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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a curve contained on a \(K3\) surface \(S\); then the Wahl map for \(C\), \(\Psi_ C:\wedge^ 2H^ 0K_ C\to H^ 0K_ C^{\otimes 3}\) has a non-trivial cokernel. -- Assume \(Pic(S)=\mathbb{Z}\), spanned by the class of \(C\); then there is a rank 2 bundle on \(S\) which induces on \(C\) an extension \(0\to L\to E\to K_ C-L\to 0\) where \(L\) is a \(g^ 1_{s+1}\) on \(C\) \((g(C)=2s)\), such that the map \(H^ 0(K_ C-L)\to H^ 1(L)\) is 0.
In this paper, the two previous facts are correlated for curves satisfying the Brill-Noether-Petri condition; the author shows that starting with an element of the cokernel of the Wahl map, one can define a linear series \(L=g^ 1_{s+1}\) on \(C\), with a nontrivial extension \(0\to L\to E\to K_ C-L\to 0\), the left map of which is surjective on global sections. -- The construction yields a new proof of the surjectivity of the Wahl map, for curves of genus \(g\geq 13\). curve on a \(K3\) surface; curves of high genus; Brill-Noether-Petri condition; linear series; surjectivity of the Wahl map Voisin, C. : '' Sur l'application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri '', Acta Math. 168 (1992) 249-272. Vector bundles on curves and their moduli, Divisors, linear systems, invertible sheaves, \(K3\) surfaces and Enriques surfaces, Curves in algebraic geometry On the Wahl map of curves with Brill-Noether-Petri condition | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an arbitrary finite Galois \(p\)-extension \(L/K\) of algebraic function fields of one variable over an algebraically closed field \(k\) of characteristic \(p>0\) as its exact field of constants with Galois group \(G=\text{Gal}(L/K)\), we obtain explicitly the Galois module structure of the \(p\)-torsion of the Jacobian variety \(J_L(p)\) associated to \(L/k\). \(p\)-torsion of the Jacobian variety; Galois module structure Arithmetic theory of algebraic function fields, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Jacobians, Prym varieties \(p\)-adic Galois representation of the 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 Let \(p\) be a prime number and set \(q=p^ m\). Let \(X_{a,i}\) be the nonsingular algebraic curve over \(\mathbb F_ q\) with affine equation \(y^ p-y=ax+1/x+b_ i\), where \(a\in\mathbb F^*_ q\) and where \(b_ i\) is a fixed element of \(\mathbb F_ q\) with trace equal to \(i\in\mathbb F_ p\). It has long been known that the number of points of a curve of the above type is given by a Kloosterman sum and this has been used by Weil to deduce the classical estimate \((| K_ a| \leq 2\sqrt{q})\) on the absolute value of such Kloosterman sums.
We study the isogeny decomposition of the Jacobians of these curves and the \(p\)-torsion of their class groups. We find that the decomposition is determined to a large extent by the strict inequality \(| K_ a| <2\sqrt{q}\). It turns out -- surprisingly -- that the \(p\)-torsion is related to the characteristic polynomial of \(a\in\mathbb F^*_ q\) with respect to \(\mathbb F_ p\). The question of the variation of the \(p\)-order of \({\#}\text{Jac}(X_{a,i}(\mathbb F_ q))\) and that of the number of isogeny factors arose in the context of coding theory [cf. \textit{R. Schoof} and \textit{M. van der Vlugt}, J. Comb. Theory, Ser. A 57, No. 2, 163--186 (1991; Zbl 0729.11065) and the authors, J. Algebra 139, No. 1, 256--272 (1991; Zbl 0729.11066)]. The methods may also be used for other exponential sums such as multiple Kloosterman sums. p-torsion of class groups; number of points of Jacobian; number of points of algebraic curve over finite field; Kloosterman sum; isogeny decomposition of the Jacobians van der Geer, Gerard; van der Vlugt, Marcel, Kloosterman sums and the \textit{p}-torsion of certain Jacobians, Math. Ann., 290, 3, 549-563, (1991) Finite ground fields in algebraic geometry, Gauss and Kloosterman sums; generalizations, Exponential sums, Jacobians, Prym varieties, Enumerative problems (combinatorial problems) in algebraic geometry Kloosterman sums and the \(p\)-torsion of certain Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a continuation of the previous work by the author [Compos. Math. 111, 51--88 (1998; Zbl 0959.22012)] and is concerned with the cohomology groups of certain coherent sheaves on a Griffiths-Schmid variety associated to an anisotropic \(\mathbb Q\)-form of the unitary group \(\text{SU}(2,1)\). The author defines transforms relating this cohomology to the coherent cohomology of some sheaves on certain threefolds, which are fibered in projective lines over Picard modular surfaces. He explicitly describes the holomorphic and anti-holomorphic parts of the 1-cohomology of the Griffiths-Schmid variety in terms of classical Picard modular forms. This description provides an explicit generating system for the part of the 2-cohomology which corresponds to those automorphic representations whose archimedean components are degenerate limits of discrete series. automorphic forms; Dolbeault cohomology; unitary groups; Picard modular surfaces; cohomology groups; coherent sheaves; Griffiths-Schmid variety; anisotropic form of the unitary group SU(2,1); coherent cohomology; automorphic representations Carayol, H., Quelques relations entre LES cohomologies des variétés de Shimura et celles de Griffiths-schmid (cas du groupe \(S U(2, 1)\)), Compos. Math., 121, 305-335, (2000) Cohomology of arithmetic groups, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Representations of Lie and linear algebraic groups over global fields and adèle rings Some relations between the cohomologies of the Shimura varieties and the Griffiths-Schmid varieties (case of the group \(\text{SU}(2,1)\)) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(P\) be a Jacobian polynomial such that \(\deg P=\deg_y P\). Suppose the Jacobian polynomial \(P\) satisfies the intersection condition satisfying \(\dim_{\mathbb C} \mathbb C[x,y]/\langle P, P_y\rangle=\deg P-1\).
In the paper under review, the author proves that the Keller map which has \(P\) as a coordinate polynomial always has its inverse. polar; class of plane curve; plane Jacobian conjecture Milnor fibration; relations with knot theory, Singularities of curves, local rings A polar, the class and plane Jacobian 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 Fix a field \(k\) and consider the class of all varieties which are smooth over \(k\). If \(W @>\pi>> \text{Spec} (k)\) is within this class, we can attach to it a relative tangent bundle or, equivalently, a locally free sheaf \(\Omega^1_{W/k}\) of relative differentials. This tangent bundle, intrinsic to \(\pi\), plays a central role in algebraic geometry. For instance
(a) in studying the birational class of \(W\),
(b) in analyzing the singular locus of a closed embedded subscheme of \(W\) (e.g. jacobian ideals).
Essential for the development of some of these problems is the fact that the class is closed by blowing up regular centers, namely has the following property:
(P) Let \(W\) be smooth over \(k\), \(C\) a regular closed subscheme of \(W\), and \(W\leftarrow W_1\) the blow-up at \(C\). Then \(W_1\) is also in the class (is also smooth over the field \(k)\).
However, this property fails to hold if we consider now the class of smooth schemes over \(\mathbb{Z}\). In this work we replace \(\text{Spec} (k)\) by a Dedekind scheme of characteristic zero (for instance \(Y=\text{Spec} (\mathbb{Z}))\), and define a class of schemes over \(Y\) such that:
(1) the class includes the smooth schemes over \(Y\),
(2) to any \(W @>\pi>> Y\) in the class there is an intrinsically defined tangent bundle,
(3) the class is closed by blowing up convenient regular centers.
As an application, in \(\S 4\), we analyze the behaviour of jacobian ideals of embedded arithmetic schemes. sheaf of relative differentials; tangent bundle; birational class; singular locus; blowing up; smooth schemes; jacobian ideals of embedded arithmetic schemes Villamayor, O.: On smoothness and blowing ups of arithmetical schemes. Math. Z. 225, 317-332 (1997) Schemes and morphisms, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic problems in algebraic geometry; Diophantine geometry On smoothness and blowing ups of 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 \(f:S\to B\) be a relatively minimal fibration of curves of genus \(g\geq 2\) with \(S\) a smooth complex projective surface and \(B\) a smooth projective curve. Set \(\chi_f:= \deg(\omega_{S/B})\) and \(\lambda_f:= (\omega_{S/ B}\cdot \omega_{S/B})/ \chi_f\) (the slope of \(f)\). Assume that \(f\) is not locally trivial. By a theorem of G. Xiao we have \(4-4/g\leq\lambda_f\leq 12\). If \(f\) is an hyperelliptic fibration there is a formula
\[
\omega_{S/B} \cdot \omega_{S/B}= 4(g-1)\chi_f/ g+\sum_{P\in B}\text{Ind} \bigl(f^{-1}(P) \bigr) \tag{1}
\]
where \(\text{Ind} (f^{-1}(P))\) is a non-negative rational number attached to each fiber and with value \(0\) on a general fiber. E. Horikawa obtained a very explicit way of computing \(\text{Ind} (f^{-1}(P))\) when \(g=2\).
In this very interesting paper the author proposes (under certain technical assumptions) a definition for a rational number \(\text{Ind} (f^{-1}(P))\) (called here the Horikawa index) and obtains (for \(g\) odd) a formula like (1) which implies \(\lambda_f\geq 6(g-1)/(g+1)\). The technical assumptions are that a certain sheaf related to the minimal free resolution of the canonical embedding of a general fiber of \(f\) is torsion and it is satisfied if Green's conjecture on canonical curves is true in full generality. The formula depends on the Clifford index of a general fiber of \(f\). minimal free resolution of a canonically embedded curve; surface of general type; syzygies; fibration of curves; slope; Horikawa index; Clifford index K. Konno, Clifford index and the slope of fibered surfaces, J. Algebraic Geom. 8 (1999), no. 2, 207-220. Fibrations, degenerations in algebraic geometry, Rational and ruled surfaces, Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry Clifford index and the slope of fibered surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a toric Calabi-Yau \(3\)-fold \(X\) the mirror theory lives on a family of curves in \((\mathbb{C}^*)^2\) called the ``mirror curve''. The Eynard-Orantin topological recursion applied to a smooth complex curve produces an infinite tower of free energies \(F_g\) and meromorphic differentials \(W_n^g\). The remodeling conjecture states that the mirror map applied to \(F_g\) and \(W_n^g\) produces the generating functions of the closed and open Gromov-Witten invariants of \(X\) respectively. The open part of the conjecture for \(X=\mathbb{C}^3\) was proved independently by Chen and Zhou. In this note the authors complete the proof for \(\mathbb{C}^3\) by demonstrating that the free energies reproduce the closed invariants given by the celebrated Faber-Pandharipande formula.
The proof relies on methods of Chen and Zhou. The mirror curve to \(\mathbb{C}^3\) is a sphere with \(3\) punctures (pair of pants), and Chen and Zhou expressed its \(W_n^g\) in terms of Hodge integrals. The authors reduce \(F_g\) to Hodge integrals as well, but the computation is more subtle. A subtlety appears also in the relation to matrix models. The Eynard-Orantin recursion applied to the spectral curve of a matrix model exactly reproduces its correlation functions. However, there is a normalization ambiguity in the computation that may lead to discrepancy in the contributions of the constant maps to free energies. The authors show explicitly that this discrepancy is in fact present for the pair of pants, and in contrast to the open part of the conjecture the matrix model machinery can not be used. In conclusion they ask if there is a more direct computation of \(F_g\) that relies on geometry of the pair of pants rather than reduction to Hodge integrals. toric Calabi-Yau; mirror curve; Gromov-Witten invariants; Eynard-Orantin recursion; Hodge integral; spectral curve of a matrix model Bouchard, V; Catuneanu, A; Marchal, O; Sułkowski, P, The remodeling conjecture and the Faber-pandharipande formula, Lett. Math. Phys., 103, 59-77, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics The remodeling conjecture and the Faber-Pandharipande formula | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Any group G of automorphisms of an algebraic variety X acts in a natural way on the module \(\Omega_ X\) of its differentials, \(\Omega_ X\) thus becoming a right kG-module. A well known result due to \textit{C. Chevalley} and \textit{A. Weil} [Abh. Math. Semin. Hamb. Univ. 10, 358-361 (1934; Zbl 0009.16001)], \textit{A. Weil}, Oeuvres Scientifiques, vol. I (1979; Zbl 0424.01027), pp. 68-71] describes this kG-module structure on \(\Omega_ X\) for the case when k is algebraically closed of characteristic zero and X is a curve over k.
The paper under review contains a strong generalization of the result for the case when the field k is algebraically closed (char(k) being arbitrary) and the factorization \(p: X\to Y=X/G\) is tamely ramified. - The result may be elegantly stated in terms of the Grothendieck group theory and reads: \([\Omega_ X]=[k\oplus (kG)^{\oplus (g_ Y-1)}\oplus \tilde R^*_ G]\), where \([\quad]\) denotes the class in the Grothendieck group, \(g_ Y\) is the genus of Y, and \(R_ G\) the (tame) ramification module. module of differentials of a curve; group of automorphisms; Grothendieck group; ramification module Kani, Ernst, The {G}alois-module structure of the space of holomorphic differentials of a curve, Journal für die Reine und Angewandte Mathematik. [Crelle's Journal], 367, 187-206, (1986) Coverings of curves, fundamental group, Group actions on varieties or schemes (quotients), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grothendieck groups (category-theoretic aspects) The Galois-module structure of the space of holomorphic differentials of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (theorem 3). Equivalent, if \(\mathbb{C}^n\) is the total space for a principal \(\mathbb{G}_a\)-bundle over some open subset of \(\mathbb{C}^{n-1}\) then the bundle is trivial. On the other hand, there is a locally trivial \(\mathbb{G}_a\)-action on a normal affine variety with non-finitely generated ring of invariants (theorem 2). group actions; locally trivial action on a factorial variety; non-finitely generated ring of invariants Deveney J. K., Transformation Groups 2 pp 137-- (1997) Group actions on varieties or schemes (quotients), Complex Lie groups, group actions on complex spaces On locally trivial \(G_\alpha\)-actions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{R. C. Gunning} [Invent. Math. 66, 377-389 (1982; Zbl 0485.14009)] gives a criterion for a principally polarized abelian variety X to be the jacobian variety of a curve by means of the trisecants of the Kummer- Wirtinger variety of X. In a previous paper [Indagationes Math. 45, 501- 520 (1983; Zbl 0542.14029)], the author gives an infinitesimalization of Gunning's criterion, and he improves his criterion in this paper. His results give a new view point to the so-called Schottky problem. In particular, \textit{E. Arbarello} and \textit{C. De Concini} [Ann. Math., II. Ser. 120, 119-140 (1984; Zbl 0551.14016)] give an analytic translation of his results, and show the existence of a close relationship between their criterion and the Novikov conjecture. jacobian variety; trisecants of the Kummer-Wirtinger variety; Schottky problem G. E. Welters, ''A criterion for Jacobi varieties,'' Ann. of Math., vol. 120, iss. 3, pp. 497-504, 1984. Jacobians, Prym varieties, Theta functions and abelian varieties, Compact Riemann surfaces and uniformization, Picard schemes, higher Jacobians A criterion for Jacobi 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 article we provide the description of the structure and the properties of representation varieties \(R_n(G(p,q))\) of the groups with the presentation \(G(p,q) = \left\langle x_1,\dots,x_2, t\mid t(x_1^2\dots x_g^2)^q\right\rangle\), where \(g\geq 3\), \(|p| > q \geq 1\). Irreducible components of \(R_n(G(p,q))\) are found, their dimensions are calculated and it is proved, that every irreducible component of \(R_n(G(p,q))\) is a rational variety. group presentation; representation variety; dimension of a variety; rational variety Representation theory for linear algebraic groups, Ordinary representations and characters, Rational and unirational varieties, Generators, relations, and presentations of groups, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations On representation varieties of some HNN-extensions of free 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 [For the entire collection see Zbl 0686.00007.]
This paper develops the algorithms introduced by one of the authors which give the cylindrical algebraic decomposition of a real algebraic plane curve and its topology by purely formal methods (without any numerical approximation of numbers). In particular the authors prove the polynomial complexity for the topology of a curve even in singular situations, generalizing thus the results of Arnon-McCallum. cylindrical algebraic decomposition of a real algebraic plane curve; complexity ROY (M.-F.) , SZPIRGLAS (A.) . - Complexity of computations with real algebraic numbers , à paraître au Journal of Symbolic Computation. Real algebraic sets, Analysis of algorithms and problem complexity, Arithmetic ground fields for curves Complexity of the computation of cylindrical decomposition and topology of real algebraic curves using Thom's lemma | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \((Y,0)\) be a germ of a complex normal surface singularity, \((X,E) \to (Y,0)\) a minimal resolution, \(E_1, \ldots,E_n\) the irreducible components of the (necessarily connected) exceptional divisor \(E\). Then
\[
{\mathcal E} = \{ D=m_1E_1 + \cdots + m_n E_n : m_i \in \mathbb{Z} , \, D.E_i \leq 0 , \, i=1, \ldots, n \}
\]
is an additive semigroup, called the \textit{Lipman semigroup} of the singularity. It has a number of interesting properties, in particular it has a unique minimal generating set \({\mathcal H}_{\mathcal E} \), called the Hilbert basis of the semigroup. The toric variety associated to \(\mathcal E \) is \(V_{\mathcal E}:={\text{Spec}}\, ({\mathbb{C}}[\mathcal E]\)). Both the semigroup \(\mathcal E\) and the variety \(V_{\mathcal E}\) are important invariants of the singularity.
