphisms from curves to varieties;
(2) various aspects of the geometry of smooth projective varieties with many rational curves;
(3) first steps toward Mori's minimal program of birational classification of higher-dimensional algebraic varieties.
More precisely, the entire text consists of seven chapters. Chapter 1, entitled ``Curves and divisors on algebraic varieties'', deals with the intersection theory of Cartier divisors, cones of curves, ampleness of divisors, nef and big divisors, and rational curves on exceptional loci of birational morphisms.
Chapter 2, still belonging to the preparatory first part of the text, discusses the space parametrizing curves on a given variety, or more precisely morphisms from a given smooth projective curve to a given smooth quasi-projective variety. The author does not reproduce Grothendieck's original and very general construction of these spaces, but he explains in some detail their classifying character and their local geometry. In order to keep the discussion as concise as possible, and to avoid too many technical intricacies, complete proofs are given only for the simplest version of these spaces, whilst additional facts are then simply sketched or listed.
Chapter 3 comes with the title ``Bend-and-break lemmas''. Here the author introduces Mori's ingenious techniques for studying deformations of curves on a projective variety passing through a fixed point. Mori's so-called bend-and-break lemmas, in their various incarnations, provide the basic toolkit for the study of projective varieties with many rational curves, and the author explains them here in a detailed manner.
Chapter 4 turns to the special case of uniruled and rationally connected varieties, including the basic facts on so-called free rational curves, very free rational curves, rationally chain-connected varieties, and smoothing trees of rational curves.
This is used, in the subsequent chapter 5, to study the rational quotient of a variety. This concept, established by F. Campana some years ago, is a very good algebro-geometric substitute for the (generally non-algebraic) quotient of a variety modulo rational chain equivalence of points. The author presents here a construction of this rational quotient, which follows the work of \textit{J. Kollár}, \textit{Y. Miyaoka} and \textit{S. Mori} (1992) in the more special case of Fano varieties rather than Campana's general method. At the end, this chapter also provides some new results on Fano varieties with high degree \((-K_X)^{\text{dim} X}\).
Whereas, chapters 3, 4, and 5 form the second main part of the book, the following chapters 6 and 7 represent the third part of it, that is an introduction to Mori's minimal model program of the birational classification of higher-dimensional algebraic varieties.
Chapter 6 studies the cone \(NE(X)\) of effective curves of a smooth projective variety \(X\). The author focuses here on proving and illustrating Mori's famous cone theorem, which gives a geometric description of the closure of \(NE(X)\) in the smooth case, and at the end of this chapter a closer study of contractions of extremal rays is carried out.
While in the smooth case the proof of the cone theorem can be obtained as an application of Mori's bend-and-break techniques (explained in chapter 3), the contraction theorem for extremal rays and the construction of a suitable ``minimal model'' for a given variety are unattainable by means of these ideas. It turns out that one has to allow some kind of singular varieties for that purpose, but Mori's techniques do not work for singular varieties. Therefore, other methods are required for tackling the minimal model program, and those are the subject of study in the concluding chapter 7.
This chapter is entitled ``Cohomological methods'', because the material presented here is mainly based on the whole machinery of cohomological vanishing theorems. Being much less geometrical than the foregoing chapters, this final chapter is by far the most involved and difficult part of the book. The author presents an approach (initiated by Y. Kawamata) which culminates in proving the cone and contraction theorems, in a unified way, as well as in doing the first steps toward the general minimal model program. This includes sections on canonical models, their singularities, singularities of pairs, the Kawamata-Viehweg vanishing theorem, the base-point-free theorem, the rationality theorem, and the length of extremal rays. Methodologically, the author uses the so-called ``logarithmic'' framework, which makes the entire treatment look more natural and conceivable.
Each chapter comes with a set of exercises, many of which point to additional, further-reaching theorems, and the more difficult ones of them are equipped with brief hints for solution. A particular feature of the text is provided by the numerous instructive examples and comments illustrating the undoubtedly demanding material.
The author has tried to keep the text as self-contained as possible, which has been a fairly difficult task, and all in all he has succeeded in writing the first introductury textbook on higher-dimensional varieties which should be accessible to any motivated reader familiar with the basics of modern algebraic geometry, about at the level of \textit{R. Hartshorne}'s celebrated standard text. In this regard, the book provides an excellent source for graduate students in algebraic geometry, seasoned mathematicians in general, and theoretical physicists using algebro-geometric methods. The exposition of the material is characterized by a very lucid, refined, and user-friendly style of writing.
Without any doubt, this book fills a gap in the existing textbook literature on algebraic geometry. higher-dimensional algebraic varieties; birational geometry; birational classification theory; minimal model program; Mori theory; cohomological vanishing theorems; cohomological nonvanishing theorems; Cartier divisors; morphisms from curves; varieties with many rational curves; rational quotient of a variety; cone theorem; contraction theorem; extremal rays Debarre O., Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York 2001. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Surfaces and higher-dimensional varieties, Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves Higher-dimensional algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The action of a finite group \(G\) on an abelian variety \(A\) induces a decomposition of \(A\) into factors related to the rational irreducible representations of \(G,\) the so called isotypical decomposition of \(A;\) when \(A = JZ\) is the Jacobian variety of a curve \(Z\) with \(G\)-action, for every subgroup \(H\) of \(G\) there is an induced canonical action of the corresponding Hecke algebra \(Q[H \backslash G/H]\) on the Jacobian of the quotient curve \(ZH= Z/H,\) and a corresponding isotypical decomposition of \(JZH\). These results have provided geometric and analytic information on the factors appearing in the isotypical decomposition of \(JZ\) and \(JZH\).
In this paper, we show that similar results hold for any abelian variety \(A\) with \(G\)-action: for every subgroup \(H\) of \(G\) there is a natural abelian subvariety \(AH\) of \(A\) fixed by \(H,\) such that the Hecke algebra \(Q[H \backslash G/H]\) acts on \(AH.\) We find the associated isotypical decomposition of \(AH,\) and the decomposition of the analytic and the rational representations of the action on \(AH.\) We also show that the notion of Prym variety for covers of curves may be extended to abelian varieties, and describe its isotypical decomposition with respect to the action of a natural induced subalgebra of its endomorphism ring. We apply the results to the decomposition of the Jacobian and Prym varieties of the intermediate cover given by \(H,\) in the case of smooth projective curves with \(G\)-action. We work out several examples that give rise to families of principally polarized abelian varieties, of Jacobian and Prym varieties, with large endomorphism rings. principally polarized abelian varieties; Jacobian variety of a curve; Hecke algebra; Prym variety for covers of curves Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Hecke algebras acting on abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth complex projective curve of genus \(g > 2\), and denote by \({\mathfrak M} (2, \xi)\) the (fine) moduli space of stable rank-2 vector bundles over \(C\) with determinant isomorphic to a fixed line bundle \(\xi \in \text{Pic} (C)\) of degree \(d\). Then any universal bundle \(U_\xi\) on \(C \times {\mathfrak M} (2, \xi)\) for this moduli problem induces, via the projection map \(p_2 : C \times {\mathfrak M} (2, \xi) \to {\mathfrak M} (2, \xi)\), a so-called Picard sheaf \(W_\xi : = (p_2)_* (U_\xi)\) on \({\mathfrak M} (2, \xi)\). In the case where \(d : = \deg (\xi) \geq 4g - 2\) and an odd number, any \(W_\xi\) is locally free and referred to as a Picard bundle on \({\mathfrak M} (2, \xi)\). In the present paper, the authors study the variation of the Picard bundles \(W_\xi\) under deformations of the universal bundle \(U_\xi\). More precisely, for any \(L \in \text{Jac} (C) = \text{Pic}^0 (C)\), define \(W_\xi (L) : = (p_2)_* (W_\xi \otimes (p_1)^* L)\). This gives a family of deformations of the Picard bundle \(W_\xi\), parametrized by the Jacobian \(\text{Jac} (C)\) of \(C\).
