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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 \(Y\) be a complete algebraic variety over a complete non-Archimedean field \(K\). If the valuation is discrete, then local heights of cycles on \(Y\) are well-known. They are given by intersection numbers in models over the discrete valuation ring. To generalize this to the non-discrete case, we replace the algebraic models by formal models over the valuation ring. First, a proper intersection product of Cartier divisors with cycles on a rigid analytic variety \(X\) is defined. Then we extend this intersection product to admissible formal models. It satisfies the usual properties and corresponds to a normalized intersection product in the algebraic situation. A Cartier divisor on a formal model induces a metrized line bundle on \(X\). The metric is called a formal metric. If \(X\) is quasi-compact and quasi-separated, then limits of roots of formal metrics are characterized as those metrics with a continuous extension to the Berkovich-compactification of \(X\). Using for \(X\) the rigid analytic variety associated to \(Y\), we define local heights of cycles as intersection numbers with Cartier divisors on an admissible formal model of \(X\). The dependence on the models is measured by the metrized line bundles. Finally, it is shown that the local heights satisfy five characteristic properties. intersection product; Cartier divisors; rigid analytic variety; local heights of cycles W. Gubler, Local heights of subvarieties over non-Archimedean fields, J. reine angew. Math. 498 (1998), 61-113. Local ground fields in algebraic geometry, Algebraic cycles Local heights of subvarieties over non-archimedean 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 \(A\) be an abelian variety defined over a global field \(K\) and \(c\) an ample symmetric divisor class on \(A\). The aim of the paper is the computation of the Néron-Tate height induced by \(c\) (a quadratic form on \(A(K)\)) \(\hat{h}_c(P)\) or, what is the same, the bilinear pairing \(\hat{h}_c(P,Q)\), see [\textit{A. Néron}, Ann. Math. (2) 82, 249--331 (1965; Zbl 0163.15205)] and their application to the computation of the regulator of \(A\) and to give numerical evidence for the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the Cartan modular curve of level 13.
The paper restricts to the case \(A\) the Jacobian of a smooth projective geometrically connected curve \(C\), \(c\) twice the class of a symmetric theta divisor \(\theta\) and \(K=\mathbb{Q}\) (although ``our algorithm can be generalized easily to work over general global fields'').
The algorithm takes advantage of an equality due to Faltings and Hriljac (Theorem 4.1): \(\hat{h}_{2\theta}([D],[E])=-\sum_{v\in M_K}\langle D,E\rangle_v\), where \(D,E\) are divisors of degree 0 on \(C\) without common component, \(M_K\) the set of places of \(K\) and \(\langle D,E\rangle_v\) the local Néron pairing at \(v\) (since \(K=Q\) \(v\) is a \(p\)-adic valuation or the archimedean one). In the case \(C\) a hyperelliptic curve that computation was already done by \textit{D. Holmes} [J. Number Theory 132, No. 6, 1295--1305 (2012; Zbl 1239.14019)] and \textit{J. S. Müller} [Math. Comput. 83, No. 285, 311--336 (2014; Zbl 1322.11074)].
Section 2 gives an algorithm to compute the non-archimedean local Néron pairings (in the authors words ``our main contribution'') and Section 3 deals with the archimedean case.
Section 4 discusses the computation of \(\hat{h}_{2\theta}([D],[E])\) using Theorem 4.1 and in particular identifies a set of places outside which the local Néron pairing of \(D\) and \(E\) vanished.
Finally Section 5 provides examples of the implementation, using the package Magma, of the algorithm for several curves including the split Cartan modular curve of level 13; it finds the regulator up to an integral square factor giving numerical evidence that the BSD conjecture holds up to an integral square. abelian variety; Jacobian; Néron-Tate height; local Néron pairing; conjecture of Birch and Swinnerton-Dyer; Cartan modular curve of level 13 Arithmetic varieties and schemes; Arakelov theory; heights, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Heights, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Abelian varieties of dimension \(> 1\) Explicit arithmetic intersection theory and computation of Néron-Tate heights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the generalization of the classical Torelli morphism from the moduli of stable curves to the moduli of principally polarized stable semi-abelic pairs. The main results are concerned with the study of the fibres and of its injectivity locus.