A natural question, addressed before by several authors, is to give an algorithm to construct the Hilbert basis \({\mathcal H}_{\mathcal E}\). In this paper the author works indirectly, by introducing an auxiliary semigroup \(\mathcal S = {\mathcal S}_{\mathcal E}\), which has a Hilbert basis \({\mathcal H}_{\mathcal S}\) that can be computed. It is proved that this basis immediately gives a Hilbert basis of \(\mathcal E\) and a parametrization of the toric variety \(V_{\mathcal E}\). The semigroup \(\mathcal S\) is defined as follows. Let \(M=M(\mathcal E)\) be the intersection matrix, i.e., its entry \(M_{ij}\) is the integer \(E_i.E_j\) and \(A=[A|I_n]\) (an \(n \times 2n\) matrix). Then,
\[
\mathcal S= \{(v_1, \ldots, v_n)\in {\mathbb{N}}^{2n}:A[v_1, \ldots,v_n]^T =0 \}\, .
\]
For semigroups of this kind (kernels of integral matrices) there are efficient combinatorial algorithms to find a Hilbert basis, e.g., see Chapter 6 of [\textit{M. Kreuzer, L. Robbiano}, Computational commutative algebra. II. Berlin: Springer. ( 2005; Zbl 1090.13021)].
The paper concludes with an example involving a singularity of type \(A_2\). Normal surface singularity; Lipman semigroup; Hilbert basis of a semigroup; toric variety; intersection matrix. Global theory and resolution of singularities (algebro-geometric aspects), Compact complex \(3\)-folds, Singularities in algebraic geometry, Local complex singularities, Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic and real-analytic geometry Hilbert basis of the Lipman semigroup | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0729.14022. complex multiplication; endomorphism ring of Jacobian; Fermat curve Jacobians, Prym varieties, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties The Jacobian of a cyclic quotient of a Fermat 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 a field and let \(A=k[X,Y]/\gamma\), \(\gamma\in k[X,Y]\), \(\gamma\) \(\neq 0\) be the affine ring of a plane algebraic curve \(C=Spec A\) over k. The author shows that SK\({}_ 1\)(A) is an abelian group with generators, certain special Mennicke symbols which can be associated to those closed points of the affine plane which do not lie on the curve C (theorem 1, which gives also the relations between the generators). As a corollary one obtains that the quadratic transformation maps between \(SK_ 1's\) of this type are always surjective. Also of interest is proposition 3, which states that \(GL_ 2(k[X,Y])\) acts transitively on the set of ordered pairs of generators of the maximal ideal (X,Y). affine ring of a plane algebraic curve; \(SK_ 1\); Mennicke symbols Krusemeyer M., Cubic curves. Comm. in Algebra 1 pp 51-- (1984) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups, \(K\)-theory and commutative rings, Curves in algebraic geometry Generators for \(SK_ 1\) of plane affine curves. I: General facts | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(F\) be an algebraically closed field of characteristic zero and \(GL(n,F)\) be the general linear group of \(n\times n\) invertible matrices with entries in \(F\). Given an \(n\times n\) matrix \(A\), its conjugacy class, \(\mathcal O(A)\), is the orbit of \(A\) under the conjugacy action of \(GL(n,F)\), i.e. \(\mathcal O(A)=\{U^{-1}AU \mid U \in GL(n,F)\}\). Denote by \(\mathcal C(A)\) the centralizer of \(A\). It is known that \(\dim \mathcal O(A) =n^2-\dim \mathcal C(A)\). Therefore, one gets three different formulae for \(\dim \mathcal C(A)\) using the existing three different formulae for \(\dim \mathcal O(A)\) when \(A\) is nilpotent. The author gives a rather direct proof of these formulae for \(\dim \mathcal C(A)\). Using the reduction to the nilpotent case, he then gets a formula for \(\dim \mathcal O(A)\) for an arbitrary matrix \(A\) yielding a nicer formula when \(A\) has no multiple nonzero eigenvalue. He finally gives a shorter proof of a theorem on irreducibility and dimension of a rank variety. dimension; conjugacy class of a matrix; rank variety; rank function; partition of a non-negative integer Group actions on varieties or schemes (quotients), Determinantal varieties On the dimension of certain \({\mathcal G}{\mathcal L}_n\)-invariant sets of matrices | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\subseteq\mathbb{P}^n\) be a quasi-projective variety and \(L\subseteq\mathbb{P}^n\) a general hyperplane such that the pair \(\left(X,X\cap L\right)\) is \(1\)-connected. The main goal of the present paper is to understand the kernel of the surjective map \[\pi_1\left(X\cap L\right)\twoheadrightarrow\pi_1\left(X\right),\] generalizing the Zariski-van Kampen theorem for \(X=\mathbb{P}^2\setminus C\) the complement of a plane algebraic curve, as well as some previous generalizations of the first author to certain singular varieties [\textit{C. Eyral}, Topology 43, No. 4, 749--764 (2004; Zbl 1060.14030)].
These previous results show that the kernel is generated by the images of certain standard monodromy operators \[\pi_1\left(X\cap L\right)\stackrel{\cong}{\rightarrow}\pi_1\left(X\cap L\right)\] associated to a general pencil \(\Pi=\left\{L_{\lambda}\right\}_{\lambda\in\mathbb{P}^1}\) of hyperplanes containing \(L\) such that the base locus \(\Pi_0\) of \(\Pi\) does not intersect \(X\).
The generalizations in this paper allow to handle the case \(\Pi_0\cap X\neq\emptyset\) by considering the \textbf{relative} monodromy operators, already introduced by the first author and \textit{D. Chéniot} [Trans. Am. Math. Soc. 358, No. 1, 1--10 (2006; Zbl 1086.14015)], which are induced by isotopies fixing the set \(\Pi_0\cap X\) pointwise.
The main result of the paper is Theorem 3.6, which states that for possibly singular \(X\) and a general pencil \(\Pi\) satisfying certain technical conditions, the kernel is indeed generated by the images of the relative monodromy operators. The case of smooth \(X\) is handled in Theorem 3.5, and actually follows from Theorem 3.6, assuming that all components of \(X\) have dimension at least \(2\). fundamental group; singular (quasi-projective) variety; pencil of hyperplane sections; relative loop; relative monodromy variation Homotopy theory and fundamental groups in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Zariski-van Kampen theorems for singular varieties -- an approach via the relative monodromy 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 In this note, the author proves that for a very ample \((2k-1)\)-spanned line bundle \(L\) on a smooth projective surface, the so-called Severi variety \(\tilde{V}_k(L)\) parameterizing all irreducible \(k\)-nodal curves \(D\in |L|\) is irreducible. Followed from the spannedness \(h^0(L)\geq 3k\), so that \(\dim \tilde{V}_k(L)=h^0(L)-k-1\) if \(h^0(L)>3k\) and \(\tilde{V}_k(L)=\emptyset\) otherwise. This result improves the same statement in [\textit{M. Kemeny}, Bull. London Math. Soc. 43, No. 1, 159--174 (2013; Zbl 1032.14005)] proved for \((3k-1)\)-very ample line bundles and is along the result in [\textit{C. Ciliberto} and \textit{Th. Dedieu}, Proc. Am. Math. Soc. 147, No. 10, 4233--4244 (2019; Zbl 1423.14192)].
Reviewer's remark: In the definition of \(k\)-spanned (\(k\)-very ample) line bundles, the rôle of \(k\) is missing. \(K3\) surface; Severi variety; nodal curve; Hilbert scheme of nodal curves Families, moduli of curves (algebraic), Surfaces and higher-dimensional varieties, Divisors, linear systems, invertible sheaves On the irreducibility of the Severi variety of nodal curves in a smooth 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 Here it is proved that the monodromy of the Riemann surfaces of genus \(g\) acts as the full symmetric group on the Weierstraß points of a general curve. The proof uses a degeneration to stable reducible curves with \(Pic^ 0\) compact introduced and developed by the authors [Invent. Math. 85, 337-371 (1986; Zbl 0598.14003)]. An example: if the curve X is union of a smooth curve Y of genus \(g-1\) and an elliptic curve E glued at a point \(p\in Y\cap E,\quad p\quad not\) a Weierstraß point of Y, the Weierstraß points on X are the ramification points on Y of the complete linear series \(| K_ Y+2p|\) and the \(g^ 2-1\) points of \(E\setminus \{p\}\) which differ from p by g-torsion. Part of the monodromy is constructed by fixing a (reducible) curve and varying its canonical series. reducible curve; canonical divisor; Weierstraß points of a general curve; ramification points Eisenbud, D. andHarris, J., The monodromy of Weierstrass points,Invent. Math. 90 (1987), 333--341. Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Classification theory of Riemann surfaces, Coverings of curves, fundamental group The monodromy of 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 [For the entire collection see Zbl 0674.00007.]
Let X be the Kummer surface of the Jacobian A of a genus two curve C. In this paper, an explicit description of a fibering \(X\to {\mathbb{R}}_ 1\) is given with generic fiber \(X_ s\) elliptic, such that the quotient map \(A\to (A/\pm 1)\) induces a double covering \(Y_ s\to X_ s\) with A as Prym variety. Kummer surface of the Jacobian; fibering; covering; Prym variety Hoyt, W.: Elliptic fiberings of Kummer surfaces. Number theory, New York, 1985/1988, Lecture notes in math. 1383, 89-110 (1989) Elliptic surfaces, elliptic or Calabi-Yau fibrations, Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces Elliptic fiberings of Kummer surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For \(M\) a very ample line bundle on a smooth complex projective variety \(Y\) and the map associated to \(M\), the author defines the property \(N_p \) for every positive integer \(p\). The problem under consideration is:
Given \(M\), a line bundle as before on \(Y\) satisfying property \(N_p,\) and fixing a natural number \(s \), for which \(k\) does the line bundle \(M^s \) satisfy the property \(N_k \)?
The author proves (proposition 1) that if \(M\) satisfies property \(N_p \) then \(M^s \) satisfies property \(N_x\) where \( x= \min \{ s,p\} \). divisors; syzygies; resolutions; line bundle on a smooth complex projective variety; property \(N_p\) Rubei E.: A note on property N p . Manuscr. Math. 101, 449--455 (2000) Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A note on property \(N_p\) | 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 A characteristic condition is given on a zero-dimensional differentiable 0-sequence \(H=(h_ i)_{i\geq 0}\), \(h_ 1\leq 3\), in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of \({\mathbb{P}}^ 3\), hence of points of \({\mathbb{P}}^ 2\) with the uniform position property. In this way an answer is given to some question stated by \textit{J. Harris} in 1982. The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in \({\mathbb{P}}^ 3\) whose generic plane section has an assigned Hilbert function satisfying that condition. Hilbert function of a generic plane section of a reduced irreducible curve; uniform position; arithmetically Cohen-Macaulay curve; space curve Maggioni, R.; Ragusa, A., The Hilbert function of generic plane sections of curves of \(\mathbf{P}^3\), Invent. Math., 91, 2, 253-258, (1988) Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Projective techniques in algebraic geometry The Hilbert function of generic plane sections of curves of \({\mathbb P}^ 3\) | 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 Let \({\mathcal E}\) be a vector bundle of rank r on a compact complex manifold M of dimension m and \(A=\det({\mathcal E})\). The \(c_ 1\)-sectional genus \(g=g(M,A)\) is defined by the relation \(2g(M,A)-2=(K+(m-1)A)A^{m-1}\) where K is the canonical bundle. The author classifies M and \({\mathcal E}\) for \(g=0,1,2.\)
If \(g=0\), \(M\cong P^ 2_{\alpha}\) and \({\mathcal E}=H_{\alpha}\oplus H_{\alpha}.\)
If \(g=1\), the classification breaks up into 5 similar parts.
When \(g=2\), the author uses the classification theory of polarized surfaces of sectional genus 2 to obtain a much more complicated 5 part classification broken up into subsections. Here, for instance, the first class contains M which are certain Jacobian varieties of smooth curves of genus 2, the second class (not known to exist) contains \(M\cong P({\mathcal F})\) with \({\mathcal F}\) a stable vector bundle of rank 2 and the fourth class consists of M which are the blowing-ups of \({\mathbb{P}}^ 2\) at 8 points. vector bundle on a compact complex manifold; \(c_ 1\)-sectional genus; classification theory of polarized surfaces; Jacobian varieties; blowing- up Fujita, T., Ample vector bundles of small \(c_{1}\)-sectional genera, J. Math. Kyoto Univ., 29, 1-16, (1989) Moduli, classification: analytic theory; relations with modular forms, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Ample vector bundles of small \(c_ 1\)-sectional genera | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Recall that the gonality of a curve is the minimal degree of a linear pencil on it thus measuring the deviation from rationality of the curve in a different way than the genus of the curve does. The gonality provides a stratification of the moduli space \(M_g\) of curves of genus \(g\). It seems natural to ask if/how the linear pencils of degree \(k\) may be used to describe the ``non-trivial'' linear series on curves in \(M_g\) of gonality \(k\). At least for the general curve of genus \(g\) and gonality \(k\) an amenable answer would help to establish a Brill-Noether theory for such curves.
In this paper we study this question for curves of low gonality -- especially for 4-gonal curves.
After passing through (and motivated by) the well-known theory of linear series on 2- and 3-gonal curves we will define in \S 1 certain types of (complete) linear series on a \(k\)-gonal curve which are well-adopted to identify and non-trivial linear series on it in the case \(k=4\). In \S 2 we apply this typification to investigate the minimal degree \(s(r)\) of birational models of the \(k\)-gonal curve in \(\mathbb{P}^r\) \((r\geq 2)\). Again, the results are complete only in case \(k=4\), for the general 4-gonal curve of genus \(g\); in that case \(s(r)=[{g+ 4r-1\over 2}]\) if \(g\) is not too small.
In \S 3 we single out irreducible components of the varieties \(W_d^r(C)\) \((d<g,\;r\geq 1)\) of special divisors on the general 4-gonal curve \(C\) obtaining for nets (i.e., \(r=2)\) a fairly complete picture. In particular, for \(d\geq s(r)\) the variety \(W_d^r(C)\) is not equi-dimensional provided that \(g\geq 4(r+1)\) \((r\geq 2)\). linear series on 4-gonal curves; gonality of a curve; stratification of the moduli space; Brill-Noether-theory \textsc{M. Coppens, } Linear series on \(4\)-gonal curves, Math. Nachr. \textbf{213} (2000), 35-55. Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus Linear series on 4-gonal 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 See the preview in Zbl 0525.55012. stable sheaf on variety of dimension bigger than one; complete intersection curve; stable bundle with zero Chern classes; irreducible unitary representation of fundamental group V.B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group. Inv. Math. 77 (1984), pp. 163--172. Sphere bundles and vector bundles in algebraic topology, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Complete intersections Restriction of stable sheaves and representations of the fundamental group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this text, the authors study the Jacobian \(J\) of the smooth projective curve \(C\) of genus \(r-1\) with affine model \[y^r=x^{r-1}(x+1)(x+t),\] over the function field \(\mathbb{F}_p(t)\), when \(p\) is prime and \(r\geq2\) is an integer prime to \(p\). When \(q\) is a power of \(p\) and \(d\) is a positive integer, they compute the \(L\)-function of \(J\) over \(\mathbb{F}_q(t^{1/d})\) and show that the Birch and Swinnerton-Dyer conjecture holds for \(J\) over \(\mathbb{F}_q(t^{1/d})\). When \(d\) is divisible by \(r\) and of the form \(p^\nu+1\), and \(K_d:=\mathbb{F}_p(\mu_d,t^{1/d})\), they write down explicit points in \(J(K_d)\), show that they generate a subgroup \(V\) of rank \((r-1)(d-2)\) whose index in \(J(K_d)\) is finite and a power of \(p\), and show that the order of the Tate-Shafarevich group of \(J\) over \(K_d\) is \([J(K_d):V]^2\). When \(r>2\), the authors prove that the new part of \(J\) is isogenous over \(\overline{F_p(t)}\) to the square of a simple abelian variety of dimension \(\phi(r)/2\) with endomorphism algebra \(\mathbb{Z}[\mu_r] ^+\). For a prime \(l\) with \(l\nmid pr\), they prove that \(J[l](L)=\{0\}\) for any abelian extension \(L\) of \(\overline{F}_p(t)\). This monograph is organized as follows. Chapter 1, deals with the curve, explicit divisors, and relations. The authors give basic information about the curve \(C\) and Jacobian \(J\) they are studying. They write down explicit divisors in the case \(d=p^\nu+1\), and we find relations satisfied by the classes of these divisors in J. These relations turn out to be the only ones, but that is not proved in general until much later in the paper. Chapter 2, deals with descent calculations. In this chapter, the authors assume that \(r\) is prime and use descent arguments to bound the rank of \(J\) from below in the case when \(d=p^\nu+1\). Chapter 3, deals with minimal regular model, local invariants, and domination by a product of curves. The authors construct the minimal, regular, proper model \(\aleph\longrightarrow\mathbb{P}^1\) of \(C/\mathbb{F}_q(u)\) for any values of \(d\) and \(r\). In particular, they compute the singular fibers of \(\aleph\longrightarrow\mathbb{P}^1\). This allows them to compute the component groups of the Néron model of \(J\). They also give a precise connection between the model \(X\) and a product of curves. Chapter 4, deals with heights and the visible subgroup. The authors consider the case where \(d=p^\nu+1\) and \(r|d\), and they compute the heights of the explicit divisors introduced in Chapter 1. This allows them to compute the rank of the explicit subgroup \(V\) and its structure over the group ring \(\mathbb{Z}[\mu_r\times\mu_d]\). Chapter 5, deals with the \(L\)-function and the \(BSD\) conjecture. The authors give an elementary calculation of the \(L\)-function of \(J\) over \(\mathbb{F}_q(u)\) (for any \(d\) and \(r\)) in terms of Jacobi sums. They also show that the \(BSD\) conjecture holds for \(J\), and we give an elementary calculation of the rank of \(J(\mathbb{F}_q(u))\) for any \(d\) and \(r\) and all sufficiently large \(q\). Chapter 6, deals with analysis of \(J[p]\) and \(NS(\aleph_d)tor\) and Chapter 7, with index of the visible subgroup and the Tate-Shafarevich group. In these technical chapters, the authors prove several results about the surface \(X\) that allow them to deduce that the index of \(V\) in \(J(K_d)\) is a power of \(p\) when \(d=p^\nu+1\) and \(r\) divides \(d\). They also use the \(BSD\) formula to relate this index to the order of the Tate-Shafarevich group. Chapter 8, deals with monodromy of \(l\)-torsion and decomposition of the Jacobian. The authors prove strong results on the monodromy of the \(l\)-torsion of \(J\) for prime to \(pr\). This gives precise statements about torsion points on \(J\) over abelian or solvable extensions of \(\mathbb{F}_p(t)\) and about the decomposition of \(J\) up to isogeny into simple abelian varieties. The paper is supported by an appendix on an additional hyperelliptic family. curve; function field; Jacobian; abelian variety; finite field; Mordell-Weil group; torsion; rank; \(L\)-function; Birch and Swinnerton-Dyer conjecture; Tate-Shafarevich group; Tamagawa number; endomorphism algebra; descent; height; Néron model; Kodaira-Spencer map; monodromy Research exposition (monographs, survey articles) pertaining to number theory, Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Algebraic functions and function fields in algebraic geometry Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A theta divisor on an abelian variety \(A\) over a field is an effective ample divisor \(\Theta\) such that the global sections of the line bundle \(O_{A}(\Theta)\) are one-dimensional. For example, the image of the \((g-1)\)st symmetric power of a smooth projective curve \(C\) of genus \(g\) with a rational point in its Jacobian is a theta divisor. Berthelot, Bloch and Esnault proved the following
Theorem 1.1. Let \(\Theta, \Theta'\) be two divisors on an abelian variety, all defined over a finite field \({\mathbb F}_{q.}\) Then \( | \Theta ({\mathbb F}_{q})| \equiv | \Theta' ({\mathbb F}_{q})| \mod q. \)
Question 1.2. Is \(| \Theta | \equiv | \Theta' | \mod {\mathbb L} \in K_{0}(\mathrm{Var}_{k})\)?