The main theorem of the paper says that, for a curve \(C\) without automorphisms, of genus \(g > 2\), the deformations of any Picard bundle \(W_\xi\) defined as above (with \(d = \deg (\xi) \geq 4g - 2\) and \(d\) odd) have the following properties:
(i) \(h^i ({\mathfrak M} (2, \xi), \text{ ad} W_\xi) = g\), if \(i = 1\);
\(\qquad h^i ({\mathfrak M} (2, \xi), \text{ ad} W_\xi) = 0\), if \(i = 0\) or \(i = 2\);
(ii) the family of deformations \(\{W_\xi (L) |L \in \text{Pic}^0 (C)\}\) defined above is injective.
This result extends the classical work of \textit{A. Mattuck} [Am. J. Math. 83, 189-206 (1961; Zbl 0225.14020)] and of \textit{G. Kempf} [Ann. Math., II. Ser. 110, 243-273 (1979; Zbl 0452.14011)] on the variations and the cohomology of the Picard bundle on the Jacobian of a curve.
In the course of the proof, the authors make extensive use of recent constructions and results of \textit{M. Thaddeus} [``Stable pairs, linear systems and the Verlinde formula'', Invent. Math. 117, No. 2, 317-353 (1994)], which are summarized at the beginning. As a by-product (and as a basic tool for the proof of their main theorem), they obtain local and global Torelli-type theorems for the Thaddeus moduli spaces of stable pairs. One of them relates the Weil-Griffiths intermediate Jacobian of any Thaddeus moduli space isomorphically to the Jacobian of the base curve, while the other one gives an explicit computation of the cohomology of the moduli spaces of stable pairs with coefficients in the tangent sheaf. Both results provide generalizations of earlier, partial results obtained by \textit{D. Mumford} and \textit{P. Newstead} [Am. J. Math. 90, 1200-1208 (1968; Zbl 0174.52902)] and by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 101, 391-417 (1975; Zbl 0314.14004)]. moduli space of stable rank-2 vector bundles over a smooth complex projective curve; Picard sheaf; deformations of the universal bundle; Torelli-type theorems; moduli spaces of stable pairs; intermediate Jacobian Balaji V and Vishwanath P R, Deformations of Picard sheaves and moduli of pairs,Duke Math. J. 76 (1994) 773--792 Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Formal methods and deformations in algebraic geometry, Picard groups, Picard schemes, higher Jacobians, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Deformations of Picard sheaves and moduli of pairs | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the Neumann system, which describes the harmonic oscillator (of arbitrary dimension) constrained to the sphere. In particular we consider the confluent case where two eigenvalues of the potential coincide, which implies that the system has \(S^{1}\) symmetry. We prove complete algebraic integrability of the confluent Neumann system and show that its flow can be linearized on the generalized Jacobian torus of some singular algebraic curve. The symplectic reduction of \(S^{1}\) action is described and we show that the general Rosochatius system is a symplectic quotient of the confluent Neumann system, where all the eigenvalues of the potential are double. Liouville integrability; Lax equation; motion of a particle; lineraization of the flow; generalized Jacobian; singular spectral curve; harmonic oscillator; confluent case; \(S^{1}\) symmetry; symplectic reduction; Rosochatius system Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, , Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties Algebraic integrability of the confluent Neumann system | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a smooth projective curve of genus \(g\geq 2\) over \({\mathbb{C}}\). Let U(r,d) (resp. \(U_ s(r,d))\) be the moduli space of algebraic semistable vector bundles (resp. the open subset corresponding to the stable bundles) of rank \(r\geq 2\) and degree d over X. It is known that \(U(r,d)\) is a normal, irreducible, projective variety. If \(gcd(g,r)\neq 1\) and one excludes also the case \(g=r=2\), d even then \(U(r,d)\) is not smooth, \(Sing(U(r,d))=U(r,d)\setminus U_ s(r,d)\) and \(co\dim_{U(r,d)}U(r,d)\setminus U_ s(r,d)\geq 2\). For \(L\in Pic(X)\), \(\deg (L)=d\) let denote by U(r,L) (resp. \(U_ s(r,L))\) the closed subvariety of \(U(r,d)\) (resp. \(U_ s(r,d))\) corresponding to the vector bundles with determinant isomorphic to L. The aim of this paper is to give a complete description of \(Pic(U(r,d))\) and \(Pic(U(r,L))\) when \(gcd(g,r)\neq 1\) and \((g,r)\neq (2,2)\), d even.
The first result is that even they are singular, \(U(r,d)\) and \(U(r,L)\) are locally factorial. Let now \(\gcd (r,d)=n\) and let \({\mathcal F}\) be a vector bundle on X such that \(\deg({\mathcal F})=(-d+r(g-1))/n\) and \(rk({\mathcal F})=r/n\). Then \(\chi({\mathcal E}\otimes {\mathcal F})=0\) for all vector bundles \({\mathcal E}\) on X of rank r and degree d. One can show that \({\mathcal F}\) above can be chosen such that there exists \({\mathcal E}\in U_ s(r,d)\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})=H^ 1(X,{\mathcal E}\otimes {\mathcal F})=0\). Then for such an \({\mathcal F}\) denote by \(\Theta^ s_{{\mathcal F}}\) (respectively \(\Theta^ s_{{\mathcal F},L})\) the set of points of \(U_ s(r,d)\) (resp. \(U_ s(r,L))\) which correspond to stable bundles \({\mathcal E}\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})\neq 0\). These are showed to be hypersurfaces in \(U_ s(r,d)\) respectively in \(U_ s(r,L)\). Their closure in \(U(r,d)\) (respectively \(U(r,L)\)) are denoted by \(\Theta_{{\mathcal F}}\) (resp. \(\Theta_{{\mathcal F},L})\) and called theta divisors.