Let us be more precise. The classical result of \textit{R. Torelli} [``Sulle varietà di Jacobi'', Rom. Acc. L. Rend. (5) 22, No. 2, 98--103, 437--441 (1913; JFM 44.0655.03)] claims the injectivity of the map from the moduli scheme of smooth projective curves of genus \(g\), \(M_g\), to the moduli scheme of principally polarized abelian varieties of dimension \(g\), \(A_g\), that maps a curve to its jacobian variety together with the theta divisor. The most common compactification of \(M_g\) is the moduli space of Deligne-Mumford stable curves, \(\bar M_g\). Thus, one is naturally concerned with the study of those compactifications of \(A_g\) together with a map from \(\bar M_g\) to it whose restriction to \(M_g\) coincides with the Torelli map.
An instance of such a pair is given by the second Voronoi toroidal compactification of \(A_g\) together with a suitable map. However, this map fails to be injective and, indeed, may have positive-dimensional fibers. We refer the reader to [\textit{Y. Namikawa}, Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics. 812. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0466.14011)] and [\textit{V. Vologodsky}, The extended Torelli and Prym maps. Univ. of Georgia PhD thesis (2003)].
In this paper, the authors deal with a second instance. Namely, they consider the coarse moduli space of principally polarized semi-abelic stable pairs introduced by Alexeev as well as an extension of the Torelli map for this case [\textit{V. Alexeev}, ``Complete moduli in the presence of semiabelian group action'', Ann. Math. (2) 155, No. 3, 611--708 (2002; Zbl 1052.14017); ``Compactified Jacobians and Torelli map'', Publ. Res. Inst. Math. Sci. 40, No. 4, 1241--1265 (2004; Zbl 1079.14019)].
The first main result is that the compactified Torelli map is injective at curves having \(3\)-edge-connected dual graph (e.g. irreducible curves, curves with two components intersecting in at least three points). The second one offers different charactizations of curves having the same image by the compactified Torelli map. Torelli map; Jacobian variety; theta divisor; stable curve; stable semi-abelic pair; compactified Picard scheme; semiabelian variety; moduli space; dual graph 9 L. Caporaso and F. Viviani, 'Torelli theorem for stable curves', \textit{J. Eur. Math. Soc.}13 (2011) 1289-1329. Jacobians, Prym varieties, Special divisors on curves (gonality, Brill-Noether theory), Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Torelli theorem for stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a nonsingular complex projective curve of genus \(g \geq 2\). Define \({\mathcal{F}}(X)\) as the collection of equivalence classes of morphisms \(f: X \rightarrow Y\) onto curves \(Y\) of genus \(g^{\prime}\geq 2\), up to isomorphisms \(Y \rightarrow Y^{\prime}\). This is a finite set, according to the De Franchis-Severi theorem.
\(Z_f\) in \(X \times X\) denotes an effective divisor associated with a morphism \(f: X \rightarrow Y\), the locally principal divisor being defined by \(f(x)=f(y)\). This is a reduced divisor. It only depends on the equivalence class \([f]\) in \({\mathcal{F}}(X)\), and the correspondence \([f] \mapsto Z_f\) is injective.
The main result about this is \textit{E. Kani}'s rigidity theorem [Invent. Math. 85, 185.-198, (1986; Zbl 0615.12017)] which states that, taking the homology class of the divisor, the map
\[
{\mathcal{F}}(X) \rightarrow H_2(X \times X, \mathbb Z), \;\;[f] \mapsto [Z_f]
\]
is still injective. The aim of the paper under review is to describe this map.
Define \({\mathcal{F}}_n(X)\) as the collection of equivalence classes of morphisms of degree \(n\). The author describes its image as a subset of an intersection of two loci: the locus of integral points \(V_n( \mathbb Z)=V_n \cap H_2(X \times X, \mathbb Z)\), where \(V_n\) is an algebraic set in \(H_2(X \times X, \mathbb C)\), and the locus of effective homology classes in the product. Here, \(V_n(\mathbb Z)\) is a finite set. The author writes explicit algebraic equations for \(V_n\) in a remarkably simple form. In the case of genus \(g=3\), certain computations are presented, which lead to a few explicit solutions in \(V_n \cap H_2(X \times X, \mathbb Q)\).