Here \(K_{0}(\mathrm{Var}_{k})\) denotes the naive Grothendieck ring of varieties over \(k,\) i.e. the free abelian group on isomorphism classes \([X]\) of varieties over \(k\) modulo the relations \([X] = [X -Y] + [Y]\) for \(Y \subset X\) a closed subvariety. The ring structure is given by product of varieties, and \({\mathbb L}\) is the class \([A^{1}_{k}]\) of the affine line. The author gives a negative answer to the above question using the fact that products of two elliptic curves can be Jacobians. symmetric power of a smooth projective curve; naive Grothendieck ring of varieties Theta functions and abelian varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Arithmetic ground fields for abelian varieties, Divisors, linear systems, invertible sheaves, Étale and other Grothendieck topologies and (co)homologies, Theta functions and curves; Schottky problem, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Elliptic curves, Finite ground fields in algebraic geometry A note on congruences for theta divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a complex algebraic curve of genus \(g\geq 2\) and let \(G\) denote an abelian group of automorphisms of \(G\). The gonality of \(C\) is defined to be the minimal integer \(d\) for which there exists a map \(f:C\rightarrow\mathbb P^1\) of degree \(d\). In [\textit{C. Maclachlan}, Proc. Lond. Math. Soc. (3) 15, 699--712 (1965; Zbl 0156.08902)], it was shown that \(|G|\leq 4g+4\) and in the special case that \(G\) is cyclic, it was shown that \(|G|\leq 4g+2\) in [\textit{A. Wiman}, Stockh. Akad. Bihang \(\text{XXI}_1\). No. 1. 23 S (1895; JFM 26.0658.02)], and later in [\textit{W. J. Harvey}, Q. J. Math., Oxf. II. Ser. 17, 86--97 (1966; Zbl 0156.08901)]. Each of the two bounds are attained for all genera by families of hyperelliptic curves, so curves with gonality \(d=2\). If the gonality \(d=3\) and genus \(g\) satisfies \(g\geq 5\), it was shown in [\textit{A. F. Costa} and \textit{M. Izquierdo}, J. Algebra 323, No. 1, 27--31 (2010; Zbl 1188.30049); corrigendum ibid. 341, No. 1, 313--314 (2010)] that when \(G\) is cyclic, \(|G|\leq 3g+3\) and moreover, if \(|G|>2g+4\), then the degree \(3\) cover \(f: C\rightarrow\mathbb P^1\) is Galois.
In the paper under review, the authors extend these results for general gonality \(d\). Specifically, they show that \(|G|\leq 2dg/(d-1)+2d\) except for the Fermat curve of degree \(d+1\); \(|G|\leq 2dg/(d-1)+d\) if \(G\) is cyclic; and if \(|G|>2g-2+2d\) and \(C\) admits a finite cover \(f: C\rightarrow\mathbb P^1\) of degree \(d\) (with \(d\) not necessarily the gonality), then \(f\) is Galois. It should be noted that for \(d=2\) and \(d=3\), these bounds imply the classical results cited. The authors also provide explicit examples to show that these bounds are attained. The method of proof primarily uses the equality given by the Riemann-Hurwitz formula which, for a given map \(\pi : C\rightarrow C'\) on a curve \(C\) to another curve \(C'\), relates genera of \(C\) and \(C'\) to the degree and the ramification data of the map \(\pi\). gonality of a curve; abelian group of automorphisms; automorphisms of compact Riemann surfaces Automorphisms of curves, Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group On the gonality of an algebraic curve and its abelian automorphism 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 In the liaison theory of space curves it is very important to consider not just reduced or integral curves but also locally Cohen-Macaulay purely one dimensional subschemes, \(C\), of \(\mathbb{P}^3\). Here the author proves the following result which is the extension to such schemes of Halphen speciality theorem bounding the first integer \(n\) with \(h^0(\mathbb{P}^3, {\mathcal I}_C(n))\neq 0\) (denoted with \(s(C))\) in terms of the speciality index \(e(C)\), i.e, the largest integer \(n\) with \(h^1(C,{\mathcal O}_C(n))\neq 0\).
Theorem. Assume characteristic 0. Let \(C\subset\mathbb{P}^3\) be a locally Cohen-Macaulay curve with index of speciality \(e\). Suppose that no subcurve \(D\) of \(C\) with \(e(D)=e\) lies on a surface of degree \(t-1\). Set \(m:=\min\{t,[(e+4)/2]\}\). Then:
(i) \(\deg(C)\geq m(e+4-m)\) with equality holding if and only if \(C\) is a specialization with constant cohomology of a complete intersection of two surfaces of degree \(m\) and \(e+4-m\);
(ii) \(s(C)\leq\deg(C)-m(e+3-m)\).
For the proof the author introduced and studied several notions related to the spectrum of \(C\). postulation; Rao module; spectrum of a space curve; Cohen-Macaulay curve; index of speciality Schlesinger, E., A speciality theorem for Cohen-Macaulay space curves, Trans. Amer. Math. Soc., 351, 2731-2743, (1999) Plane and space curves, Linkage A speciality theorem for Cohen-Macaulay space 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 higher Chow groups of a regular scheme were defined by Bloch, furnishing a geometric interpretation for the higher \(K\)-groups. The sheaves of higher \(K\)-groups recover the full ordinary Chow groups, including torsion: formula of Bloch-Quillen, spectral sequence of Gersten. In the paper these relationships are generalized to the higher Chow groups. Relative Chow groups, with respect to a closed subscheme, and relative higher Chow groups are defined, to which the relationships extend as well. For divisors, the relative Chow group is expressed in terms of Picard groups, using the generalized Bloch formula. algebraic \(K\)-theory; Bloch-Quillen formula; Gersten spectral sequence; sheaves of higher \(K\)-groups; torsion; higher Chow groups; relative Chow group; Picard groups S. E. Landsburg, ''Relative Chow groups,'' Illinois J. Math., vol. 35, iss. 4, pp. 618-641, 1991. \(K\)-theory of schemes, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles Relative Chow groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author introduces the paper under review as a twin article of his previous paper [Pac. J. Math. 268, No. 2, 283--312 (2014; Zbl 1312.11033)]. To a modular cuspidal eigenform \(f\) of weight \(2\), a well-known construction due to Eichler and Shimura attaches an abelian variety \(A_f\) defined over \({\mathbb{Q}}\) of dimension the degree of the field generated by the coefficients of \(f\) over \({\mathbb{Q}}\). The author considers the Mordell-Weil group \(A_f(k)\) for \(k\) a fixed number field. He produces control theorems which give information on how the group \(\hat{A}_f(k)=A_f(k)\otimes_{\mathbb{Z}} {\mathbb{Z}}_p\) varies as \(f\) varies among those cuspidal eigenforms \(f\) in the family. He also considers the same question where the number field \(k\) is replaced by an \(\ell\)-adic field.
The organization of the paper is the following. After the introduction, section 2 deals with sheaves associated to abelian varieties. Section 3 deals with \(U (p)\)-isomorphisms. The structure of \(\Lambda\)-BT groups over number fields and local fields is described in section 4. Section 5 is about abelian factors of modular Jacobians. In section 6, the author studies the structure of ind-\(\Lambda\)-MW groups over number fields and local fields. The last section 7 deals with the closure of the global \(\Lambda\)-MW group in the local case. modular curve; Hecke algebra; modular deformation; analytic family of modular forms; Mordell-Weil group; modular Jacobian Hecke-Petersson operators, differential operators (one variable), Modular correspondences, etc., Counting solutions of Diophantine equations, Elliptic curves over global fields, Abelian varieties of dimension \(> 1\), Arithmetic aspects of modular and Shimura varieties, Jacobians, Prym varieties Limit Mordell-Weil groups and their \(p\)-adic closure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 general k-gonal curve, M an image of A in some \({\mathbb{P}}^ n\), and fix integers g, d, x, a and b verifying that \(k\geq n+2\) and \(\rho(d,g,n)\geq 0\) (the Brill-Noether number).
Consider each of the following conditions: (i) M is an embedding of degree d and genus g; (ii) M is an embedding of degree \(d+3x\) and genus \(g+2x\) having a x-secant line; (iii) M is an embedding of degree \(d+a\) and genus g except for b nodes and \((n+1)a\geq nb.\)- Then the author proves that, in each of the above cases, M is a smooth point of the component W of \(Hilb({\mathbb{P}}^ n)\) where it lies. Also, in a neighborhood of M in W, all embeddings of A in the same conditions form a subscheme of the right dimension.
Using the same techniques, the author also proves that a set of points Z in \({\mathbb{P}}^ n\) is the intersection of \({\mathbb{P}}^ n\) with some k- dimensional scroll in \({\mathbb{P}}^{n+k}\) if and only if Z is curvilinear. prints of projective space as intersection with a scroll; k-gonal curve; Brill-Noether number --, On special linear systems on curves.Comm. in Algebra 18 (1990), 279--284. Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry On special linear systems on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the authors prove that the unboundedness of ranks in a certain family of Jacobian varieties is equivalent to the divergence of a certain infinite series. For a fixed odd prime \(p\), and for any \(p\)-th power free integer \(m\), let \(J_m^p\) denote the Jacobian variety of the twisted Fermat curve \(C_m^p : x^p+y^p=m\). Fix a nonzero integer \(m'\) such that \(C_{m'}^p(\mathbb{Q})\neq\emptyset\), and choose \(\alpha\in\overline{\mathbb{Q}}\) satisfying \(\alpha^p=m/m'\). Let \(\phi:C_m^p\rightarrow C_{m'}^p\) denote the isomorphism over \(\mathbb{Q}(\alpha)\) defined by \(\phi((x,y,z))=(x,y,\alpha z)\). Let \(D_{m'}^p\in \text{Div}(J_{m'}^p)\) be a fixed symmetric positive divisor defined over \(\mathbb{Q}\), and let \(\hat{h}_m^p:J_m^p(\bar{\mathbb{Q}})\rightarrow \mathbb{R}\) be the canonical height with respect to the divisor \(3\phi^*D_{m'}^p\). For any nonnegative real numbers \(i, j\), let
\[
T_p(i,j)=\sum_{m\in \mathbb{N}^{(p)}}|m|^{-i}\sum_{P\in J_m^p(\mathbb{Q})-J_m^p(\mathbb{Q})_{tor}}\hat{h}_m^p(P)^{-j},
\]
where \(\mathbb{N}^{(p)}\) denotes the set of \(p\)-th power free integers. Their main result states that \(\text{rank}(J_m^p(\mathbb{Q}))<2j\) for every \(m\in\mathbb{N}^{(p)}\) if and only if \(T_p(i,j)\) converges for \(i=1\). Fermat curve; Jacobian variety; elliptic curve; canonical height Dąbrowski, A.; Jędrzejak, T.: Ranks in families of Jacobian varieties of twisted Fermat curves, Canad. math. Bull. 53, 58-63 (2010) Abelian varieties of dimension \(> 1\), Elliptic curves over global fields, Heights, Rational points, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Special algebraic curves and curves of low genus, Arithmetic ground fields for abelian varieties Ranks in families of Jacobian varieties of twisted 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 classical generalized Reed-Muller codes introduced by \textit{T. Kasami}, \textit{S. Lin} and \textit{W. W. Peterson} [IEEE Trans. Inf. Theory IT-14, 807-814 (1968; Zbl 0214.471)], and studied also by \textit{P. Delsarte}, \textit{J. M. Goethals} and \textit{F. J. Mac Williams} [Inform. Control 16, 403-442 (1970; Zbl 0267.94014)], are defined over the affine space \(\mathbb{A}^ n(\mathbb{F}_ q)\) over the finite field \(\mathbb{F}_ q\) with \(q\) elements. Moreover \textit{G. Lachaud} [Discrete Math. 81, No. 2, 217- 221 (1990; Zbl 0696.94015)], following \textit{S. G. Vladuţ} and \textit{Yu. I. Manin} [J. Sov. Math. 30, 2611-2643 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 209-257 (1984; Zbl 0629.94013)], has considered projective Reed-Muller codes, i.e. defined over the projective space \(\mathbb{P}^ n(\mathbb{F}_ q)\).
In this paper, the evaluation of the forms with coefficients in the finite field \(\mathbb{F}_ q\) is made on the points of a projective algebraic variety \(V\) over the projective space \(\mathbb{P}^ n(\mathbb{F}_ q)\). Firstly, we consider the case where \(V\) is a quadratic hypersurface, singular or not, parabolic, hyperbolic or elliptic. Some results about the number of points in a (possibly degenerate) quadric and in the hyperplane sections are given, and also is given an upper bound of the number of points in the intersection of two quadrics. -- In application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters. -- Secondly, we take \(V\) as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound). If \(V\) is of degree \(h\), we give parameters of Reed-Muller codes of order \(d<h\), associated to \(V\). number of points in a quadric; generalized Reed-Muller codes Y. Aubry, Reed-Muller codes associated to projective algebraic varieties. Lecture Notes in Math. 1518 (1992), 4-17. Zbl0781.94004 MR1186411 Geometric methods (including applications of algebraic geometry) applied to coding theory, Computational aspects of higher-dimensional varieties, Algebraic coding theory; cryptography (number-theoretic aspects), Finite ground fields in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Reed-Muller codes associated to projective algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Nash problem asks for the bijectivity of the Nash map that is an injective map from the set of irreducible components of the space of arcs passing through the singular locus to the set of the essential divisors. This problem was negatively solved by the reviewer and \textit{J. Kollár} in genera [Duke Math. J. 120, No. 3, 601--620 (2003; Zbl 1052.14011)]; i.e., the Nash map is not surjective in general. Therefore the next problem is to determine the image of the Nash map. In this paper, the authors prove that a non-uniruled exceptional divisor belongs to the image of the Nash map. arcs; wedges; resolution of singularities; Nash map; essential divisors; uniruled variety M. Lejeune-Jalabert and A. J. Reguera-López, Exceptional divisors which are not uniruled belong to the image of the Nash map, 2008. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Exceptional divisors that are not uniruled belong to the image of the Nash map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Cartier divisor \(L\) is said to be strictly nef if \(LC > 0\) for all curves \(C\). Such divisors are not always ample, so we consider small perturbations of \(L\). The question is posed on whether \(L + \varepsilon K\) is \(\mathbb{Q}\)-ample for small enough rational numbers \(\varepsilon > 0\), \(K\) denoting a canonical divisor. An affirmative answer is given for Gorenstein surfaces and in most cases of smooth threefolds. In particular, it is shown that a smooth threefold is Fano provided \(- K\) is strictly nef. Fano threefolds; perturbations of Cartier divisor; ampleness of strictly nef divisors; Gorenstein surfaces Serrano F.: Strictly nef divisors and Fano threefolds. J. Reine Angew. Math. 464, 187--206 (1995) Divisors, linear systems, invertible sheaves, \(3\)-folds, Fano varieties Strictly nef divisors and Fano threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0651.00010.]