The line bundle \({\mathcal O}(\Theta_{{\mathcal F},L})\) is independent of the choice of \({\mathcal F}\) and \(Pic(U(r,L))\) is isomorphic to \({\mathbb{Z}}\) having \({\mathcal O}(\Theta_{{\mathcal F},L})\) as generator. Let \(I^{(d)}\) be the Jacobian of the line bundles of degree d on X. Then, through the canonical morphism \(\det: U(r,d)\to I^{(d)},\) \(Pic(I^{(d)})\) is seen as a subgroup of Pic(U(r,d)) and one has the isomorphism \(Pic(U(r,d))\cong Pic(I^{(d)})\oplus {\mathbb{Z}}{\mathcal O}(\Theta_{{\mathcal F}})\). Here \({\mathcal O}(\Theta_{{\mathcal F}})\) is dependent on the choice of \({\mathcal F}:\) \({\mathcal O}(\Theta_{{\mathcal F}'})\cong {\mathcal O}(\Theta_{{\mathcal F}})\otimes \det^*(\det {\mathcal F}'\otimes (\det {\mathcal F})^{-1}).\)
The paper also contains a complete description of the dualizing sheaves of \(U(r,L)\) and \(U(r,d)\) and a proof of the nonexistence of Poincaré bundles on open subsets of the moduli space \(M_ s({\mathbb{P}}_ 2({\mathbb{C}}),r,c_ 1,c_ 2)\) in case r, \(c_ 1\) and \(\chi\) are not prime to each other. factoriality of moduli space of algebraic semistable vector bundles; Picard group; smooth projective curve; determinant; theta divisors; Jacobian Drézet, J.-M.; Narasimhan, M. S., Groupe de Picard des variétés de modules de faisceaux semi-stables sur les courbes algébriques, Invent. Math., 97, 53-94, (1989) Picard groups, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques. (Picard groups of moduli varieties of semi- stable bundles on algebraic curves) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0711.00011.]
Let \(N\) be a positive integer, and let \(J=J_ 0(N)\) be the Jacobian variety of a modular curve \(X_ 0(N)_ \mathbb{C}\). The cotangent space of \(J\) is the vector space \(S\) of weight-two cusp forms on \(\Gamma_ 0(N)\). According to the theory of Atkin and Lehner, \(S\) may be written as a direct sum \(S_{old}\oplus S_{new}\), where \(S_{old}\) is a sum of copies of spaces of cusp forms whose levels are the positive divisors of \(N\) which are less than \(N\). Using ``degeneracy operators'' among modular forms, one may mirror the decomposition of \(S\) by writing \(J\) as a sum \(J_{old}+J_{new}\), where \(J_{old}\) and \(J_{new}\) are abelian subvarieties of \(J\) whose cotangent spaces are \(S_{old}\) and \(S_{new}\). The intersection \(\Delta:=J_{old}\cap J_{new}\) is a finite subgroup of \(J\) whose order and structure are unknown, in general. To study \(\Delta\) is to investigate the interaction between the decomposition \(S=S_{old}\oplus S_{new}\) and the lattice in \(S\) consisting of elements of \(S\) with integral Fourier coefficients. --- The author studies \(\Delta\) in the case where \(N\) is the product of two distinct primes. He obtains, in particular, a formula for the odd part of the order of \(\Delta\). The methods used in this article are arithmetic, and rely on the author's work on Serre's conjectures [Invent. Math. 100, No. 2, 431-476 (1990)]. old subvariety of the Jacobian variety of a modular curve; integral Fourier coefficients; Serre's conjectures Kenneth A. Ribet, The old subvariety of \?\(_{0}\)(\?\?), Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 293 -- 307. Modular and Shimura varieties, Jacobians, Prym varieties The old subvariety of \(J_ 0(pq)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Cartier divisors on normal varieties with the action of a reductive group \(G\). We give criteria for a divisor to be Cartier, globally generated and ample, and apply them to a study of the local structure and the intersection theory of a \(G\)-variety. In particular, we prove an integral formula for the degree of an ample divisor on a variety of complexity 1 and apply this formula to computing the degree of a closed 3-dimensional orbit in any SL\(_2\)-module. Most of our results generalize those of \textit{M. Brion} [Duke Math. J. 58, 397--424 (1989; Zbl 0701.14052)], where the theory of Cartier divisors on a spherical variety is developed. action of a reductive group; Cartier divisors; spherical variety Timashev, DA, Cartier divisors and geometry of normal \(G\)-varieties, Transform. Groups, 5, 181-204, (2000) Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients) Cartier divisors and geometry of normal \(G\)-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 Manin-Mumford Conjecture states that if \(X\) is a nonsingular projective curve of genus \(>1\) over an algebraically closed field of characteristic 0, and \(A\) is the Jacobian variety of \(X\) then \(X\cap \text{ Tor}(A)\) is finite. In 1982 M. Raynaud proved that if \(A\) is an abelian variety in characteristic 0, and \(X\) is a subvariety of \(A\) then \(X\cap \text{ Tor}(A)\) is a finite union of cosets (proving in particular the conjecture). In 1987 M. Hindry proved the same thing, but with \(A\) an arbitrary commutative algebraic group. In 1996 E. Hrushovski used the model theory of algebraically closed fields with a generic automorphism to prove the (extended) Manin-Mumford Conjecture over number fields. Let \(A\) be a connected commutative algebraic group, and \(X\) be a subvariety of \(A\), both defined over a number field. Then \(X\cap \text{ Tor}(A)\) is a finite union of sets of the form \(\text{ Tor}(B)+a\), where \(B\) is a connected algebraic subgroup of \(A\), and \(a\in A\). (In fact, there is an effective bound to the number of such sets in the union depending on \(\text{ dim}(A)\) and \(\text{ deg}(X)\).) In the paper under review the author gives Hrushovski's proof in the case where \(A\) is an abelian variety, without worrying about the effective bounds. The proof makes use of facts presented by \textit{Z. Chatzidakis} in the same volume [``Groups definable in ACFA'', ibid. 25-52 (1997; see the review above)]. abelian variety; theory ACFA; group definable in a model; model theoretic stability; 1-basedness; model companion of the theory of fields with an automorphism; Manin-Mumford conjecture; projective curve; Jacobian variety Model-theoretic algebra, Abelian varieties of dimension \(> 1\), Model theory of fields, Rational points ACFA and the Manin-Mumford conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian of a non-singular, compact Riemann surface is the group \(\mathrm{Pic}^{0}\) of divisors of degree zero factored out by principal divisors. As such, \(g\) being the genus of \(C,\) the Jacobian is \({\mathbb{C}}^{g}/\Lambda,\) \(\Lambda\) being the \(g-\)dimensional lattice of periods. In this case of the genus 2 (hyperelliptic) surface, there are coordinates \(x: C \rightarrow {\mathbb{P}}^{1}\) and \(y: C \rightarrow {\mathbb{P}}^{1}\) with poles of orders 2 and 5 respectively which satisfy a relation of the form
\[
y^{2} = 4x^{5} + \lambda_{4} x^{4} + \lambda_{3} x^{3} + \lambda_{2} x^{2} + \lambda_{1} x + \lambda_{0},
\]
the \(\lambda_{i}\) being constants in the ground field. Functions associated with more general special divisors provide us with other models. Such models are related by birational transformations. Thus we will be concerned with (singular) models of the genus 2 curve in the form
\[
y^{2} = g_{6} x^{6} + 6 g_{5} x^{5} + 15 g_{4} x^{4} + 20 g_{3} x^{3} + 15 g_{2} x^{2} + 6 g_{1} x + g_{0},
\]
which are related amongst themselves and to the quintic by simple Moebius maps.