In the last years the authors studied the topology of the complement of a plane curve, in order to get applications to the study of complex surfaces of general type. The main point was to regard the algebraic curve embeddings in the projective plane in analogy with the knots in \(S^3\). As a main application, the authors constructed first examples of simply-connected surfaces of general type with positive and zero indices.
The present paper contains two parts. The first one is a detailed and self-contained presentation of the main braid group techniques in complex case. There are 5 sections concerning the following: Some algebraic background; remarks about fundamental groups of punctured 2-disks; frame of a braid group; E. Artin's finite presentation of braid groups; positive braids. - In the second part of the paper, containing 4 sections, one establishes an explicit formula for braid monodromy corresponding to specific line arrangements in \(\mathbb{P}^2\) (a line arrangement is a union of projective lines). fundamental group; topology of the complement of a plane curve; braid group; braid monodromy; line arrangement Moishezon, B; Teicher, M, Braid group technique in complex geometry, I, line arrangements in \(\mathbb{C}\mathbb{P}^2\), Contemp. Math., 78, 425-555, (1988) Coverings of curves, fundamental group, Projective techniques in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects) Braid group technique in complex geometry. I: Line arrangements in \({\mathbb{C}}P^ 2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The modular curve \(X_r=X_1(Np^r)\), its Jacobian \(J_r\) and the \(p\)-divisible part \(J_{r,p}=J_r[p^{\infty}]\) of the Jacobian are considered. Then \(J_{r,p}\) is a module over the Hecke algebra \(h_r\) of the curve \(X_r\). The limits taken with respect to the natural coverings \(X_r\to X_s\) (for \(r\geq s\geq 1)\) \(J_{\infty,p}=\varinjlim_ rJ_{r,p}\), \(h_{\infty}=\varprojlim_r h_r\), are considered. The structure of \(J_{\infty,r}\) as a \(h_{\infty}\)-module is considered. More exactly, constructions of \textit{H. Hida} [Invent. Math. 85, 545--613 (1986; Zbl 0612.10021)] are generalized and, for every local factor \(R\) of the ordinary part of the Hecke algebra, the structure of the \(R\)-module \(J_{\infty,p}(R)\)
\[ J_{\infty,p}(R) \cong R\otimes_{\Lambda}\Hom(\Lambda,\Pi_p) \oplus\Hom(R,\Pi_p), \]
where \(\Pi_p=\mathbb Q_P/\mathbb Z_P\), \(\Lambda =\mathbb Z_p[[\Gamma ]]\) is the Iwasawa algebra, is completely described. p-divisible part of the Jacobian; modular curve; Hecke algebra; Iwasawa algebra Tilouine, J., Un sous-groupe \textit{p}-divisible de la jacobienne de \(X_1(N p^r)\) comme module sur l'algèbre de Hecke, Bull. Soc. Math. France, 115, 3, 329-360, (1987) Jacobians, Prym varieties, Holomorphic modular forms of integral weight, Arithmetic ground fields for curves A certain \(p\)-divisible subgroup of the Jacobian of the curve \(X_1(Np^r)\) as a module over the Hecke algebra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If a variety \(V \subset \mathbb{P}^ n\) is arithmetically Cohen-Macaulay (aCM) then the same is true of the general degree \(d\) hypersurface section \(Z\) of \(V\). The converse holds if \(\dim V \geq 2\). In the case of curves, it is interesting to ask what conditions on \(Z\) may force \(V\) to be aCM. The first step was proved by Strano: if \(n = 3\), \(V\) is integral not lying on a quadric surface, \(d = 1\) and \(Z\) is a complete intersection then \(V\) must be aCM (hence a complete intersection). This was generalized by Re (allowing higher \(n)\), Huneke-Ulrich (allowing higher \(n\) and \(d\), and \(Z\) Gorenstein), in that order. This paper is a continuation of this project. \(Z\) is allowed to have higher Cohen- Macaulay type (recall that ``Gorenstein'' means Cohen-Macaulay type 1), but it is assumed that the coordinate ring of \(Z\) is \textit{level}; that is, the last free module in a minimal free resolution of \(I_ Z\) is a direct sum of copies of the same rank one free module \(R (-m)\) (for some \(m)\).
Positive results are obtained here in the following situations:
(a) \(d\) is large (from the point of view of the deficiency module of \(V)\);
(b) \(V\) is arithmetically Buchsbaum; and
(c) \(V\) is a curve of maximal rank. (Recall that Ballico and Ellia, in a series of papers, proved that the ``general'' curve of degree \(d\) and genus \(g\) has maximal rank.)
Also, many examples are given of what can and cannot be expected in the way of counterexamples. In preparation for the discussion of Buchsbaum curves, a new result is proved about the minimal free resolution of curves obtained by certain liaison techniques. arithmetically Cohen-Macaulay variety; arithmetically Buchsbaum variety; hypersurface section; curve of maximal rank; liaison Migliore, J. C.; Nagel, U., On the Cohen-Macaulay type of the general hypersurface section of a curve, Math. Z., 219, 2, 245-273, (1995) Plane and space curves, Complete intersections, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Linkage On the Cohen-Macaulay type of the general hypersurface section of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a non-singular rational surface X over an algebraically closed field k, let -K denote an anticanonical divisor. The anti-Kodaira dimension \(\kappa^{-1}(X)\) is defined to be \(\kappa\) (-K,X). The main result is stated as follows: Let X be a non-singular rational surface with \(\kappa^{-1}(X)=2\). Then the anticanonical ring \(R^{- 1}(X)=\oplus_{m\geq 0}H^ 0(X,0(-mK))\) is finitely generated over k, and the anticanonical model \(Y=\Pr oj R^{-1}(X)\) satisfies the following properties: (i) Y has only isolated rational singularities, (ii) \(-K_ Y\) is an ample Q-Cartier divisor. Conversely, if a normal projective surface Y has the above properties (i) and (ii), then its minimal resolution X is a rational surface with \(\kappa^{-1}(X)=2.\)
The proof is based on the theory of the Zariski decomposition of divisors (pseudo effective) established by \textit{O. Zariski} (Ann. Math., II. Ser. 76, 560-615 (1962; Zbl 0124.370)] and by \textit{T. Fujita} [Proc. Jap. Acad., Ser. A 55, 106-110 (1979; Zbl 0444.14026)]. It turns out that for a rational surface X, -K is pseudo effective if and only if \(\kappa^{- 1}(X)\geq 0\). If so, there exists the Zariski decomposition: \(-K=P+N\), where P is a numerically effective Q-divisor. The negative part N is related to the singularities of the anticanonical model Y (in case \(\kappa^{-1}(X)=2)\). In order to analyse it, we need some local properties of surface singularities. For this, combined with a vanishing theorem, a local version of the notion of the Zariski decomposition is useful.
In addition, a dimension formula for the m-th anti-genus is considered, where it appears a term depending on the singularities of Y. Define \(d(X)=P^ 2\) and call it the degree of X. The effect of blowing ups on the degree is discussed. We remark that there are many interesting examples of rational surfaces with \(\kappa^{-1}(X)=2\). In this article, the structures of rational surfaces with \(\kappa^{-1}(X)=0\) and 1 are also described. ample Cartier divisor; pseudo effective divisor; anti-Kodaira dimension; rational surface; isolated rational singularities; minimal resolution; Zariski decomposition of divisors; dimension formula; anti-genus Sakai, F, Anticanonical models of rational surfaces, Math. Ann., 269, 389-410, (1984) Families, moduli, classification: algebraic theory, Special surfaces, Rational and unirational varieties, Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties Anticanonical models of rational surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give an inductive way to obtain the vanishing cycles which occur in the ``second Lefschetz theorem'' about pencils of hyperplane sections of a non-singular projective variety. In fact we do this for the more general case of a non-singular quasi-projective variety. As a corollary, we show that the Lefschetz hyperplane section theorem for a non-singular quasi-projective variety can be made more precise in the middle dimension by a statement which recalls the theorem of von Kampen on the fundamental group of the complement of a plane projective curve. second Lefschetz theorem; vanishing cycles; pencils of hyperplane sections; fundamental group of the complement of a plane projective curve D. Chéniot, Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety, Proc. London Math. Soc. (3) 72 (1996), no. 3, 515 -- 544. Algebraic cycles, Pencils, nets, webs in algebraic geometry, Hypersurfaces and algebraic geometry Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review the author proves various results on the Koszul cohomology (such as duality theorem, vanishing theorems, Lefschetz theorems etc.) thus developing algebraic techniques for describing the equations defining the image of a complex manifold under an embedding in a projective space. As an application, the author describes the structure of a minimal free resolution of the ideal of the embedding of a smooth curve C by a complete linear system of divisors of a sufficiently large degree. Another application concerns the so-called Arbarello-Sernesi module AS(X,L): if X is a compact complex manifold and L is an analytic line bundle over X, then \(AS(X,L)=\oplus_{q\in {\mathbb{Z}}}H^ 0(X,K_ X\otimes L^ q)\) viewed as a module over \(S(H^ 0(X,L))\) (where S denotes the symmetric algebra). The author shows that if \(| L|\) does not have fundamental points and maps X onto an n-dimensional variety \((n=\dim X)\), then, with a few exceptions, AS(X,L) is generated in degree \(\leq n-1\) and its relations are generated in degrees \(\leq n\) (this generalizes Petri's result for curves). Other applications include various local Torelli theorems. Koszul cohomology; duality theorem; vanishing theorems; Lefschetz theorems; image of a complex manifold; embedding; minimal free resolution of the ideal of the embedding of a smooth curve; Arbarello-Sernesi module; local Torelli theorems M. Green, Koszul cohomology and the cohomology of projective varieties, J. Differential Geom. 19 (1984), 125-171. (Co)homology theory in algebraic geometry, Analytic sheaves and cohomology groups, Complex manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Transcendental methods, Hodge theory (algebro-geometric aspects), Vanishing theorems Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians An island of a nonsingular real algebraic curve of even degree is an outermost oval of the real algebraic curve, together with all ovals contained in it. An island is even if it consists of an even number of ovals. \textit{V. A. Rokhlin} derived inequalities for even degree nonsingular real algebraic curves and introduced the notion of complex orientations on such curves. An extremal property of Rokhlin's inequality is derived for those real algebraic curves in which every island is even. This extremal property disqualifies certain complex orientations on such real algebraic curves. Rokhlin's inequality; island of a real curve; oval Special algebraic curves and curves of low genus, Topology of real algebraic varieties, Projective techniques in algebraic geometry An extremal property of Rokhlin's inequality for real algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(E\) be an elliptic curve over the finite field \(\mathbb F_{q}\), \(P\) a point in \(E(\mathbb F_{q})\) of order \(n\), and \(Q\) a point in the group generated by \(P\). The discrete logarithm problem on \(E\) is to find the number \(k\) such that \(Q = kP\). In this paper we reduce the discrete logarithm problem on \(E[n]\) to the discrete logarithm on the group \(\mathbb F^*_{q}\), the multiplicative group of nonzero elements of \(\mathbb F_q\), in the case where \(n|q-1\), using generalized Jacobian of \(E\). elliptic curve; discrete logarithm problem; generalized Jacobian Elliptic curves, Cryptography, Elliptic curves over global fields Generalized Jacobian and discrete logarithm problem on elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be an irreducible complex smooth curve of genus \(g\). Let \(\pi: \widetilde{C}\to C\) be a connected unramified double covering of \(C\). The Prym variety associated to the covering is, by definition, the component of the origin of the kernel of the norm map \(P(\widetilde{C},C)= \text{Ker(Nm}_\pi)^0\subset J\widetilde{C}\). It is a principally polarized abelian variety (p.p.a.v.) of dimension \(g(\widetilde{C})-g= g-1\). One defines the Prym map
\[
\begin{aligned} P_g:{\mathcal R}_g &\to {\mathcal A}_{g-1},\\ (\widetilde{C}@>\pi>>C) &\mapsto P(\widetilde{C},C), \end{aligned}
\]
where \({\mathcal R}_g\) is the coarse moduli space of the coverings \(\pi\) as above and \({\mathcal A}_{g-1}\) stands for the coarse moduli space of p.p.a.v.'s of dimension \(g-1\).
In this note we characterize the fibres of positive dimension of the Prym map. To state our theorem we need some notation: Let \({\mathcal {RB}}_g\) be the coarse moduli space of the unramified double coverings \(\pi:\widetilde{C}\to C\) such that \(C\) is a smooth bi-elliptic curve of genus \(g\). This variety has \([\frac{g-1}{2}]+2\) irreducible components
\[
{\mathcal {RB}}_g=\Biggl( \bigcup_{t=0}^{[\frac{g-2}{2}]} {\mathcal {RB}}_{g,t}\Biggr)\cup {\mathcal {RB}}_g'
\]
[see \textit{J.-C. Naranjo}, J. Reine Angew. Math. 424, 47-106 (1992; Zbl 0733.14019)].
Theorem. Assume \(g\geq 13\). A fibre of \(P_g\) is positive dimensional at \((\widetilde{C},C)\) if and only if \(C\) is either hyperelliptic or \((\widetilde{C},C)\in \bigcup_{t\geq 1}{\mathcal {RB}}_{g,t}\). Jacobian; Prym variety; Prym map; bi-elliptic curve J. C. Naranjo, The positive dimensional fibres of the Prym map, Pac. J. Math. 172 (1) (1996), 223--226. Jacobians, Prym varieties, Picard schemes, higher Jacobians, Elliptic curves The positive dimensional fibres of the Prym map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 birational rigidity of multidimensional algebraic varieties with a pencil of double quadrics of index 1 sufficiently twisted with respect to the base is established. Fano variety; birational automorphisms; ample divisors; birationally superrigid Fano fibration; pencil of double quadrics Rational and birational maps, Automorphisms of surfaces and higher-dimensional varieties, Fano varieties Birational automorphisms of algebraic varieties with a pencil of double quadrics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a proper smooth surface and \(U\subset K\) a dense open subset. Let \(D_ 1,D_ 2,...,D_ r\) be integral divisors on X and \(m=\sum^{r}_{i=1}n_ jD_ j \) for \(n_ j\geq 0\). Let G be a commutative algebraic group. Let \(Cat_ m\) be the category of rational maps \(\alpha:\quad X\to G\) with domain U admitting m as modulus. A generalized albanese pair \((G_{U_ m},\alpha)\) is an object \(\alpha:\quad G_{U_ m}\to G\) in \(Cat_ m\) such that any object in \(Cat_ m\) factorizes uniquely through \(\alpha\). The existence of generalized albanese pairs is shown and concrete models are given in special cases. The projective limit of the system \(G_{U_ m}\) is related to the abelian fundamental group of U. integral divisors; category of rational maps; generalized albanese pair; abelian fundamental group Hurşit Önsiper, On generalized Albanese varieties for surfaces, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 1 -- 6. Surfaces and higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Abelian varieties and schemes On generalized Albanese varieties for surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0745.00034.]
The author considers the problem of determining the topology of a complex algebraic variety which is given in some explicit way. The author considers the notion of stable diffeomorphity and its criteria in the general setting of normal maps. Then he gives a list of several algebraic geometrical situations where some topological situations are applicable modelling the fulfilment of the Lefschetz theorem. topology of a complex algebraic variety; stable diffeomorphity; Lefschetz theorem N. Yu. Netsvetaev, ''Diffeomorphism criteria for smooth manifolds and algebraic varieties,''Contemporary Math.,431 (3), 453--459 (1992). Topological properties in algebraic geometry, Differential topological aspects of diffeomorphisms Diffeomorphism criteria for smooth manifolds and algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R=\oplus R_ n\), \(n\in {\mathbb{Z}}\), be a reduced graded ring with finitely many minimal primes. For d a positive integer, let \(R(d)=\oplus R_{nd}\), \(n\in {\mathbb{Z}}\). If \({}^+R\) is the seminormalization of R, it is shown that \({}^+R\) is a graded ring, and that \((^+R)(d)\) is the seminormalization of R(d). Using these and related facts, the paper then shows that the weak normalization of a projective variety is a projective variety. weak normalization of a projective variety DOI: 10.1155/S0161171285000254 General commutative ring theory, Varieties and morphisms, Graded rings and modules (associative rings and algebras) Seminormal graded rings and weakly normal projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to construct curves C in \({\mathbb{P}}^ 3\) having the genus large with respect to the degree d of C and with respect to the smallest degree k of a surface containing C. This partially answers to a problem already considered by Halphen: to find, for any \(d,k>0\), the maximum genus of a non singular connected curve in \({\mathbb{P}}^ 3.\)
In a letter to the authors (1978) Joe Harris has communicated how to prove the existence of curves having the same genus found in the present paper, but his reasoning uses hypotheses of general position. The analysis of such hypotheses in the context of the reflexive sheaves instead of curves as in the paper of \textit{A. Hirschowitz} [Publ. Math., Inst. Hautes Étud. Sci. 66, 105-137 (1988; Zbl 0647.14004)], has originated the work under review.
The method used here is to construct certain good reflexive sheaves of rank 2 on \({\mathbb{P}}^ 3\) and afterwards to deduce the existence of good non singular curves associated to the sheaf as locus of zeroes of global sections. curves in projective 3-space; degree; maximum genus of a non singular connected curve [HH 2] Hartshorne, R., Hirschowitz, A.: Nouvelles courbes de genre éléve dans ?3 (en préparation) Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry Nouvelles courbes de bon genre dans l'espace projectif. (New curves of good genus in projective space) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Picard group of a scheme and related functors are studied. The representability problem for functors is of particular interest. Picard group of a scheme Picard groups, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, Riemann-Roch theorems Abelsche Schemata und Darstellbarkeit von algebraischen Gruppenräumen | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0655.00011.]