If \(\mathrm{Pic}^{0}\) is identified with \(\mathrm{Pic}^{2}\) and \(\mathrm{Jac}(C)\) is constructed as a quadric variety in \({\mathbb{P}}^{15},\) the locus of seventy two linearly independent quadratic identities. Sixteen homogeneous coordinates on \({\mathbb{P}}^{15}\) are chosen to be symmetric functions in two points on the curve. The purpose of the current paper is to use a little representation theory to oil the wheels of this machinery and to uncover some structure intrinsic to the collection of quadratic identities. The idea is that the coordinates on \(\mathrm{Jac}(C)\) can be chosen to belong to irreducible \(G-\)modules where \(G\) is a group of birational transformations. Quadratic functions arise by tensoring up these modules and decomposing into irreducibles. Next he is presenting the Lie algebraic action of the coordinate transformations on the variables and the coefficients of the curve and define the construction of a highest weight element that use for a component of the decomposition. Thus author present a treatment of the algebraic description of the Jacobian of a general genus two plane curve which exploits an \(\mathrm{SL}_{2}(k)\) equivariance and clarifies the structure of Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety. Jacobian of a non-singular; compact Riemann surface; hyperelliptic surface; special divisors; birational transformations; homogeneous coordinates; Kummer variety Athorne C., On the equivariant algebraic Jacobian for curves of genus two, J. Geom. Phys., 2012, 62(4), 724--730 Picard groups, Riemann-Roch theorems, Rational and birational maps, Families, moduli of curves (algebraic), Jacobians, Prym varieties On the equivariant algebraic Jacobian for curves of genus two | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is well-known that the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\) is an irreducible quasiprojective subvariety of the moduli space \({\mathcal A}_g\) of principally polarized Abelian varieties of dimension \(g\), whose closure \(\overline M_g\) -- in the four-dimensional case -- turns out to be the exact zero-set of a theta-function. This theta-function is in fact a cusp form of weight 8 of the Siegel modular group, the so-called Schottky modular form. In the present paper, this fine result is achieved quite naturally by thorough geometric considerations involving the Gauss map on the theta divisor.
To be more precise, for a \(g\)-dimensional principally polarized Abelian variety \((A,\beta)\) with marked odd theta characteristic \(\xi\), the Taylor expansion about the origin of the theta function
\[
\vartheta[\xi](z, \Omega)= l(z, \Omega)+ m(z, \Omega)+\cdots
\]
yields a linear form \(l\) and a cubic form \(m\) in \(z\) such that all the ideals \((l),(l, m),\dots\subset \mathbb{C}[z]\) transform by the same automorphy factor under the action of a suitable subgroup of finite index of the Siegel modular group \(\text{Sp}(2g, \mathbb{Z})\). Especially, if \((A, \theta)\) is the Jacobian of a generic curve of genus \(g\), then for any choice of odd theta characteristics the restriction of \(\overline m\) to the hyperplane \(l= 0\) is a Fermat cubic, i.e. the sum of \(g- 1\) cubes.
In a second step, to any homogeneous polynomial invariant \(\varphi\) on cubic forms a natural globalization \(G_\varphi\) is constructed in such a way that the Siegel modular form \(G_\varphi(\overline m)\) vanishes on Jacobians if the invariant \(\varphi\) vanishes on the Fermat cubic.
Restricting to dimension form, the Fermat ideal generated by all Siegel modular forms coming from those invariants which vanish on the Fermat cubic is principal. Its zero set is exactly the closure of the locus of period matrices of genus four Riemann surfaces and a generator of this ideal equals, up to a constant multiple, the Schottky modular form. Siegel modular forms; cubic hypersurfaces; zero set of the Fermat ideal; Schottky modular form; Gauss map on the theta divisor; principally polarized Abelian variety; theta function; Siegel modular group; Jacobian of a generic curve; Fermat cubic; cubic forms Mccrory, C.; Shifrin, T.; Varley, R.: Siegel modular forms generated by invariants of cubic hypersurfaces. J. algebraic geom. 4, 527-556 (1995) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta functions and curves; Schottky problem, Modular and Shimura varieties, Abelian varieties of dimension \(> 1\) Siegel modular forms generated by invariants of cubic hypersurfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 a generic complex abelian variety of dimension \(a\); the aim of this article is to prove that the geometric genus of a curve on \(A\) is greater than \(a (a - 1)/2\), and that the geometric genus of a curve on the Kummer variety \(K_A\) is at least \((a - 1) (a - 2)/2\).
This is done as follows. Let \(C\) be a smooth curve with a non-constant morphism \(f : C \to A\). We denote by \(C^+\) an Abel-Jacobi image of \(C\) in its Jacobian \(J(C)\), and by \(C^-\) the image of \(C^+\) by multiplication by \(-1\). The 1-cycle \(C^+ - C^-\) is homologically equivalent to 0, but it is often not algebraically equivalent to 0 (as shown by Ceresa). More precisely, if one considers the primitive intermediate Jacobian \(P(J(C))\) of \(J(C)\) (that is, the quotient of its intermediate Jacobian by a maximal abelian subvariety) and the Abel- Jacobi map \(AJ\), the element \(AJ(C^+ - C^-)\) of \(P(J(C))\) is usually non torsion. On the other hand, it was shown by Nori that for \(a\geq 4\), the Abel-Jacobi image of any homologically trivial 1-cycle in \(A\) is torsion in \(P(A)\). The idea is to use this contrast. If one considers a family \(F : {\mathcal C} \to {\mathcal A}\) of deformations of the map \(f\), over an open subset \(U\) of the Siegel space, the Abel-Jacobi maps globalize to define normal functions \(\nu_F\) and \(\nu\), attached to the cycles \(C^+_t - C^-_t\), which are sections of the families \({\mathcal P} (A) \to U\) and \({\mathcal P} (J(C)) \to U\) of primitive intermediate Jacobians. Let \(P^{p,q} (A)\) be the primitive Hodge spaces of \(A\); the tangent space to \(U\) is isomorphic to \(T = \text{Sym}^2 (H^{0,1} (A))\), and the Kodaira-Spencer map defines a map \(\beta : T \otimes P^{2,1} (A) \to P^{1,2}(A)\), whose kernel is denoted by \(K(A)\). The Griffiths infinitesimal invariant is a linear functional \(\delta \nu_F\) on \(K(A)\); we know from Nori's theorem that it vanishes. It factorizes as \(K(A)\hookrightarrow K(J(C)) \to (\mathbb{C})\), where the second map is \(\delta \nu\). In other words, \(\delta \nu\) vanishes on \(K(A)\). Using a formula from the author's joint paper with \textit{A. Collino} in Duke Math. J. 78, No. 1, 59-88 (1995)], the author uses this fact to construct \((a - 1) (a - 2)/2\) independent elements in \(H^{1,0} (C)\) orthogonal to \(f^* H^{1,0} (A)\). This implies the bound \(g(C) \geq (a - 1) (a - 2)/2 + a\). A more refined analysis shows that \(g(C) = a(a - 1)/2 + 1\) can only happen when \(a = 4\) or 5 and \(C\) is a Prym curve, together with the statement about curves on generic Kummer varieties. curve lying on a generic complex abelian variety; geometric genus of a curve; Kummer variety; Abel-Jacobi image; intermediate Jacobian; Prym curve G. P. Pirola, Abel-Jacobi invariant and curves on generic abelian varieties. In: \textit{Abelian varieties (Egloffstein}, 1993), 237-249, de Gruyter 1995. MR1336610 Zbl 0837.14036 Picard schemes, higher Jacobians, Jacobians, Prym varieties, Algebraic cycles Abel-Jacobi invariant and curves on generic 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 Let \(A\) be a complex abelian variety of dimension \(g\geq3\), \(\Theta\) a symmetric theta divisor defining a principal polarization on \(A\), \(| 2\Theta|_ \infty\) the linear system of all those divisors in \(|2\Theta|\) with multiplicity greater or equal to 4 at the origin of \(A\), and \(V(|2\Theta|_ \infty)\) the scheme-theoretic intersection of all divisors in \(|2\Theta|_ \infty\). A conjecture due to \textit{B. van Geemen} and \textit{G. van der Geer} [cf. Am. J. Math. 108, 615-642 (1986; Zbl 0612.14044){]} states that the latter subscheme of \(A\) is concentrated at the origin if \(A\) is not the Jacobian of a curve. G. Welters has shown that if \(A\) is the Jacobian of a curve \(C\) of genus \(g\neq4\), then the subscheme \(V(|2\Theta|_ \infty)\) is set-theoretically equal to the surface \(C-C\subseteq C\times C\), and equal to \(C-C\) plus two other isolated points in the case of \(g=4\) [cf. \textit{G. E. Welters}, ``The surface \(C-C\) on Jacobi varieties and 2nd order theta functions'', Acta Math. 157, 1-22 (1986)].
In the first part of the present paper, the author provides a certain refinement of Welters' result. More precisely, he shows that if \(A\) is the Jacobian of a non-hyperelliptic curve \(C\) of genus \(g\geq3\), then \(V(|2\Theta|_ \infty)\) is smooth off the origin, hence even scheme-theoretically equal to \(C-C\) off the origin for \(g\neq4\). --- The second part of the paper is devoted to the infinitesimal study of the subscheme \(V(|2\Theta|_ \infty)\). It is shown that for a non- hyperelliptic Jacobian \(A=J(C)\) the infinitesimal scheme \(V_{inf}(|2\Theta|_ \infty)\), regarded as a subscheme of \(\mathbb{P}^{g-1}\), is either equal to the canonical image of the curve \(C\) itself, or equal to \(C\) plus one additional point. principal polarization on a complex abelian variety; symmetric theta divisor; theta functions; Jacobian of a non-hyperelliptic curve IZADI (E.) . - Fonctions thêta du second ordre sur la jacobienne d'une courbe lisse , Math. Ann., t. 289, 1991 , n^\circ 2, p. 189-202. MR 92a:14024 | Zbl 0735.14029 Theta functions and abelian varieties, Theta functions and curves; Schottky problem, Jacobians, Prym varieties Second order theta functions on the Jacobian 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 Let A be an abelian variety and T a (linear) torus, defined over the complex numbers, and let V be an algebraic sub-variety of the algebraic group \(G=A\times T\). The author obtains a description of the torsion points of G which lie in V: qualitatively, he shows that the torsion points of V are all contained in a finite number of cosets of algebraic subgroups of G which lie in V.
The method of proof actually provides a quantitative version of the result. For this, G is defined over a number field K and well embedded in a product of projective spaces \(P^ M\times P^ N\). The author uses a refined Bezout's theorem which applies to such a product to bound the degrees of the subgroups (and the order of the coset representatives) of the finite number of cosets such that the torsion points of G over the algebraic closure L of K which lie in V are in the cosets. When A is of CM type, the bound can be given explicitly. The author gives some applications of this when A is the Jacobian of a curve.
This paper is a seminar report with sketches of proofs; complete proofs will appear elsewhere. abelian variety; torus; torsion points; Bezout's theorem; Jacobian of a curve Algebraic theory of abelian varieties, Arithmetic ground fields for abelian varieties Points de torsion sur les sous-varietés de groupes algébriques. (Torsion points on subvarieties of algebraic groups) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A system of coordinates for the elements on the Jacobian variety of Picard curves is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an efficient algorithm for the reduction and addition of divisors. Exploiting the geometry of the Picard curves, a completely effective reduction algorithm is developed, which works for curves defined over any ground field \(k\), with \(\text{char}(k)\neq 3\).
In the generic case, the algorithm works recursively with the system of coordinates representing the divisors, instead of solving for points in their support. Hence, only one factorization is needed (at the end of the algorithm) and the processing of the system of coordinates involves only linear algebra and evaluation of polynomials in the definition field of the divisor \(D\) to be reduced. The complexity of this deterministic reduction algorithm is \(O(\text{deg}(D))\). The addition of divisors may be performed iterating the reduction algorithm.
See also the papers by the first two authors and \textit{J. A. Piñeiro Barceló}, Math. Nachr. 208, 149-166 (1999; Zbl 0960.14032) and the second author and \textit{J. A. Piñeiro Barceló}, Decoding of codes in Picrd curves, Preprint Nr. 96-30 Humboldt Univ. Berlin (1996). system of coordinates; Jacobian variety of Picard curves; efficient algorithm; reduction and addition of divisors; complexity E. Barreiro, J. Sarlabous, J. Cherdieu, Efficient reduction on the jacobian variety of Picard curves, Coding Theory, Cryptography and Related Areas, Springer, Berlin, 2000, pp. 13 -- 28. Jacobians, Prym varieties, Computational aspects of algebraic curves, Number-theoretic algorithms; complexity Efficient reduction on the Jacobian variety of 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 Almost all papers on moduli spaces of vector bundles on curves need to assume the genus of the curve to be at least two. The paper under review contains a systematic study of the case of elliptic curves. The starting point is the classical paper on vector bundles over an elliptic curve by \textit{M. F. Atiyah} [Proc. Lond. Math. Soc., III. Ser. 7, 414-452 (1957; Zbl 0084.173)]. It turns out that the main ingredients in the theory have a very easy and nice description:
Let \({\mathcal M}_{n,d}\) be the moduli space of equivalence classes of semistable rank \(n\) vector bundles of degree \(d\) over an elliptic curve \(C\). Then \({\mathcal M}_{n,d}\) identifies with \(S^ hC\), the \(h\)-th symmetric power of the curve \((h\) being the greatest common divisor of \(n\) and \(d)\). -- The determinant map \({\mathcal M}_{n,d}\to J_ d(C)\) (the latter being the Jacobian of line bundles of degree \(d\) on \(C)\) identifies with the Abel-Jacobi map \(S^ hC\to J_ h(C)\). In particular, for any \(L\in J_ d\), the space \({\mathcal M}_{n,L}\) of classes of semistable vector bundles with determinant \(L\) is a projective space of dimension \(h-1\). -- The appropriate Brill-Noether loci, \(W^ r_{n,0}(\forall)\) (resp. \(W^ r_{n,0}(\exists)\), defined as the set of equivalent classes with all representatives (resp. at least one representative) having \(r+1\) independent sections, are also identified with symmetric powers of \(C\). The same holds for the theta divisors. -- The corresponding Brill-Noether loci in \({\mathcal M}_{n,L}\) and theta divisors can also be identified with linear subspaces of it.