From the introduction: There is a classical problem whether there exist curves in the complex projective plane \(\mathbb P^ 2\) with assigned numerical characters satisfying the genus formula of Clebsch. More than half a century ago Lefschetz and Zariski studied such a problem for Plückerian characters. Concerning the problem we consider the following case. Let \((d,e)\) [resp. \(<d,e>]\) be a pair of positive integers, then it is said to be effective if there is a curve \(C\) in \(\mathbb P^ 2\) with \(C- \{P\}\cong \mathbb A^ 1\) [resp. \(\mathbb G_ m]\) for some point \(P\), where \(d=\deg (C)\) and \(e=\text{mult}_ PC\). Then there is the following conjecture:
If a pair \((d,e)\) [resp. \(<d,e>\)] is effective, then \(d<3e\) [resp. \(d\leq 3e\)].
In the former case Tsunoda has shown that \(d\leq 3e+2\) by using the logarithmic version of Miyaoka's inequality. In this note we shall prove the following theorem by considering the double covering of \(\mathbb P^ 2\) branched along \(C\):
A pair \((d,2)\) [resp. \(<d,2>\)] is effective if and only if \(d<6\) [resp. \(d\leq 6\)].
Moreover we have the following proposition: Let \(C\) be an irreducible rational curve with at most double points. Then the number of non- cuspidal singular points is at least \(k^ 2-3k+1\), where \(d=2k\) or \(2k+1.\)
Remark. A pair \((7,3)\) is not effective. Hence the converse assertion of the conjecture is not true. plane curves; multiplicity of a singular point; degree of curve; genus formula; number of non-cuspidal singular points Yoshihara, H.: A note on the existence of some curves, 801-804 (1988) Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Special algebraic curves and curves of low genus A note on the existence of some 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 \(Y\) be an integral curve with \(m\) nodes (ordinary double points) over \(\mathbb C\), and \(p:X\longrightarrow Y\) be the normalization. Let \(\overline J(Y)\) denotes the compactified Jacobian of rank one torsion-free sheaves of degree \(-1\) on \(Y\), and \(h: \widetilde J(Y) \longrightarrow\overline J(Y)\) be the normalization. One can identify \(\widetilde J(Y)\) with the moduli space of generalized parabolic line bundles on \(X\) [\textit{U. Bhosle}, Ark. Mat. 30, No. 2, 187--215 (1992; Zbl 0773.14006)]. Let \(U_Y(r,d)\) be the moduli space of semi-stable torsion-free sheaves of rank \(r\) and degree \(d\) on \(Y\), and let \(P(r,d)\) denotes the moduli space of \(\alpha\)-semi-stable \((0\leq\alpha < 1)\) generalized parabolic bundles (GPBs in short) of rank \(r\) and degree \(d\) on \(X\) [loc. cit.]. Assume \(\alpha\) is close to \(1\) (e.g. \(1-\frac{1}{r(r-1)}<\alpha<1\)). Then there is a surjective birational morphism \(f: P(r,d) \longrightarrow U_Y(r,d)\).
Let \(E_0\) be a stable vector bundle on \(Y\) of rank \(r\) degree \(d+r, -r < d \leq 0\) such that \(p^*E_0\) is stable, and let \(\overline{E_0'}\) (\(E_0'= p^*E_0\)) be a corresponding GPB over \(X\). Consider the morphisms \(\alpha_Y: \overline{J}(Y) \longrightarrow U_Y(r,d)\) given by \(L \mapsto L\otimes E_0\) and \(\widetilde{\alpha}_Y: \widetilde{J}(Y) \longrightarrow P(r,d)\) given by \(\overline{L}\mapsto \overline{L}\otimes \overline{E_0'}\) (this is a tensor product of two GPBs). In this article, the author shows that, if \(E_0'=p^*E_0\) has no line bundle direct summands, then both \(\alpha_Y\) and \(\widetilde{\alpha}_Y\) are embedddings.
Let \(\Theta_{\overline{J}(Y)}, \Theta_{U_Y(r,d)}\) be the theta divisors on \(\overline{J}(Y)\) and \(U_Y(r,d)\) respectively. Set \(\Theta_{\widetilde{J}(Y)}=h^*\Theta_{\overline{J}(Y)}\) and \(\Theta_{P(r,d)}=f^*\Theta_{U_Y(r,d)}\). With the assumption that \(p^*E_0\) has no line bundle direct summands, the author obtains several relations between the theta divisors on pull backs under the morphisms \(\alpha_Y\) and \(\widetilde{\alpha}_Y\).
Let \(\text{gcd}(r,d)=1\), and \(E_{r,d}\) be the Picard bundle on \(U_Y(r,d), r \geq 2\) and \(d\geq (2g-2)r\). Choose a stable vector bundle \(E_0\) on \(Y\) such that \(\alpha_Y:\overline{J}(Y) \longrightarrow U_Y(r,d), L \mapsto L\otimes E_0\) is an embedding. The author proves that the restriction of \(E_{r,d}\) to \(\overline{J}(Y)\) is stable with respect to any theta divisor \(\Theta_{\overline{J}(Y)}\) if \(d > r(2g-1)\) and semi-stable if \(d=r(2g-1)\). Similar results for \(r=1\) were obtained in [\textit{U. N. Bhosle} and \textit{A. J. Parameswaran}, Int. Math. Res. Not. 2014, No. 15, 4241--4290 (2014; Zbl 1303.14042)].
The author also shows that if the genus of \(X\) is at least two, then any general stable bundle of rank \(\leq 2\) (of any degree) has natural cohomology; also provided sufficient conditions on vanishing of first and second cohomology of a general stable bundle of rank \(r\geq 3\).
In the case of smooth curves, similar results were obtained in [\textit{Y. Li}, Int. J. Math. 2, No. 5, 525--550 (1991; Zbl 0751.14019)].
Reviewer's Comment: In the course of the proof of Theorem 1.1, the following fact is used: \(\mathcal{E}nd^0(p^*E_0)\), the trace zero endomorphisms, has no line bundle direct summands (Proposition 3.1). It's not clear that the assumption of Theorem 1.1 is sufficient to conclude this. But the author proves that for a general stable bundle \(E_0\), this condition is satisfied (Proposition 2.9). compactified Jacobian; theta divisors; moduli space; Picard bundle Vector bundles on curves and their moduli Embedding of a compactified Jacobian and theta divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0741.00067.]
This paper is devoted to prove the Riemann-Kempf singularity theorem using geometric-enumerative arguments. Let us assume that \(C\) is a non- hyperelliptic curve of genus \(g\), \(D\) a positive divisor of degree \(d\leq g-1\), \(W_ d\) the subvariety of the Jacobian, \(J\), representing linear equivalence classes of positive divisors of degree \(d\) and \(u(D)\) the point of \(W_ d\) defined by \(D\). The starting point is Kempf's description of the tangent cone \({\mathcal C}_ D\) to \(W_ d\) at \(u(D)\): \({\mathcal C}_ D=\bigcup_{D_ \lambda\in| D|}\overline D_ \lambda\subset\mathbb{P}_{g-1}=\mathbb{P}(T_{u(D)}J)\), \(\overline D_ \lambda\) being the linear subspace of \(\mathbb{P}_{g-1}\) spanned by \(D_ \lambda\). Let \(i=h^ 1({\mathcal O}_ C(D))\) be the index of speciality of \(D\) and \(r=h^ 0({\mathcal O}_ C(D))-1\). Let \(p_ 1,\ldots,p_ i\) be \(i\) points of \(C\) in general position on \(C\) and in general position with respect to the linear systems \(| D|\) and \(| K-D|\) and let \(L=\overline{p_ 1+\cdots+p_ i}\) be the linear span of these points. The author proves that \(L\) intersects \({\mathcal C}_ D\) transversally and that \({\mathcal C}_ D\cap L=\bigcup\overline{p_{\nu_ 1}+\cdots+p_{\nu_ r}}\), (\(p_{\nu_ 1}+\cdots+p_{\nu_ r}\) being all choices of \(r\) points from the \(i\) points \(p_ 1,\ldots,p_ i)\). From this result the degree of \({\mathcal C}_ D\) can be trivially computed and, therefore, the multiplicity of \(W_ d\) at \(u(D)\). singularities; Riemann-Kempf singularity theorem; non-hyperelliptic curve; linear equivalence classes of positive divisors; index of speciality Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry A note on the Riemann-Kempf theorem 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 [For part I see Compos. Math. 60, 115-132 (1986; Zbl 0607.13009).]
Let A be an affine normal domain of dimension \(d\geq 2\) over a universal domain. In this paper, a necessary and sufficient condition is obtained for the existence of projective A-modules of rank d which have no direct summand of rank 1. The condition is expressed in two equivalent ways:
(i) the Chow group of zero cycles of A is non-zero;
(ii) the subgroup of \(K_ 0(A)\), the Grothendieck group of projective A- modules, generated by the classes of residue fields of smooth points of Spec(A), is nonzero.
In the course of the proof, the following structure theorem is obtained for the divisor class group of a normal projective variety over an algebraically closed field k: It is an extension of a finitely generated abelian group by the group of k-points of an abelian variety. indecomposable projective modules; Chow group; Grothendieck group; divisor class group of a normal projective variety Picard groups, Projective and free modules and ideals in commutative rings, Parametrization (Chow and Hilbert schemes), Grothendieck groups, \(K\)-theory and commutative rings Indecomposable projective modules on affine domains. 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 First the author determines the \(\Lambda\)-module structure of the Mordell-Weil group \(A(k_{\infty})\), where A is an abelian variety defined over a \({\mathfrak p}\)-adic number field k with good reduction and \(\Lambda\) the Iwasawa algebra for the \({\mathbb{Z}}_ p\)-extension \(k_{\infty}/k\). The main tool to prove is the result of \textit{P. Schneider} [Invent. Math. 87, 587-602 (1987; Zbl 0608.14034)] concerning the group of universal norms of a commutative formal group and a description of the \(k_{\infty}\)-rational points of a toroidal formal group due to \textit{J. Lubin} and \textit{M. I. Rosen} [J. Algebra 52, 236-240 (1978; Zbl 0417.14035)]. The author also shows that there are only finitely many \({\mathbb{Z}}_ p\)-extensions \(k_{\infty}/k\) such that the torsion of \(A(k_{\infty})\) is infinite.
Next the author considers a simple abelian variety A defined over a global number field k and a \({\mathbb{Z}}_ p\)-extension \(k_{\infty}/k\) such that \(Tor(A(k_{\infty}))\) is infinite and proves that A admits complex multiplication by a CM-field K and \(k_{\infty}\) is \(k(A({\mathfrak p}))\) for a prime \({\mathfrak p}| p\) of K of degree 1. Mordell-Weil group; abelian variety; Iwasawa algebra; universal norms of a commutative formal group; rational points of a toroidal formal group; \({\mathbb{Z}}_ p\)-extension; complex multiplication Wingberg, K, On the rational points of abelian varieties over \({\mathbb{Z}}_{p}\)-extensions of number fields, Math. Ann., 279, 9-24, (1987) Arithmetic ground fields for abelian varieties, Rational points, Cyclotomic extensions, Complex multiplication and abelian varieties, Formal groups, \(p\)-divisible groups On the rational points of abelian varieties over \({\mathbb{Z}}_ p\)- extensions of number fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a curve of genus 2 defined over a number field. For points \(P\) on \(C\), let \(\overline P\) denote its conjugate under the hyperelliptic involution. It is already well known that the set of rational points on \(C\) that are integral with respect to \(P\) and \(\overline P\) can be effectively bounded. This paper shows that effective bounds exist as functions of the height of \(P\); in addition it shows that if the height of \(P\) is sufficiently large (greater than a certain effectively computable constant), then \(C\) has no \(K\)-rational points which are integral with respect to both \(P\) and \(\overline P\). (Of course by Mordell's conjecture the heights of such \(P\) are bounded, but not effectively.)
The proof proceeds by relating these integral points to points on the Jacobian \(J\) of \(C\) that are integral with respect to \([2]_* \Theta\), where \(\Theta\) is the theta divisor (which in this case is just the image of \(C\) in \(J)\) and [2] denotes multiplication by 2 under the group law on \(J\). Then it is shown that the set of points on \(J\) that are integral with respect to \([2]_* \Theta\) can be effectively bounded; this is the first example of effectively bounding the heights of integral points on a possibly simple abelian surface.
In addition, the paper discusses the issue of bounding points on \(C\) that are integral with respect to just one point \(P\). Such points are related to \(\Theta\)-integral points on \(J\). Some conjectural approaches to bounding the heights to the latter are discussed. hyperelliptic curves; integral points; curve of genus 2; rational points; effective bounds; Jacobian; theta divisor Grant, D.: Integer points on curves of genus two and their Jacobians. Trans. amer. Math. soc. 344, No. 1, 79-100 (1994) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Special algebraic curves and curves of low genus, Abelian varieties of dimension \(> 1\) Integer points on curves of genus 2 and their Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper completes the classification of algebras of finite type over a field which are Euclidean domains, begun by Armitage, Samuel, Leitzel, Madan, Queen, Lenstra, and the author. The main point proved here, left open by the previous work, is that Euclidean algebras of finite type over infinite ground fields are coordinate rings of affine open subschemes of the projective line \({\mathbb{P}}^ 1\). - A classification of Euclidean rings of S-integers in algebraic number fields was given, conditionally upon the generalised Riemann hypothesis, by Cooke, Weinberger, and Lenstra.
Several consequences of the classification are given; in particular, a question of Samuel is answered affirmatively: namely, that a principal ideal domain, which is an algebra of finite type over a field, is Euclidean if and only if the (so-called) minimal algorithm does not terminate at the second stage. - The proof of the main result uses Diophantine geometry. More precisely, it is based on Siegel's finiteness theorem for integral points on curves over number fields and an analogue proved in this paper for curves over fields of positive characteristic. Euclidean algorithm; generalised Jacobian varieties; algebras of finite type over a field; Euclidean domains; Diophantine geometry; integral points on curves Brown, M.L., Euclidean rings of affine curves, Math. Z., 208, 3, 467-488, (1991) Commutative Artinian rings and modules, finite-dimensional algebras, Euclidean rings and generalizations, Rational points Euclidean rings of affine curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theory of \(\mathbb Q\)-Cartier divisors on the space of \(n\)-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of \(\mathbb Q\)-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in \(\mathbb P^r\) is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree \(d\) plane curves is explicitly evaluated. enumerative geometry; Cartier divisors; \(n\)-pointed, genus 0, stable maps; intersection products of \(\mathbb{Q}\)-divisors; characteristic numbers of rational curves; 1-cuspidal rational locus R. Pandharipande, Intersections of \(\({ Q}\)\)-divisors on Kontsevich's moduli space \(\({\overline{M}}_{0, n}({ P}^{r}, d)\)\) and enumerative geometry. Trans. Am. Math. Soc. 351(4), 1481-1505 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Birational geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves, Picard groups Intersections of \(\mathbb{Q}\)-divisors on Kontsevich's moduli space \(\overline M_{0,n}(\mathbb P^r,d)\) and enumerative geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0653.00006.]
Let N be an ample invertible sheaf on a complex abelian variety X, let P be a Poincaré sheaf on \(X\times \hat X\) and let \(V\{N\}=\pi_{2*}(\pi_ 1^*N\otimes P)\) where \(\pi_ i\) are the projections from \(X\times \hat X\) to X and \(\hat X\) respectively. It is explained how to construct a canonical Hermitian metric on \(V\{N\}\). This metric provides canonical Hermitian metrics on the sheaves \(W_{ng}:=p_{2*}L_{ng}\) where \(L_{ng}\) is an invertible sheaf on \(C\times J\) (C a curve of genus g and J its Jacobian) which is a universal family of invertible sheaves of degree \(ng\) \((n\geq 2)\). Picard bundles; sheaf on a complex abelian variety; Hermitian metric Analytic theory of abelian varieties; abelian integrals and differentials, Picard groups, Divisors, linear systems, invertible sheaves Some metrics on Picard bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(p\neq 3\) be an odd prime and \(k\) be a field containing a primitive \(p\)th root of unity. We construct cubic polynomials \(f(t)\), \(g(t)\), \(h(t)\) with coefficients in \(K\), a certain transcendental extension field of \(k\), such that all of the ranks of Jacobian variety of the curves \(f(t)y^p=f(x)\), \(g(t)y^p=f(x)\), \(h(t)y^p=f(x)\), \(g(t)h(t)y^p=f(x)\) over \(K(t)\) are positive. superelliptic curve; Jacobian variety; twist theory Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Plane and space curves A certain quadruple family of superelliptic curves associated with \((p,p)\)-extension | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 generic initial ideal (gin) of a curve \(C\) is obtained from the associated homogeneous ideal \(I_C\) in a ``general'' choice of coordinates, using monomial reduction in the reverse lexicographic term ordering. gin\((I_C)\) preserves several informations on the geometry of the curve so it proves useful for studying combinatorically the properties of \(C\).
An important tool in the theory of projective curves is the connections between \(C\) and its general hyperplane sections \(Z\). For space curves, \(Z\) is a set of points in \(\mathbb P^2\) and gin\((I_Z)\) allows to reconstruct the ``character'' of \(Z\), arising from the degrees of a free resolution of the ideal; this is a fundamental invariant for points in the plane, whose extension to the case of curves in \(\mathbb P^3\) is straightforward only in the arithmetically Cohen-Macaulay case. For general curves, the author proposes here a definition of the character which generalizes the definition given for points via the generic initial ideals. In fact, the character of a space curve is defined from the monomial ideal obtained by \(I_C\) by taking first the non saturated section \(I=I_C\) mod(\(x_4\)) and then looking at the ``partial projections'' \(I_k= I:x_3^k\).
The author proves that the character of a space curve, as in the case of general points in the plane, is connected. This allows to extend easily some results on the classification of curves whose general hyperplane sections are in special position. generic initial ideal; lexicographic term; character of a space curve; general hyperplane sections Cook, M.: The connectedness of space curves invariants. Compositio Math. 111, 221--244 (1998) Plane and space curves, Complete intersections The connectedness of space curve invariants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It was conjectured by M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. This conjecture is open even for Gorenstein local rings corresponding to numerical semigroups, namely to monomial curves in affine 4-space. In the paper under review, the authors give a method to construct new large families of monomial curves in affine 4-space with corresponding Gorenstein local rings in embedding dimension 4 supporting Rossi's conjecture. By starting with any monomial curve in affine 2-space, this constructive method provides monomial curve families in affine 4-space having non-Cohen-Macaulay tangent cones with non-decreasing Hilbert functions. monomial curve; symmetric numerical semigroup; gluing; Gorenstein; Hilbert function of a local ring; Rossi's conjecture Arslan, F.; Sipahi, N.; Şahin, N., Monomial curve families supporting Rossi's conjecture, J. symbolic comput., 55, 10-18, (2013) Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Semigroups, Symbolic computation and algebraic computation Monomial curve families supporting Rossi's conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The interrelation between two formulas: a formula for the Todd class of a toric variety and a formula for the number of lattice points inside a lattice polyhedron, discovered by \textit{V. I. Danilov} [Russ. Math. Surv. 33, No. 2, 97--154 (1978); translation from Usp. Math. Nauk 33, No. 2(200), 85--134 (1978; Zbl 0425.14013)] and \textit{A. G. Khovanskii} [Funct. Anal. Appl. 11, 289--296 (1978); translation from Funkts. Anal. Prilozh. 11, No. 4, 56--64 (1977; Zbl 0445.14019)] is a source of many results both in algebraic geometry and combinatorics.
In the paper under review the author proves a formula of Danilov type and, moreover, makes it more explicit using the residue theorem of P. Baum and R. Bott [\textit{M. F. Atiyah} and \textit{R. Bott}, Topology 23, 1--28 (1984; Zbl 0521.58025)]. He also shows the relation of these formulas to a classical theorem on the area of a convex plane polygon due to Pick. Todd class of a toric variety; number of lattice points; lattice polyhedron R. Morelli, Pick's theorem and the Todd class of a toric variety, Adv. Math. 100(2), 183--231 (1993). MR 1234309 (94j:14048). Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Group actions on varieties or schemes (quotients) Pick's theorem and the Todd class of a toric 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 Let \(X\) be a smooth complex projective variety of dimension \(n\). \textit{S. L. Kleiman} [Ann. Math. 84, 293--344 (1966; Zbl 0146.17001)] showed that the dual of the cone of numerical classes of ample divisors is the closure of the cone of numerical classes of effective curves. On the other hand, Boucksom, Demailly, Paun, and Peternell proved that the dual of the cone of numerical classes of effective divisors on \(X\) is the closure of the cone of numerical classes of movable curves.
In this paper, the author considers the case where the dual of the cone of divisors on a complete \(\mathbb Q\)-factorial toric variety whose stable base loci have dimension less than \(k\), and obtains the following result which is an affirmative answer of a question asked by Debarre and Lazarsfeld in the toric case: The dual of the cone of divisors on a complete \(\mathbb Q\)-factorial toric variety \(X\) whose stable base loci have dimension less than \(k\) is generated by curves on small modifications of \(X\) that move in families sweeping out the birational transforms of \(k\)-dimensional subvarieties of \(X\). Stable base loci; toric variety; cone of divisors Payne S.: Stable base loci, movable curves, and small modifications, for toric varieties. Math. Z. 253(2), 421--431 (2006) Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies Stable base loci, movable curves, and small modifications, for toric 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 an integral curve over \(\mathbb C\) which is rational and has only nodes as singularities. The author studies the moduli space of stable sheaves \(E\) of rank 2 on \(C\) with \(\chi(E)=1\). The main result of this article says that the topological Euler number of this moduli space is equal to the number of nodes of \(C\). The main tool is a stratification of this moduli space by the type of the sheaf at the singular points. A local computation shows that each node contributes to the topological Euler number through one stratum which has Euler number one. All the other strata have vanishing Euler number. This is shown with the aid of fixed point free actions of cyclic subgroups of the Jacobian of \(C\) on these strata. stable sheaf; rank 2 vector bundles; integral curve; number of nodes; stratification of moduli space; actions of cyclic subgroups of Jacobian; Euler number; torsion free sheaf; rational curve Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Euler number of the moduli space of sheaves on a rational nodal curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper presents a rather detailed investigation of the topology of the Milnor fibre \(F\) and the monodromy for a plane curve singularity.
Starting from the map-germ \(f:(\mathbb{C}^ 2,0)\to(\mathbb{C},0)\) take a local embedded resolution \(\pi:X\to Y\) of \(f^{-1}0\), with exceptional fibre \(E_ +=\bigcup E_ i\). An \(e\)-fold covering of \((\mathbb{C},0)\) gives a semistable reduction, and \(X'\) is the normalisation of \(X\times_ \mathbb{C}\mathbb{C}'\); the exceptional fibre here is \(D_ +=\bigcup D_ i\). A model for \(X'\) can be constructed by plumbing neighbourhoods of the \(D_ i\) along intersections \(B_{ij}\). There is a map \(F\to D_ +\) which is a homeomorphism outside the \(D_ i\cap D_ j\); this induces a decomposition \(F=\bigcup F_ i\). Then \(H_ 1F\) is filtered by \(M_{- 3}=0\), \(M_{-2}=\) image of \(\oplus H_ 1(F_ i\cap F_ j)\), \(M_{- 1}=\) image of \(\oplus H_ 1F_ i\), \(M_ 0=H_ 1F\).
There is a corresponding dual filtration of cohomology; a major result of the paper is that when tensored with \(\mathbb{Q}\) this coincides with Steenbrink's mixed Hodge structure [\textit{J. H. M. Steenbrink} in Real and complex Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 525- 563 (1977; Zbl 0373.14007)]. However these filtrations are defined over \(\mathbb{Z}\); the graded group is torsion free and is described explicitly by generators and relations. After a full development of this theory, it is illustrated with explicit calculations on certain examples, including one where the graded modules are not isomorphic but become so after tensoring with \(\mathbb{Q}\). topology of the Milnor fibre; monodromy for a plane curve singularity; filtration of cohomology; mixed Hodge structure Du Bois, Ph.; Michel, F.: Filtration par le poids et monodromie entière. Bull. soc. Math. France 120, 129-167 (1992) Singularities of curves, local rings, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of differentiable mappings in differential topology, Germs of analytic sets, local parametrization, Milnor fibration; relations with knot theory, Mixed Hodge theory of singular varieties (complex-analytic aspects) Weight filtration and integral monodromy | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with the deformation theory of isolated hypersurface singularities of dimension three with a small resolution, i.e. a resolution of singularities whose exceptional set has dimension 1. If X denotes the germ of such a singularity and \(\tilde X\) a small resolution of it, there is an induced morphism of deformation functors Def \(\tilde X\to Def X\). It is shown that this morphism is unramified and its image is studied. Next the global problem is considered. Let X denote either a generalized Fano variety or a (singular) threefold with trivial dualizing sheaf. Precise results are obtained in either case when all singularities are ordinary double points. As a corollary of the result on Fano threefolds, the maximum number of nodes on a quartic threefold in \({\mathbb{P}}^ 4\) is shown to be 45. The results on varieties with trivial dualizing sheaf are used to construct examples of non-Kähler threefolds with trivial canonical bundle. Finally, there are some applications to the behavior of rational curves of low degree on quintic threefolds. deformation theory of isolated hypersurface singularities; small resolution; generalized Fano variety; threefold with trivial dualizing sheaf; maximum number of nodes; rational curves of low degree on quintic threefolds Losev, A., Nekrasov, N., Shatashvili, S.: \textit{Testing Seiberg-Witten solution}. In: Strings, branes and dualities (Cargèse, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. \textbf{520}. Dordrecht: Kluwer Acad. Publ., 1999, pp. 359-372 Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Deformations of singularities, Formal methods and deformations in algebraic geometry, Special surfaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities of surfaces or higher-dimensional varieties Simultaneous resolution of threefold double 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 [For the entire collection see Zbl 0588.00017.]
Quoting from the abstract in the paper: ''A formalism due to Huber is considered with a view to introducing a notion of relative homotopy group to a subcategory of locally ringed \(T_ 0\) spaces. For X an elliptic curve over \({\mathbb{C}}\) the formalism leads to a trivial group, but with a suitably altered homotopy relation it leads to the group (End(X,\b{O}),\(+)\) of endomorphisms of X keeping \b{O}\(\in X\) fixed. Viewed as partial submonoids of End(X,\b{O}) the fundamental groups of affine models of X are calculated in several cases. Some other calculations of homotopy groups in the context of locally ringed spaces are described.''
For developing a homotopy theory internal to algebraic varieties this work showed the importance of finding up to homotopy a monoid with zero in algebraic varieties to play the role of the unit interval. A monoid in algebraic varieties with zero is rational and this would restrict the interest of an associated homotopy theory. relative homotopy group; locally ringed \(T_ 0\) spaces; elliptic curve; fundamental groups of affine models; homotopy theory internal to algebraic varieties; monoid in algebraic varieties with zero Homotopy groups of special types, Homotopy theory and fundamental groups in algebraic geometry, Elliptic curves, Formal methods and deformations in algebraic geometry Fundamental groups of elliptic curves internal to locally ringed spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this ``Docteur en troisième cycle'' thesis the author extends results of \textit{C. Andradas} [Commun. Algebra 13, 1151-1169 (1985; Zbl 0563.12021) and ``Specialization chains of a real valuation rings'' (Preprint, Madrid 1983)], and of \textit{L. Bröcker} and \textit{H.-W. Schülting} [``Valuations of function fields from the geometrical point of view'' (Preprint, 1984); see also J. Reine Angew. Math. 365, 12-32 (1986; Zbl 0577.12021)] dealing with valuations of function fields over real closed fields.
A prime cone P of a commutative ring A is a subset P of A, additively and multiplicative closed, containing all \(a^ 2, a\) in A, with -1\(\not\in P\) and with the property that xy\(\in P\) implies ``x\(\in P''\) or ``-y\(\in P''\). The support of P, supp(P), is the prime ideal \(P\cap -P\) of A. After a very readable summary of other work in this area, the following main theorem is proved:
Let W be an irreducible affine variety over a real closed field. Denote the coordinate ring of W by A, the function field of W by K, and the dimension of W by n. Let \(Q_{m-1}\subset...\subset Q_ 1\subset Q_ 0\) denote a chain of prime ideals of A with \(\dim (Q_ i)=d_ i\), such that there is a chain of prime cones in A, \(P_{m-1}\subset...\subset P_ 1\subset P_ 0\) with \(\sup p(P_ i)=Q_ i\). Denote further by \(f_ 1,...,f_ t\) elements of A and by \((s_ 0,...,s_{m-1})\) and \((r_ 0,...,r_{m-1})\) two sequences of integers such that, for \(0\leq i\leq (m-1): \) \(P_{m-1}\supset (\sum K^ 2[f_ 1,...,f_ t] \cap A)\), \(0\leq s_ 0<...s_{m-1}\leq n\), \(0<r_{m-1}<...<r_ 0\), and \(s_ i+r_ i\leq n\); \(r_ i\geq m_ i\), \(d_ i\leq s_ i\). Then there exists an order B of K in which the \(f_ i\) are positive, and a chain \(\{V_ i\}\), \(0\leq i\leq m-1\), of valuation rings of K with formally real residue class fields, all compatible with B and containing A such that: \(\dim (V_ i)=s_ i\), \(rank(V_ i)=m-i\), \(rational\quad rank(V_ i)=r_ i,\) the centre of \(V_ i\) in A is \(Q_ i\). valuation rings of function fields; coordinate ring of affine; variety over a real closed field; prime cone Real algebraic and real-analytic geometry, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Valuations and their generalizations for commutative rings Constructions de places réelles dans géométrie semialgébrique. (Constructions of real places in semialgebraic geometry). (Thèse) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 article is to sharpen a theorem by Kempf about deformation of the symmetric product \(C^{(r)}\) of a non-hyperelliptic curve \(C\). The author proves the following:
Theorem: Let \(C\) be a smooth curve of genus \(g\geq 2\), \(r\geq 2\) an integer. Then the natural map \(\text{Def}(C)\to\text{Def}(C^{(r)})\) is an isomorphism if and only if \(g\geq 3\).
To prove this result we have to compare the deformation of the product \(C^r\) with the deformation of the symmetric product \(C^{(r)}\), i.e. the deformation of a smooth variety with the deformation of its quotient by a finite group: the symmetric group \(S_r\).
Lemma: If the genus of \(C\) is at least 2, the natural map \((\text{Def}(C))^r\to\text{Def}(C^r)\) is an isomorphism, it induces an isomorphism \(\text{Def}(C)\to\text{Def}(C^r)^{S_r}\). The functors \(\text{Def}(C^r)\) and \(\text{Def}(C^r)^{S_r}\) are unobstructed. -- By Künneth and by vanishing of \(H^0(C,\theta_C)\), we get \(H^1(C^r,\theta_{C^r})\simeq\bigoplus H^1(C,\theta_C )\). Hence the natural map \((\text{Def}(C))^r\to\text{Def}(C^r)\) is an isomorphism on tangent spaces and, as \((\text{Def}(C))^r\) is unobstructed, it is an isomorphism.
The tangent space of \(\text{Def}(C^r)^{S_r}\) is the \(S_r\)- invariant subspace of \(\bigoplus H^1(C,\theta_C)\), hence the map \(\text{Def}(C)\to\text{Def}(C^r)^{S_r}\) is an isomorphism on tangent spaces and the unobstructness of \(\text{Def}(C)\) implies that it is an isomorphism. Then the theorem is equivalent to: \(\text{Def}(C^r)^{S_r}\to\text{Def}(C^{(r)})\) is an isomorphism. As \(\text{Def}(C^r)^{S_r}\) is unobstructed, it is enough to have an isomorphism on the tangent spaces.
Let \(B\) be the branch locus of the quotient map \(\pi:C^r\to C^{(r)}\), we define the subsheaf \(\theta_{C^{(r)}}(-\log B)\) of derivations of \(\theta_{C^{(r)}}\) respecting the ideal of \(B\) in \(C^{(r)}\) and the equisingular normal sheaf \({\mathcal N}_{B/C^{(r)}}'\) as the quotient \({\mathcal N}_{B / C^{(r)}}' = \theta_{C^{(r)}} / (\theta_{C^{(r)}} (- \log B))\). We have an isomorphism \(\pi_*^{S_r } (\theta_{C^r}) \simeq \theta_{C^{(r)}} (- \log B)\) and the map on tangent spaces of \(\text{Def} (C^r)^{S_r} \to \text{Def} (C^{(r)})\) fits into an exact sequence:
\[
H^0 \bigl( B, {\mathcal N}_{B/C^{(r)} }' \bigr) \to H^1 \bigl( C^r, \theta_{C^r} \bigr)^{S_r} \to H^1 \bigl( C^{(r)}, \theta_{C^{(r)}} \bigr) \to H^1 \bigl( B, {\mathcal N}_{B/C^{(r)}}' \bigr) \to H^2 \bigl( C^r, \theta_{C^r} \bigr)^{S_r}.