These identifications allow to extend to elliptic curves results of Drezet and Narasimhan on the Picard group of \({\mathcal M}_{n,d}\) and \({\mathcal M}_{n,L}\) for \(n\geq 2\), and formulas by Beauville, Narasimhan and Ramanan, Verlinde, and Bott and Szenes about theta divisors. moduli space; symmetric power of a curve; Jacobian of line bundles; Brill-Noether loci; theta divisors \beginbarticle \bauthor\binitsL. \bsnmTu, \batitleSemistable bundles over an elliptic curve, \bjtitleAdv. Math. \bvolume98 (\byear1993), page 1-\blpage26. \endbarticle \OrigBibText L. Tu, Semistable bundles over an elliptic curve , Adv. Math. 98 (1993), 1-26. \endOrigBibText \bptokstructpyb \endbibitem Elliptic curves, Families, moduli of curves (algebraic), Vector bundles on curves and their moduli Semistable bundles over an elliptic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(W_ r\), 1\(\leq r\leq g-1\), be the image of the Abel map for the r- fold symmetric product of a non-singular algebraic curve C of genus g; and \(W^-_ r\) be the ''inverse set'' of \(W_ r\), i.e. \(W^-_ r\) is the image of \(W_ r\) under the involution \(w\mapsto -w.\) As cycles, \(W_ r\) and \(W^-_ r\) are homologically equivalent on the Jacobian J(C). Moreover, it is well known that \(W_ r\) and \(W^-_ r\) are algebraically equivalent on J(C) when \(r=g-1\) (the Riemann symmetry of the \(W_{g-1}=\Theta\)-divisor!) or, for all r, when C is hyperelliptic. The paper under review shows that for 1\(\leq r\leq g-2\) on a generic Jacobian variety the cycles \(W_ r\) and \(W^-_ r\) are algebraically independent. The proof uses an ''inversion theorem'' for Abelian varieties and is done by reduction to a singular Abelian case. The crucial step is the case \(g=3\), \(r=1\). This result implies that Poincaré's formula is not valid for the algebraic equivalence ring of J(C) with C generic. Jacobian variety; Abel map; r-fold symmetric product of a non-singular algebraic curve; algebraic equivalence ring Ceresa, G., \(C\) is not algebraically equivalent to \(C^{-}\) in its Jacobian, Ann. Math. (2), 117, 285-291, (1983) Jacobians, Prym varieties, Cycles and subschemes C is not algebraically equivalent to \(C^-\) in its 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 For any integer \(m>1\), let \(X_ m\) be the Fermat curve over the complex number field \(\mathbb{C}\) defined by \(x^ m+y^ m+z^ m=0\). Let \({\mathfrak A}_ m=\{(a,b,c)\mid a,b,c\in\mathbb{Z}/m\mathbb{Z}\backslash\{0\}, a+b+c=0\}\), and let \({\mathfrak S}_ m\) denote the orbit space \((\mathbb{Z}/m\mathbb{Z})^ \times\backslash{\mathfrak A}_ m\), induced by the natural action \(t\cdot(a,b,c)\mapsto (ta,tb,tc)\) of the group \((\mathbb{Z}/m\mathbb{Z})^ \times\) on \({\mathfrak A}_ m\). Then we have an isogeny \(\pi:J(X_ m)\to\prod A_ S\), \(S\in{\mathfrak S}_ m\), where \(A_ S\) is an abelian variety of CM type in the sense of Shimura and Taniyama. There arise the following two natural questions:
(Q1) When are \(A_ S\) and \(A_{S'}\) isogenous over \(\mathbb{C}\) (for two distinct orbits \(S\) and \(S')\)?;
(Q2) When is \(A_ S\) absolutely simple?
In Can. J. Math. 30, 1183-1205 (1978; Zbl 0399.14023), \textit{N. Koblitz} and \textit{D. Rohrlich} gave the answer to these questions in three typical cases:
(i) g.c.d.\((m,6)=1\),
(ii) \(m=2^ n\),
(iii) \(m=3^ n\).
In this paper the author improves these results considerably and gives an ``almost complete answer'' to (Q1) and (Q2). Jacobian of a Fermat curve; isogenous abelian variety of CM type; Fermat curve N. Aoki, Simple factors of the Jacobian of a Fermat curve and the Picard number of a product of Fermat curves, Amer. J. Math., 113 (1991), 779-833. JSTOR: Jacobians, Prym varieties, Special algebraic curves and curves of low genus, Isogeny Simple factors of the Jacobian of a Fermat curve and the Picard number of a product of Fermat curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give a simple proof of the theorem of Beauville on the classification of four dimensional principally polarized Abelian varieties whose theta divisor is singular. That is: (1) In the moduli space \(A_ 4\) of four dimensional principally polarized Abelian varieties, the hypersurface \(N_ 0\) of those having singular theta divisor consists of two irreducible components \(J_ 4\) and \(\theta_{null}\). (2) At a general point of \(J_ 4\) the corresponding (Jacobian) theta-divisor has exactly two ordinary double points; at a general point of \(\theta_{null}\) the theta-divisor has exactly one ordinary double point, corresponding to one vanishing even theta-null.
The proof consists in showing that the multiplicity of a principally polarized Abelian variety as a point of \(N_ 0\) equals the sum of the Milnor numbers of its singular points. This allows one to identify the locus where \(J_ 4\) meets other components of \(N_ 0\) as the set of Jacobian theta-divisors with Milnor number greater than two. This locus is irreducible and at a generic point of it the Milnor number of \(\theta\) is three. It follows that at most one other component of \(N_ 0\) meets \(J_ 4\) and the Milnor number on that component is generically one. Since \(Pic(A_ 4)\cong {\mathbb{Z}}\), this ends the proof. Schottky problem; four dimensional principally polarized Abelian varieties; singular theta divisor; ordinary double points; multiplicity of a principally polarized Abelian variety; Milnor numbers; Jacobian theta-divisors Roy Smith and Robert Varley, On the geometry of \?\(_{0}\), Rend. Sem. Mat. Univ. Politec. Torino 42 (1984), no. 2, 29 -- 37 (1985). Theta functions and abelian varieties, \(4\)-folds, Jacobians, Prym varieties, Picard schemes, higher Jacobians On the geometry of \(N_ 0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors prove the following theorem: Let \(J\) be a general Jacobian variety of dimension \(g\geq 4\) defined over \({\mathbb{C}}\). If \(\chi: J'\to J\) is an isogeny and J' is a Jacobian then there is an isomorphism of principally polarized abelian varieties \(\mu: J\to J'\) and \(\chi\circ \mu\) is the multiplication by an integer.
The proof is based on a \((3g-3)\)-variables degeneration of \(J\) to a generalized Jacobian of an irreducible stable curve with two nodes, on the analysis of the extension classes coming from each node and on the comparison of the Gauss maps associated to the two surfaces of extension classes coming from the corresponding degenerations of \(J\) and of \(J'\) respectively.