\]
For \(i = 0, 1\), \(H^i (B, {\mathcal N}_{B/C^{(r)}}')\) is canonically isomorphic to \(H^i (C,2 \theta_C)\), and by Künneth decomposition we have \(H^2 (C^r, \theta_{C^r}) \simeq \bigoplus_{i \neq j} H^1 (C_i, \theta_{C_i})\otimes H^1(C_j,{\mathcal O}_{C_j})\). Then \(H^1(C^r,\theta_{C^r})^{S_r}\to H^1(C^{(r)},\theta_{C^{(r)}})\) is an isomorphism and we get the theorem. deformation of the symmetric product of a non-hyperelliptic curve; genus B. Fantechi, Déformations of symmetric products of curves.Contemporary Math. 162 (1994), 135--141. Curves in algebraic geometry, Formal methods and deformations in algebraic geometry Deformations of symmetric products of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a beautiful report on a beautiful subject. Sklyanin algebras are graded noncommutative algebras having strong regularity properties [\textit{M. Artin, J. Tate} and \textit{M. van den Bergh}, Invent. Math. 106, 335-388 (1991; Zbl 0763.14001); \textit{M. Artin} and \textit{W. Schelter}, Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)]. Initially, they arose from Baxter's solution of the Yang-Baxter equation [\textit{E. K. Sklyanin}, Funkts. Anal. Prilozh. 16, 27-34 (1982; Zbl 0513.58028)]. A geometrical construction and generalizations were provided by \textit{A. Odeskij} and \textit{B. Feigin} [Funkts. Anal. Prilozh. 23, 45-54 (1989; Zbl 0687.17001)] and more recently by \textit{J. Tate} and \textit{M. van den Bergh}, Homological properties of Sklyanin algebras (preprint 1993)]. Let us fix a Sklyanin algebra \(A = A(E,\tau)\) of dimension 4; it is constructed from an elliptic curve \(E\), a line bundle of degree 4 over \(E\) and a point \(\tau \in E\) not of order 4. (Sklyanin algebras of dimension 3 are defined in a similar way, taking instead a line bundle of degree 3.) The main part of the survey under review deals with the classification of all the irreducible finite dimensional \(A\)-modules. Work in this direction was done, besides the already mentioned authors, by Levasseur, Smith, Stafford, Staniszkis. The main steps are: (1) The problem is reduced to the classification of all the irreducible objects in the category \(\text{Proj }A\) (this is the category of finitely generated graded modules localized at the subcategory of finite dimensional ones). Indeed, a simple finite dimensional \(A\)-module is always the quotient of an irreducible from \(\text{Proj }A\). One should also decide which of those irreducibles from \(\text{Proj }A\) have a finite dimensional quotient. (2) The first approximation to the irreducibles of \(\text{Proj }A\) is to classify the point modules. These are cyclic modules having the same Hilbert series as a polynomial ring in one variable, and for a commutative graded algebra, these are in bijective correspondence with the points of the associated projective variety. In the present case, the point modules are in bijective correspondence with the points of \(E\) plus 4 more points. (3) The irreducible modules which are not point ones are called fat. There is also the notion of line modules: cyclic modules having the same Hilbert series as a polynomial ring in two variables. In this case, they are in correspondence with the lines secant to \(E\), and any fat module is a quotient of a line module. If \(\tau\) is not of finite order, the classification can now be worked out (see the following review Zbl 0809.16052). If \(\tau\) is of finite order, the classification requires more work and was done by the author [The four dimensional Sklyanin algebra at points of finite order (Preprint 1992)]. The difference between the two cases has its roots at the center of the Sklyanin algebra. Indeed, in the infinite case, the center is generated by 2 elements first found by Sklyanin, whereas in the second case \(A\) is a finite module over the center. Sklyanin algebras; graded noncommutative algebras; regularity; Yang- Baxter equation; elliptic curve; line bundle; survey; irreducible finite dimensional \(A\)-modules; category of finitely generated graded modules; point modules; cyclic modules; Hilbert series; projective variety; irreducible modules Smith, S. P., The four-dimensional Sklyanin algebras, \(K\)-Theory. Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992), 8, 1, 65-80, (1994) Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Homological dimension in associative algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Noetherian rings and modules (associative rings and algebras), Finite rings and finite-dimensional associative algebras The four-dimensional Sklyanin algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abelian varieties. In addition, it briefly describes recent important developments, most significant of which is the proof of the Strange Duality conjecture due to Belkale and Marian-Oprea. generalized theta divisors; moduli spaces of vector bundles on curves Mihnea Popa, Generalized theta linear series on moduli spaces of vector bundles on curves, Handbook of moduli. Vol. III, Adv. Lect. Math. (ALM), vol. 26, Int. Press, Somerville, MA, 2013, pp. 219 -- 255. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Generalized theta linear series on moduli spaces of vector bundles on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{M. Baker} and \textit{S. Norine} [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)] proved a Riemann-Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of
\textit{A. Björner} et al. [Eur. J. Comb. 12, No. 4, 283--291 (1991; Zbl 0729.05048)
]. We use this connection to prove Riemann-Roch-type results on directed graphs. We give a simple proof for a Riemann-Roch inequality on Eulerian directed graphs, improving a result of \textit{O. Amini} and \textit{M. Manjunath} [Electron. J. Comb. 17, No. 1, Research Paper R124, 50 p. (2010; Zbl 1277.05105)]. We also study possibilities and impossibilities of Riemann-Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework. Riemann surface; Riemann-Roch theorem; Jacobian of a finite graph; chip-firing games Directed graphs (digraphs), tournaments, Paths and cycles, Riemann surfaces; Weierstrass points; gap sequences Chip-firing based methods in the Riemann-Roch theory of directed graphs | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Der Index eines Punktes in Bezug auf einen Kegelschnitt ist die Zahl \(\frac{ ma \cdot mb}{d^2}\), wenn \(a\) und \(b\) die Schnittpunkte des Kegelschnitts mit einer durch \(m\) gehenden Transversalen, und \(d\) die Länge des dieser Transversalen parallelen Halbmessers ist. Für den Index einer Geraden in Bezug auf einen Kegelschnitt, eines Punktes, einer Geraden, und einer Ebene in Bezug auf eine Fläche zweiter Ordnung werden ähnliche metrische Definitionen aufgestellt, welche es dem Verfasser ermöglichen, eine grosse Anzahl metrischer Relationen aufzustellen. Index of a point; index of a curve Questions of classical algebraic geometry, Curves in algebraic geometry, Polynomials and matrices Index theory for a curve and a second order surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be an algebraically closed field of characteristic zero. The paper presents the main ideas of the recent solution of the famous problem on the existence of a decomposition of a birational mapping between algebraic, complete smooth varieties over \(K\) into a sequence of blowing ups and downs at smooth centers [given by \textit{D. Abramovich, K. Karu, K. Matsuki} and \textit{J. Włodarczyk}, J. Am. Math. Soc. 15, 531--572 (2002; Zbl 1032.14003) and by \textit{J. Włodarczyk}, ``Toroidal varieties and the weak factorization theorem'' (http://arXiv.org/abs/math.AB/9904076)]. blowing up; toric variety; birational cobordism; decomposition of a birational mapping Bonavero, L.: Factorisation faible des applications birationnelles. Astérisque 282 (2002) Rational and birational maps Weak factorization of birational maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J_ 0(N)/ \mathbb{Q}\) denote the Jacobian of the modular curve \(X_ 0(N)/ \mathbb{Q}\). Fix a prime \(p \geq 5\), and let \(K\) denote the maximal unramified extension of \(\mathbb{Q}_ p\). Given an abelian variety \(A/K\), the connected component of zero of the special fiber of its Néron model is an extension of an abelian variety of dimension \(a_ k (A)\) by the product of a torus of dimension \(t_ k (A)\) and a unipotent group of dimension \(u_ k (A)\). Using a description of a regular model for \(X_ 0 (N)/K\), with \(\text{gcd} (N,\sigma) = 1\), due to \textit{Edixhoven}, the author is able to compute the numbers \(t_{K_ d} (J_ 0(p^ 2))\) and \(a_{K_ d} (J_ 0(p^ 2))\), where \(K_ d\) is a certain extension of \(K\) of degree \(d \in J: = \{2,3,4,6\}\). Using this he shows that the number of isogeny classes of elliptic curves \(E/\mathbb{Q}_ p\), having potentially good reduction at \(p\), and which are quotients of the Jacobian \(J_ 0 (p^ 2)\), is bounded by \((p/4+4)\). Jacobian of modular curve; number of isogeny classes of elliptic curves; quotients of the Jacobian Jacobians, Prym varieties, Isogeny, Elliptic curves, Elliptic curves over local fields, Local ground fields in algebraic geometry On modular elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians (Siehe auch JFM 10.0111.01, JFM 10.0111.02, JFM 10.0111.03) Die dritte Notiz der erst genannten Broschüre: Généralisation d'un théorème de M. Smith ist ein Auszug aus den Ann. Soc. Sc. Brux. 2, 211--224 und ist die Abhandlung des englischen Gelehrten im Messenger (2) 7, 81--82 (s. JFM 08.0074.03) entstanden. Der Hauptsatz heisst: Eine Determinante
\[
\sum \pm a_{11}a_{22}\dots a_{nn},
\]
wo
\[
a_{ik}= a_{ki}=a_{i-k, k} \quad \text{oder} \quad a_{i,k-i}
\]
ist eine einfache Function ihrer Diagonalen. Corollar 1). Jedes Product \(x_1 x_2\dots x_n\) kann in die Form einer Determinante dieser Art gebracht werden, wo
\[
a_{11}=x_1 +x_d+ \cdots +x_{d'} +x_i
\]
und \(1, d,\dots d',i\) alle Theiler von \(i\) sind. 2) Jede Determinante dieser Form wird mit \(+1, -1\) oder 0 multiplicirt, wenn man die Elemente einer Linie durch die entsprechenden Elemente der zweiten Diagonale der Determinante ersetzt. Die Arbeit von Catalan enthält einen etwas veränderten Beweis des Hauptsatzes und Corollars 1); die von Le Paige einen directen Beweis des Corollars 1), nur mit Hülfe des Princips der Addition der Reihen. Die zweite Notiz von Mansion enthält den Beweis desselben Corollars durch Multiplication zweier Determinanten, die nach dem oben bezeichnten Princip gebildet sind, so dass alle Elemente oberhalb der Hauptdiagonale Null sind. Theorems of Mansion and Smith; determinant; function of the diagonals; product; divisors; line; proof of a corollary; principle of addition of rows; elements above the principal diagonal Research exposition (monographs, survey articles) pertaining to linear algebra, Determinants, permanents, traces, other special matrix functions, Products, amalgamated products, and other kinds of limits and colimits, Divisors, linear systems, invertible sheaves, Proof theory in general (including proof-theoretic semantics) On a theorem of M. Mansion. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a finite field of characteristic \(p>0\) and \(K\) be an algebraic function field of one variable over \(k\). Consider an elliptic curve \(E\) over \(K\) with \(j\)-invariant transcendental over \(\mathbb{F}_p\). In this paper the author proves that there exists a finite set of primes \(S\), with \(p\in S\), such that for every positive integer \(n\) not divisible by any of the primes of \(S\), the pure \(n\)-torsion points of the fibers of the Néron model of \(E\) describe after completion a smooth irreducible projective curve \(C_n\). The Selmer group \(S_n(E/K)\) embeds canonically into \(_n\text{Pic}^0 (C_n)\), and a necessary and sufficient condition is given for an element of \(_n\text{Pic}^0 (C_n)\) to belong to \(S_n(E/K)\). Moreover, the induced embedding \(E(K)/nE(K) \hookrightarrow {_n\text{Pic}^0} (C_n)\) is described explicitly. Shafarevich-Tate group; Picard groups; transcendental \(j\)-invariant; finite field; algebraic function field; elliptic curve; fibers of the Néron model; irreducible projective curve; Selmer group; embedding Elliptic curves over global fields, Global ground fields in algebraic geometry, Elliptic curves, Arithmetic ground fields for curves Selmer groups and Picard 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 If X is a 0-dimensional subscheme of a smooth quadric \(Q\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) we investigate the behaviour of X with respect to the linear systems of divisors of any degree (a,b). This leads to the construction of a matrix of integers which plays the role of a Hilbert function of X; we study numerical properties of this matrix and their connection with the geometry of X. Further we put into relation the graded Betti numbers of a minimal free resolution of X on Q with that matrix, and give a complete description of the arithmetically Cohen- Macaulay 0-dimensional subschemes of Q. postulation; arithmetically Cohen-Macaulay 0-dimensional subschemes of a smooth quadric; linear systems of divisors; Hilbert function Giuffrida, S; Maggioni, R; Ragusa, A, On the postulation of \(0\)-dimensional subschemes on a smooth quadric, Pac. J. Math., 155, 251-282, (1992) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the postulation of 0-dimensional subschemes on a smooth quadric | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an affinoid algebra A over a non-archimedean valued complete and stable field k, this paper gives a method for the calculation of the Picard group Pic(A) of A. The main result is: ''If the reduction \(\bar A\) of A has no zero-divisors then the canonical map \(\beta:Pic(\bar A)\to Pic(A)\) is injective. If moreover \(\bar A\) is non-singular then \(\beta\) is an isomorphism.'' This result was already known for fields with a discrete valuation. There is a list of examples, with \(\bar A\) singular, for which the cokernel of \(\beta\) is calculated. In one example one uses the structure of the formal group of an elliptic curve over a valuation ring. affinoid space; rigid analytic geometry; non-archimedean valued ground field; affinoid algebra; Picard group; formal group of an elliptic curve E. Heinrich , M. van der PUT . '' Uber die Picardgruppen affinoider Algebren ''. Math Z. 186 , 9 - 28 ( 1984 ). Article | MR 735047 | Zbl 0543.14011 Local ground fields in algebraic geometry, Picard groups, Non-Archimedean valued fields, Formal groups, \(p\)-divisible groups, Special algebraic curves and curves of low genus Über die Picardgruppen affinoider Algebren | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 The author shows that cotangent bundles of moduli spaces of vector bundles over a Riemann surface are algebraically completely integrable Hamiltonian systems. More precisely, let G be a complex semisimple Lie group, let N be the moduli space of stable G-bundles with prescribed topological invariants on a compact Riemann surface and let n be the dimension of N. The cotangent space to N at the point represented by a G- bundle P is \(H^ 0(M;ad(P\otimes K))\) where ad(P) is the bundle associated to P via the adjoint representation of G on its Lie algebra g. Thus a choice of basis \(p_ 1,...,p_ k\) for the ring of invariant polynomials on g induces a holomorphic map \(\phi: T*N\to \oplus H^ 0(M;K^{d_ i})\) where \(d_ i\) is the degree of \(p_ i\). The components of \(\phi\) are n functionally independent Poisson-commuting functions on T*N, and when G is a classical group the generic fibre of \(\phi\) is an open set in an abelian variety on which the Hamiltonian vector fields defined by the components of \(\phi\) are linear. This is what it means to say that T*N is an algebraically completely integrable Hamiltonian system. The abelian varieties occurring are either Jacobian or Prym varieties of curves covering M. cotangent bundles of moduli spaces of vector bundles over a Riemann surface; completely integrable Hamiltonian systems; adjoint representation; Jacobian; Prym varieties N. Hitchin, \textit{Stable bundles and integrable systems}. Duke Math. J. 54 (1987), no. 1, 91--114. Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Stable bundles and integrable systems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a projective nonsingular variety over a real closed field \(R\) such that the set \(X(R)\) of \(R\)-rational points of \(X\) is nonempty. Let \(\text{Cl}_ R(X)=\text{Cl}(X)/\Gamma(X)\), where \(\text{Cl}(X)\) is the group of classes of linearly equivalent divisors on \(X\) and \(\Gamma(X)\) is the subgroup of \(\text{Cl}(X)\) consisting of the classes of divisors whose restriction to some neighborhood of \(X(R)\) in \(X\) is linearly equivalent to 0. It is proved that the group \(\text{Cl}_ R(X)\) is isomorphic to \((\mathbb{Z}/2)^ s\) for some nonnegative integer \(s\). Moreover, an upper bound on \(s\) is given in terms of the \(\mathbb{Z}/2\)- dimension of the group cohomology modules of \(\text{Gal}(C/R)\), where \(X=R(\sqrt{-1})\), with values in the Néron-Severi group and the Picard variety of \(X_ C=X\times_ RC\). variety over a real closed field; divisors Divisors, linear systems, invertible sheaves, Real algebraic sets, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Divisors on varieties over a real closed 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 Let \(X\) and \(Y\) be affine nonsingular real algebraic varieties. A general problem in real algebraic geometry is to try to decide when a continuous map \(f:X\to Y\) can be approximated by regular maps in the space of \(C^0\) mappings from \(X\) to \(Y\), equipped with the \(C^0\) topology. This note solves this problem when \(X\) is the connected component containing the origin of the real part of a complex abelian variety and \(Y\) is the standard 2-dimensional sphere. algebraic cycles; approximation by regular mappings; real part of a complex abelian variety Joglar-Prieto, N.; Kollár, J., Real abelian varieties with many line bundles, Bull. Lond. Math. Soc., 35, 79-84, (2003) Topology of real algebraic varieties, Approximation by other special function classes, Real algebraic sets Real Abelian varieties with many line bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0725.00006.]
A cubic graph is a graph whose vertices have valence three; if \(G\) is a cubic graph, one can define a unique stable curve \(X_ G\) from \(G\) as follows: \(X_ G\) is the union of \(\nu\) (= number of vertices) rational curves which meet in ordinary nodes according to the edges of \(G\). \(X_ G\) is a stable curve of genus \(g=\nu/2+1\). On a smooth curve \(X\), we have the Gaussian map \(\varphi\) from \(\bigwedge^ 2H^ 0(X,\omega_ X)\) to \(H^ 0(X,\Omega^ 1_ X\otimes\omega^ 2_ X)\), which can be defined also on stable curves, hence on curves such as \(X_ G\). The main result of the paper is to show that, when \(G\) is planar (i.e. embeddable in a 2- sphere) plus some extra hypotheses, the Gaussian map \(\varphi\) of \(X_ G\) has corank 1. --- The interest of such result lies in the fact that for the general curve of genus \(g\geq 12\) or \(g=10\), \(\varphi\) is surjective. On the other hand, this is not true for all curves, e.g. for the hyperelliptic ones having \(\text{coker} \varphi\) of dimension \(3g-2\). It would then be interesting to stratify \({\mathcal M}_ g\) into strata \({\mathcal M}_{g,k}\), where by \({\mathcal M}_{g,k}\) is the locus with \(\dim(\text{coker} \varphi)=k\).