If \(J\) is the Jacobian \(J(C)\) of a smooth curve \(C\), \(\Phi: C\to J\) is some fixed Abel Jacobi embedding, and \(\Phi _ n: C\to J\) is the map \(n\circ \Phi\) the theorem above has the following corollary: The irreducible curves of geometric genus \(g\) contained in a general Jacobian \(J(C)\) of dimension \(g\geq 4\) are the curves \(\Phi _ n(C)\) and all their translates. In particular they are all birationally equivalent. general Jacobian variety of dimension \(g\geq 4\); generalized Jacobian of an irreducible stable curve with two nodes; isogeny; geometric genus Bardelli, F., Pirola, G.P.: Curves of genusg lying on ag-dimensional Jacobian variety, Invent. Math.95, 263--276 (1989) Picard schemes, higher Jacobians, Jacobians, Prym varieties Curves of genus \(g\) lying on a \(g\)-dimensional Jacobian 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 We compute traces of the group structure on the Kummer variety of a hyperelliptic curve of genus 3 defined by a polynomial of degree 7. This allows to define a height function on the Jacobian of such curves and will give algorithms for computing the torsion subgroup and for infinite descent on these Jacobians. Kummer variety of a hyperelliptic curve of genus 3; height function on the Jacobian; algorithms for computing the torsion subgroup; infinite descent Heights, Jacobians, Prym varieties Heights and infinite descent on hyperelliptic 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 An embedding of the Jacobian variety of a curve \({\mathcal C}\) of genus 2 is given, together with an explicit set of defining equations. A pair of local parameters is chosen, for which the induced formal group is defined over the same ring as the coefficients of \({\mathcal C}\). It is not assumed that \({\mathcal C}\) has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field (of characteristic \(\neq 2, 3\), or 5). embedding of the Jacobian variety of a curve; local parameters; formal group Flynn, Eugene Victor, The Jacobian and formal group of a curve of genus \(2\) over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc., 107, 3, 425-441, (1990) Jacobians, Prym varieties, Formal groups, \(p\)-divisible groups, Computational aspects of algebraic curves The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 of this paper see J. Algebr. Geom. 1, No. 1, 5-14 (1992; Zbl 0783.14015).]
If the principally polarized abelian variety \(X\) is a Jacobian of a curve then the Kummer variety \(K(X)\) has infinitely many trisecant lines; in particular \(K(X)\) has a trisecant line. Welters' conjecture [see \textit{G. E. Welters}, Ann. Math., II. Ser. 120, 497-504 (1984; Zbl 0574.14027)]) is that the existence of a trisecant line of \(K(X)\) should imply that \(X\) is a Jacobian of a curve. The aim of this article is to improve the results of part I, where a weak version of this conjecture is proved under some additional hypotheses. More precisely, let \(\Theta\) be the theta divisor of \(X\), and let \(K:X \rightarrow K(X)\) be the natural map onto the Kummer variety, defined by the linear system \(|2 {\Theta} |\). Assume that \(K(X)\) admits a trisecant line \(L\), and let \(L \cap K(X) = \{ a_0, b_0, c_0 \}\). The results of this part II imply the following version of Welters' conjecture: \(X\) is the Jacobian of a curve provided the points \(a_0,b_0,c_0\) are distinct, and for some \(a,b,c \in X\), such that \(a_0 = K(a)\), \(b_0 = K(b)\), \(c_0 = K(c)\), the subgroup of \(X\) generated by \(a-b, b-c\) is dense in \(X\). principally polarized abelian variety; Jacobian of a curve; trisecant; Schottky problem; theta divisor Debarre O., Compisito Math. 107 pp 177-- Theta functions and curves; Schottky problem, Theta functions and abelian varieties, Jacobians, Prym varieties Trisecant lines and Jacobians. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors adapt the theory of adjoint divisors to the case of Gorenstein curves. They obtain an infinitesimal Torelli-type theorem for line bundles on an integral Gorenstein curve \(C\). If \(C\) is irreducible with either genus \(2\) or genus \(\geq 3\) and it is not hyperelliptic, they prove that an element of \(H^1(\omega _C^\vee)\) inducing a zero-adjoint is trivial (this is a Torelli theorem for \(C\) with respect to the subspace \(H^1(\omega _C^\vee)\) of \(\mathrm{Ext}^1(\Omega _C,\mathcal {O}_C)\), i.e. to the locally trivial deformations of \(C\)). They give examples showing that the results are false for reducible curves. extension class of a line bundle; infinitesimal Torelli problem; Gorenstein curve; generalized divisors Rizzi, L., Zucconi, F.: On Green's proof of infinitesimal Torelli theorem for hypersurfaces, Preprint Torelli problem, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Curves over finite and local fields A note on Torelli-type theorems for Gorenstein curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth, projective and connected curve (over an algebraically closed field of characteristic zero) of genus \(g(X)\geq 2\). Let \(x\) be a (closed) point of \(X\) and \(SU_X(r,d)\) the moduli space of semi-stable vector bundles on \(X\) of rank \(r\geq 2\) and determinant \({\mathcal O}(dx)\). As usual, the geometric points of \(SU_X(r,d)\) correspond to polystable bundles, namely direct sums \(E= \bigoplus E_i\) where \(E_i\) is stable of slope \(\mu(E_i)= d/r\) (and \(\bigotimes_i \det(E_i)= {\mathcal O}(dx))\).
Definition. The number of stable summands in the preceding sum is called the length \(l\) of the polystable bundle \(E\).
The singular locus of \(SU_X(r,d)\) consists exactly of the non stable points (except if \(r= g(X)= 2\) and \(d\) even). In this case, \(SU_X(r,d)= \mathbb{P}^3\). In particular, except in the exceptional case above, \(SU_X(r,d)\) is smooth if and only if \(r\) and \(d\) are relatively prime. General facts about the action of reductive groups ensure that \(SU_X(r,d)\) is Cohen-Macaulay, normal and that the singularities are rational. The principal aim of this paper is to give additional information about the singularities, essentially the description of the completion of the local ring at a singular point of \(SU_X(r,d)\) and to compute the multiplicity and the tangent cones at those singular points \(E\) which are not too bad, i.e. \(l(E)= 2\) (or equivalently \(\Aut (E)= G_m\times G_m\)). Further, we give a complete description in the rank 2 case.
\textit{M. S. Narasimhan} and \textit{S. Ramanan} [in: Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 415-427 (1987; Zbl 0685.14023)] discovered the link between the rank 2 vector bundles on a canonical curve \(X\) of genus 3 and the geometry of the Kummer variety \(\kappa(X)\) of \(JX\). The crucial point is that \(SU_X(2)\) is in this case canonically isomorphic to the so-called Coble quartic. We get the local form of this quartic and prove that the \(\kappa(X)\) is schematically defined by 8 cubics, the partials derivatives of the Coble quartic (theorem IV.6), although the corresponding homogeneous ideal is not generated by these cubics.
In the last part of the paper, we compute the multiplicity of a generalized theta divisor of \(SU_X(2,{\mathcal O})\) at a point \([L \oplus L^\vee]\), where \(L^2\neq{\mathcal O}\). In fact, this computation could be done with only minor changes for a point \(E\) of any rank with \(\det(E)={\mathcal O}\) and \(\Aut(E)= G_m\times G_m\). connected curve; moduli space of semi-stable vector bundles; length of the polystable bundle; singular locus; local ring at a singular point; multiplicity; tangent cones; Kummer variety; Coble quartic; multiplicity of a generalized theta divisor DOI: 10.1007/BF02566426 Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties, Theta functions and curves; Schottky problem Local structure of the moduli space of vector bundles over curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A generalized jacobian of a curve represents isomorphism classes of invertible sheaves algebraically equivalent to zero (AEZ) with a trivialization along a closed subscheme. The universal vector extension of its jacobian represents isomorphism classes of AEZ invertible sheaves with a connection. The bridge between these two schemes is a scheme which represents isomorphism classes of AEZ invertible sheaves with a connection and a `` horizontal trivialization'' along a closed subscheme.
The author makes explicit the connection between the theory of the universal vectorial extension and the theory of generalized jacobians, for curves over an arbitrary base. In particular, he proves those results needed in his paper ``The universal vectorial bi-extension and \(p\)-adic heights'' [Invent. Math. 103, No. 3, 631-650 (1991)] to verify the equality of Mazur-Tate height and the height defined in his joint paper with \textit{B. H. Gross} ``\(p\)-adic heights on curves'' in Adv. Stud. Pure Math. 17, 73-81 (1989). generalized jacobian of a curve; horizontal trivialization; universal vectorial extension Coleman, R.: Vectorial extensions of Jacobians. Ann. inst. Fourier (Grenoble) 40, No. 4, 769-783 (1990) Jacobians, Prym varieties Vectorial extensions of Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we present algorithms, suitable for computer use, for computation in the Jacobian of a hyperelliptic curve. We present a reduction algorithm which is asymptotically faster than that of Gauss when the genus \(g\) is very large. algorithms; computation in the Jacobian of a hyperelliptic curve D. G. Cantor, \textit{Computing in the Jacobian of a hyperelliptic curve}, Math. Comp., 48 (1987), pp. 95--101, . Jacobians, Prym varieties, Software, source code, etc. for problems pertaining to algebraic geometry, Software, source code, etc. for problems pertaining to field theory, Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Special algebraic curves and curves of low genus Computing the Jacobian of a hyperelliptic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobians of Fermat curves are of CM type. Is there a systematic way to construct other curves with this property ? The authors have this problem in mind and ask if the canonical lifting, due to Serre and Tate, of the Jacobian of an ordinary curve over a perfect field k of characteristic \(p>0\) is again the Jacobian of some curve. In this paper it is shown that, when p is odd and the genus is \(\geq 4\), the answer is ''no'' for most curves, even if one works \(mod p^ 2.\) The same problem is independently treated by \textit{F. Oort} and \textit{T. Sekiguchi} [J. Math. Soc. Japan 38, 427-437 (1986; Zbl 0605.14031)], and the results considerably overlap in both works. But the general ideas of the arguments are quite different from each other. Our authors proceed by ''pure thought'', while the others follow a very concrete way. moduli space; principally polarized abelian variety; crystalline cohomology; canonical lifting; Jacobian of an ordinary curve Dwork, B.; Ogus, A., \textit{canonical liftings of Jacobians}, Compositio Math., 58, 111-131, (1986) Picard schemes, higher Jacobians, Jacobians, Prym varieties, Families, moduli of curves (algebraic) Canonical liftings of Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y\) be an \((n-1)\)-dimensional smooth subvariety of the Grassmannian variety \(G(1,n)\) of lines in the complex projective \(n\)-space. In: Projective geometry with applications, Lect. Notes Pure Appl. Math. 166, 43-66 (1994; Zbl 0839.14015), \textit{E. Arrondo}, \textit{M. Bertolini} and \textit{C. Turrini} classified all such varieties admitting a scroll structure over a curve. In this paper we study whether \(Y\) admits a structure of a quadric fibration over a smooth curve. The first observation is that the bigger \(n\) is, the fewer \((n-2)\) varieties of small degree (degree one for studying scrolls and two for quadric bundles) there are in \(G(1,n)\). In particular, we get that, in case \(n\geq 7\), a quadric bundle \(Y\) must have a fundamental curve, so we can use our general result in the mentioned paper. It also happens that the case \(n=3\) turns out to be very difficult. This case has been studied by \textit{M. Gross} [Math. Z. 212, No. 1, 73-106 (1993; Zbl 0812.14033); pp. 89-93], who can only give partial results. Hence we will work out only the cases with \(n\geq 4\).
The main tool to limit the number of possible cases, is to use in a suitable way Castelnuovo's bound for the genus of projective curves, as well as a generalization of it obtained by L. Giraldo for curves in an arbitrary Grassmannian variety, which appears in the Appendix to this work. We give the whole list of possible varieties, although the actual existence of some of them remains still open. This classification, after the one of scrolls over a curve, is another step in the classification of those \(Y\) that do not behave well under the adjunction process (i.e. the adjoint bundle of which is not nef and big). Grassmannian variety; quadric fibration over a smooth curve; Castelnuovo's bound for the genus of projective curves Arrondo, E.; Bertolini, M.; Turrini, C.: Quadric bundle congruences in \(G(1,n)\). Forum math. 12, 649-666 (2000) Grassmannians, Schubert varieties, flag manifolds, Low codimension problems in algebraic geometry, Fibrations, degenerations in algebraic geometry Quadric bundle congruences in \(G(1,n)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author determines the Mordell-Weil groups over \({\mathbb{Q}}\) of the abelian variety \(J_ 0(81)\), and of each of its \({\mathbb{Q}}\)-simple factors. Ligozat has shown that the two elliptic curve factors each have rank zero. Utilizing Mazur's descent techniques, the author shows that the simple two-dimensional factor \(J_ 0(81)^{new}\) also has rank zero. The computation of the torsion subgroups of the Mordell-Weil groups follows the approach taken by \textit{A. P. Ogg} [in Analytic Number Theory,Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 221-231 (1973; Zbl 0273.14008)]. This involves counting rational points on the abelian varieties over various finite fields, and then using the fact that reduction modulo p is injective on the prime-to-p torsion points (for \(p\neq 3)\). Finally, the author notes that the finiteness of \(J_ 0(81)^{new}({\mathbb{Q}})\) is predicted by the conjecture of Birch and Swinnerton-Dyer. modular curve; jacobian; abelian variety; \({\mathbb{Q}}\)-simple factors; torsion subgroups of the Mordell-Weil groups; conjecture of Birch and Swinnerton-Dyer Rational points, Holomorphic modular forms of integral weight, Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Special algebraic curves and curves of low genus, Elliptic curves, Arithmetic ground fields for abelian varieties Points rationnels sur \({\mathbb{Q}}\) de la jacobienne de la courbe modulaire \(X_ 0(3^ 4)\). (Rational points over \({\mathbb{Q}}\) of the jacobian of the modular curve \(X_ 0(3^ 4))\) | 0 |
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