Since curves like the \(X_ G\)'s appear as hyperplane sections of unions of planes, if one could show that such union can be smoothed to a \(K3\) surface \(S\), then the general curve on \(S\) would have a corank 1 Gaussian map. This would in turn imply that the generic element in \({\mathcal M}_{g,1}\) is a curve on a \(K3\) surface. moduli of curves; stable curve according to a cubic graph; Gaussian map; curve on a \(K3\) surface Rick Miranda, The Gaussian map for certain planar graph curves, Algebraic geometry: Sundance 1988, Contemp. Math., vol. 116, Amer. Math. Soc., Providence, RI, 1991, pp. 115 -- 124. Families, moduli of curves (algebraic), Graph theory, Special algebraic curves and curves of low genus, \(K3\) surfaces and Enriques surfaces The Gaussian map for certain planar graph curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a report on a discovery of a counterexample for Igusa's conjecture on a \(p\)-adic integral of a complex power function over the integral points. Such integrals are called generalized Igusa local zeta functions. The author of this paper proved that the generalized Igusa local zeta function associated to the representation \(\rho\) of \(\text{SL}_7\), where \(\rho\) is the Cartan product of the first, third and fifth fundamental representations of \(\text{SL}_7\), is explicitly computable and shown not to satisfy the functional equation expected by Igusa's conjecture. \(p\)-adic integral of a complex power function; Igusa's conjecture; generalized Igusa local zeta functions Zeta functions and \(L\)-functions, Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) A counterexample in the theory of local 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 The Krichever correspondence between algebraic curves with additional structures and certain commutative algebras of differential operators, on which the Mulase-Shiota proof of the Novikov conjecture and solution of the problem of Riemann-Schottky are based, is refined and generalized to form an equivalence between a fibered category over the category of algebraic curves and some category of commutative algebras of matrix pseudo-differential operators, with the aim of a characterization of special (e.g. d-gonal) curves by differential equations for their theta functions. A generalization and reinterpretation of the generalized Jacobian and the singularization procedure of Rosenlicht-Serre is used to show that a deformation of the curve data is reflected by the matrix KP hierarchy on the operator side. Krichever correspondence; problem of Riemann-Schottky; matrix pseudo- differential operators; theta functions; generalized Jacobian; matrix KP hierarchy Theta functions and curves; Schottky problem, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Jacobians, Prym varieties, Pseudodifferential operators, General theory of partial differential operators, Partial differential equations of mathematical physics and other areas of application, Theta functions and abelian varieties Über den Kričever-Funktor in der Theorie der algebraischen Kurven. (On the Krichever functor in the theory of algebraic curves) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\Sigma \) be the spinor tenfold of degree 12 in \({\mathbb P}^{15}\), i.e., the unique closed orbit of Spin(10) in its projectivized half-spinor representation. The so called fundamental form allows one to identify \(\Sigma \) to its dual variety \({\Sigma}^{\vee}\subset {\mathbb P}^{15\vee}\). \textit{S. Mukai} [in: Algebraic geometry and commutative algebra. In honor of Masayoshi Nagata, Vol.I, Kinokuniya, Tokyo, 357--377 (1988; Zbl 0701.14044)] showed that a nonsingular section of \(\Sigma \) by a linear subspace \({\mathbb P}^m\subset {\mathbb P}^{15}\), \(m=6,\;7\), resp., 8, is a canonical curve, a \(K3\) surface, resp., a prime Fano threefold and that, conversely, a generic canonical curve of genus 7, a generic \(K3\) surface of degree 12 and, resp., any nonsingular prime Fano threefold of degree 12 (i.e., \(\text{Pic}X={\mathbb Z}(-K_X)\) and \((-K_X)^3=12\)) can be obtained as such a linear section.
Now, let \(X={\mathbb P}^8\cap \Sigma \) be a prime Fano threefold of degree 12. The orthogonal linear section \(\Gamma := {\mathbb P}^{8\vee}\cap {\Sigma}^{\vee}\) is a canonical curve of genus 7 and the intermediate Jacobian \(\text{J}(X)\) is isomorphic to the Jacobian of \(\Gamma \). In their previous paper [Adv. Geom. 4, No. 3, 287--318 (2004; Zbl 1074.14039)], \textit{A. Iliev} and \textit{D. Markushevich} proved that, for a generic \(X\), the moduli space \({\text{M}}_X(2;1,5)\) of stable rank 2 vector bundles on \(X\) with \(c_1=-K_X\) and \(\text{deg}c_2=5\) is isomorphic to \(\Gamma \). \textit{A. Kuznetsov} [Math. Notes 78, No. 4, 537--550 (2005; Zbl 1111.14038)] remarked that this moduli space is fine and provided a natural universal bundle on it.
In the paper under review, the authors show that, for a generic \(X\), the moduli space \({\text{M}}_X(2;1,6)\) is birational to the Brill-Noether locus \({\text{W}}^1_6(\Gamma )\) which is nothing else but the singular locus of the theta divisor \(\Theta \subset {\text{Pic}}^6\Gamma \) and that \({\text{M}}_X(2;1,6)\) is irreducible.
In order to prove this result, they show that any \(E\in {\text{M}}_X(2;1,6)\) is generated by global sections at the generic point of \(X\) and that the zero locus of a non-zero global section of \(E\) lies in the closure of the family of elliptic sextics on \(X\). In order to handle the later family, they prove some auxiliary results about families of lower degree curves on \(X\). For example, they identify the curve of lines in \(X\) to \({\text{W}}^1_5(\Gamma )\) and show that the surface of conics in \(X\) is isomorphic to the symmetric power \({\Gamma}^{(2)}\). The proof of the irreducibility of the family of elliptic sextics in \(X\) uses a result of \textit{N. Perrin} [Courbes elliptiques sur la variété spinorielle de dimension 10, electronic preprint, \url{arXiv:math.AG/0409125}] on the irreducibility of the family of elliptic curves of fixed degree on \(\Sigma \).
A technical tool used extensively throughout the paper is the Iskovskikh-Prokhorov-Takeuchi birational transformation \(X\dashrightarrow Q_3\subset {\mathbb P}^4\) that can be obtained by a blow-up with center a conic \(q\), followed by a flop and a contraction of one divisor onto a curve \({\Gamma}_q\subset Q_3\) isomorphic to \(\Gamma \). Using this transformation, one can, for example, associate to any degree \(d\) curve \(C\) in \(X\) its Abel-Jacobi image \(\text{AJ}(C)\in {\text{Pic}}^{5d} \Gamma \). Fano threefold; moduli of vector bundles; spinor variety; intermediate Jacobian; theta divisor Iliev A., Markushevich D., atParametrization of sing {\(\Theta\)} for a Fano 3-fold of genus 7 by moduli of vector bundles, Asian J. Math., 2007, 11(3), 427--458 \(3\)-folds, Fano varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Parametrization of sing \(\Theta\) for a Fano 3-fold of genus 7 by moduli of vector | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated. finitely generated ring of global sections; complete toric variety; Cartier divisor Elizondo, E. J., The ring of global sections of multiples of a line bundle on a toric variety, Proc. Am. Math. Soc., 125, 9, 2527-2529, (1997) Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies The ring of global sections of multiples of a line bundle on a toric 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 [For the entire collection see Zbl 0642.00029.]
The author studies the algebraic geometry codes of V. D. Goppa that can be derived from the projective curve \(x^ r+y^ r+z^ r=0\) over the finite field \(GF(p^{2h})\), where h is an arbitrary positive integer, \(r=p^ h+1,\) and the prime p satisfies \(p=2\) or \(p\equiv 1\) mod 4. It is shown how to construct parity-check matrices for such codes. Furthermore, the subcodes over the prime field and the asymptotics for their code parameters are considered. A basic role in this work is played by a connection between the rational points of the curve and the prime divisors of degree 1 of the corresponding algebraic function field. algebraic geometry codes of V. D. Goppa; projective curve; finite field; parity-check matrices; rational points; prime divisors of degree 1; algebraic function field Linear codes (general theory), Arithmetic ground fields for curves Codes on Hermitian 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 It is known that a vector bundle \(E\) on a smooth projective curve \(Y\) defined over an algebraically closed field is semistable if and only if there is a vector bundle \(F\) on \(Y\) such that both \(H^{0}(X,E \otimes F\)) and \(H^{1}(X,E \otimes F\)) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let \(X\) be a geometrically irreducible smooth projective curve defined over a perfect field \(k\), and let \(E\) be a vector bundle on \(X\). We prove that \(E\) is semistable if and only if there is a vector bundle \(F\) on \(X\) such that \(H^i(X,E \otimes F)=0\) for all \(i\). We also give an explicit bound for the rank of \(F\). moduli space; vector bundles on a curve; generalized theta divisor Biswas, I.; Hein, G.: Generalization of a criterion for semistable vector bundles, (2008) Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Generalization of a criterion for semistable vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author characterizes the possible Hilbert functions of 0-dimensional subschemes of \(\mathbb{P}^3\) lying on an irreducible curve \(C\) which is contained in a smooth quadric surface \(Q\subseteq\mathbb{P}^3\) solely using the type \((a,b)\) of the curve. More precisely, he shows that in case \(1\leq a\leq b\), there is such a 0-dimensional scheme \(X\), if and only if its Hilbert function \(H_X\) satisfies
(1) \(\Delta H_X(i)=2i+1\) for \(i=0,\dots,b-1\),
(2) \(\Delta H_X(i)=a+b\) for \(i=b,\dots,s-1\) with some \(s\geq b\),
(3) \(\Delta H_X(s-1)>\Delta H_X(s)>\Delta H_X(s+1)+1>\cdots>\Delta H_X(s+e-1)+e-1>0\) for some \(0\leq e\leq a\), and
(4) \(\Delta H_X(i)=0\) for \(i\geq s+e\).
His proof uses an ``ad hoc'' construction and is partly based on prior work by \textit{G. Raciti} [cf. Commun. Algebra 18, No. 9, 3041-3053 (1990; Zbl 0721.14002) and in: Curves Semin. Queen's, Vol. VI, Queen's Pap. Pure Appl. Math. 83, Exposé J (1989; Zbl 0714.14035)]. curves on a surface; type of curve; Hilbert functions; 0-dimensional subschemes Zappalà G.,0-dimensional subschemes of curves lying on a smooth quadric surface, Le Mathematiche,52 (1997), 115--127. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Plane and space curves \(0\)-dimensional subschemes of curves lying on a smooth quadric surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(p>3\) be prime. Let \(J_ 0(p)\) be the Jacobian of the modular curve \(X_ 0(p)\) and let \({\mathcal J}_{\mid\mathbb{Z}}\) be the Néron model of \(J_ 0(p)\). Let \(U_ p\), \(T_ \ell\in M_ g(\mathbb{Z})\) be the matrices of the Atkin-Lehner operator and the Hecke operators acting on \(S_ 2(\Gamma_ 0(p),\mathbb{Z})\) for all primes \(\ell\neq p\). Form the formal power series \(\sum^ \infty_{n=1}A_ n\cdot n^{-s}=(I_ g-U_ p\cdot p^{-s})^{-1}\cdot\prod_ \ell(I_ g-T_ \ell\cdot p^{- s}+I_ g\cdot p^{1-2s})^{-1}\). Let \(L(X,Y)\) be the \(g\)-dimensional formal group law with logarithm \(f(x)=\sum^ \infty_{n=1}{1\over n}A_ nX^ n\in\mathbb{Q}[X_ 1,\ldots,X_ g]^ g\). The author then establishes the following theorem:
The group law \(L(X,Y)\) is defined over \(\mathbb{Z}\) and is isomorphic to the formal completion of \({\mathcal J}\) along the zero section.
The proof is based on certain results of B. Mazur. --- Similar objects to the above exist in the theory of rank-2-Drinfeld modules and one wonders if an analogous result may be established. Jacobian of the modular curve; Néron model; Hecke operators; formal group law E. Nart : The formal completion of the Néron model of J0(p) , Publ. Mat. 35 (1991), 537-542. Jacobians, Prym varieties, Modular and Shimura varieties, Formal groups, \(p\)-divisible groups, Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties The formal completion of the Néron model of \(J_ 0(p)\) | 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 \(S_ n\) be the set of isomorphism classes of complex curves of genus \(2\) having a ``maximal'' map of degree \(n\) onto an elliptic curve (maximal means that the map does not factorize through an unramified covering of the elliptic curve). The purpose of this article is to prove that the class of a curve belongs to \(S_ n\) if and only if the class of its Jacobian belongs to the Humbert surface \(H_{n^ 2}\) in the moduli space of principally polarized abelian surfaces. This amounts to proving that it has a period matrix of the form \((\begin{smallmatrix} z_ 1 & 1/n & 1 & 0 \\ 1/n & z_ 2 & 0 & 1 \end{smallmatrix} )\) with \(\text{Im} z_ 1\), \(\text{Im} z_ 2 > 0\). moduli space of curves; covering of the elliptic curve; Jacobian; Humbert surface; moduli space of principally polarized abelian surfaces; period matrix Murabayashi, N.: The moduli space of curves of genus two covering elliptic curves. Man. Math. 84, 125--133 (1994) Elliptic curves, Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification, Coverings of curves, fundamental group The moduli space of curves of genus two covering elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi:K\to A\) be the Kummer variety of a q-dimensional Abelian variety over \({\mathbb{C}}\). Let C be a smooth curve of genus \(g,\) and \(\phi:C\to K\) be a nonconstant morphism. - The author shows that:
1. \((\phi,C)\) is rigid if \(g\leq q-2\). (In particular: Kummer surfaces are not uniruled.)
2. \(g\geq q-2\) if A is general among the Abelian varieties carrying a polarization of any given type.
3. As a consequence, if \(q\geq 3\) a general Abelian variety does not contain any hyperelliptic curve. non-existence of hyperelliptic curve; Kummer variety; Abelian variety G.\ P. Pirola, Curves on generic Kummer varieties, Duke Math. J. 59 (1989), no. 3, 701-708. Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces, Jacobians, Prym varieties Curves on generic Kummer 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 \(Y/Z\) be a finite Galois covering of complete irreducible nonsingular algebraic curves defined over a field \(k=\mathbb{F}_q\) of \(q=p^n\) elements with Galois group \(G\), whose order \(m\) is not divisible by \(p\). Let \(k\) be sufficiently large to contain the \(m\)-th roots of unity and suppose that \(Y\) has genus greater than zero. In Transl., II. Ser., Am. Math. Soc. 45, 254--264 (1965); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802), \textit{Yu. I. Manin} computes the reduction modulo \(p\) of the numerator of the zeta-function of \(Y\) in terms of the characteristic polynomial of the Cartier operator acting on the space of regular differentials of \(Y\). In [J. Number Theory 22, 177--189 (1986; Zbl 0581.14025)], \textit{H.-G. Rück} generalizes this result, computing, for each irreducible character \(\chi\) of \(G\), the reduction of the \(\chi(1)\)-th power of the \(L\)-series \(L(t,Y/X,\chi)\) of \(Y/X\) of character \(\chi\). From both results follow relations between the trace of the Cartier operator and character sums.
The first aim of this paper is to show that, when \(G\) is abelian, Rück's result follows from Manin's result. The general case is reduced to the abelian one through a classical theorem of Brauer on characters.
The second aim of this paper is to refine the previous expression of the \(\chi(1)\)-th power of \(L(t,Y/X,\chi)\) modulo \(p\) calculating exactly \(L(t,Y/X,\chi)\) modulo \(p\). Galois covering of complete irreducible nonsingular algebraic curves over a finite field; regular differentials; \(L\)-series; trace of the Cartier operator; character sums Coverings of curves, fundamental group, Finite ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Divisors, linear systems, invertible sheaves, Curves over finite and local fields, Exponential sums and character sums The trace of the Cartier operator and character sums | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a curve over a field \(k\) with a rational point \(e\). We define \(\Delta _ e\in Z^ 2(X^ 3)_{\text{hom}}\), a canonical cycle. Suppose that \(k\) is a number field and that \(X\) has semi-stable reduction over the integers of \(k\) with fiber components non-singular. We construct a regular model of \(X^ 3\) and show that the height pairing \(\langle \tau _ *(\Delta _ e),\tau '_ *(\Delta _ e)\rangle \) is well defined where \(\tau \) and \(\tau '\) are correspondences. The paper ends with a brief discussion of heights and \(L\)-functions in the case that \(X\) is a modular curve. diagonal cycle; triple product of a pointed curve; regular models; semi-stable reduction; height pairing; \(L\)-functions; modular curve Gross, B. H.; Schoen, C., \textit{the modified diagonal cycle on the triple product of a pointed curve}, Ann. Inst. Fourier (Grenoble), 45, 649-679, (1995) Arithmetic varieties and schemes; Arakelov theory; heights, Rational points, Arithmetic ground fields for curves, Modular and Shimura varieties, Algebraic cycles, Arithmetic aspects of modular and Shimura varieties The modified diagonal cycle on the triple product of a pointed 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 employ the theory of Manin and Drinfeld on p-adic theta functions to obtain formulae expressing the p-adic period matrix of a Mmford-Schottky curve of genus two in terms of the coefficients of an equation for the curve. Using these formulae, we approximate the p-adic period matrix of certain modular curves and gather further evidence in support of the ``exceptional zero conjecture'' described in the paper ``On p-adic analogues of the conjecture of Birch and Swinnerton-Dyer'', Invent. Math. 84, 1-48 (1986) by \textit{B. Mazur}, \textit{J. Tate} and the author. We also obtain an explicit p-adic analytic parametrization of a portion of the boundary of the moduli space of curves of genus two. p-adic theta functions; p-adic period matrix of a Mmford-Schottky curve of genus two; exceptional zero conjecture Teitelbaum, J., \(p\)-adic periods of genus two Mumford-Schottky curves, J. Reine Angew. Math., 385, 117-151, (1988) Arithmetic ground fields for curves, Local ground fields in algebraic geometry, Theta functions and abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) p-adic periods of genus two Mumford-Schottky curves | 0 |
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