Different but related approaches to the subject have been developed by Martens, Tanabe, and Naranjo-Pirola. Genus of a curve; locally principal divisors; morphisms; homology Curves in algebraic geometry, (Co)homology theory in algebraic geometry Morphisms on an algebraic curve and divisor classes in the self product | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper there is given a formula for the calculation of the Picard group of an arbitrary compact toric variety \(X_ \Sigma\). This improves the standard result in that \(X_ \Sigma\) may be singular and a previous result of the author in that \(X_ \Sigma\) may be non-projective. The main difference between the singular and the nonsingular case is that in the nonsingular case the rank of the Picard group of the variety \(X_ \Sigma\) is known to be determined by the combinatorial structure of \(\Sigma\) whereas in the singular case \(\text{Pic} X_ \Sigma\) may depend on metrical properties of the fan.
The proof of the formula is based on the geometric description of special \(T\)-invariant (i.e. invariant under the action of the torus \(T)\) Cartier divisors by a set of points which is called a Cartier set of \(\Sigma\). It is proved that these special divisors always exist and metric properties of the points in the corresponding Cartier set are used in order to develop the formula. An example of two combinatorially equivalent fans \(\Sigma_ 1\) and \(\Sigma_ 2\) is given such that \(X_{\Sigma_ 1}\) is projective with a non-trivial Picard group and \(X_{\Sigma_ 2}\) is non-projective with a surprisingly trivial Picard group. Furthermore, it is shown that \(\Sigma_ 2\) is a fan which cannot be spanned by any topological sphere which is the union of \((d-1)\)-polytopes such that the polytopes correspond exactly to the full dimensional cones of \(\Sigma_ 2\). Picard group; compact toric variety; Cartier divisors; fan M. Eikelberg, ''The Picard group of a compact toric variety,''Res. Math. 22, 509--527 (1992) (see also his paper ''Picard groups of compact toric varieties and combinatorial classes of fans,'' ibid.,23, 251--293 (1993). Toric varieties, Newton polyhedra, Okounkov bodies, Picard groups, Divisors, linear systems, invertible sheaves, Polyhedral manifolds The Picard group of a compact 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 We develop explicit formulas for multiplication by an integer \(r\) in the Jacobian of a hyperelliptic curve. We obtain recurrence relations for such calculations, similar to those for the division polynomials of an elliptic curve. These may be used in an extension of Schoof's algorithm to calculate the zeta function of the curve. The formulas are developed using the theory of Padé approximations to find appropriate functions on the curve. Jacobian of a hyperelliptic curve; division polynomials; zeta function; Padé approximations Cantor, D. G.: On the analogue of the division polynomials for hyperelliptic curves, Journal für die reine und angewandte Mathematik 447, 91-145 (1994) Elliptic curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Jacobians, Prym varieties, Padé approximation, Elliptic curves over global fields On the analogue of the division polynomials for hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a number field, \(\overline{k}\) an algebraic closure and \(C\) a non-singular complete curve defined over \(k\) of genus \(g>1\). If we let \(f\) denote a canonical embedding of \(C\) in its Jacobian variety \(J\), then via \(f\) nontrivial automorphisms of \(C\) induce automorphisms of \(J\), which are stable on the image of \(C\). The author studies the restriction this places on the distribution of \(C(\overline{k})\) in \(J(\overline{k})\) via \(f\). This is done by studying the Neron-Tate height function \(\langle \cdot,\cdot\rangle\) and the corresponding norm \(\| \cdot\| \) on \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), which are associated to a theta divisor on \(\overline{J}=J\times_{\text{Spec} k}\text{Spec} \overline{k}\).
A central result shows there exists a canonically defined scalar product on \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), which is preserved by isomorphisms of Jacobians over \(\overline{k}\) induced by isomorphisms of curves over \(\overline{k}\). In this way a canonically defined height function on a curve of genus \(g>1\) over a number field, which is invariant under isomorphisms over the algebraic closure, is obtained.
As an application of his work the author obtains a new proof of the classical result asserting that the number of fixed points of a non-trivial automorphism of a curve of genus \(g>1\) over a number field is at most \(2g+2\).
Finally he applies his results concretely to plane curves over \(k\) defined by \(C:X^4+Y^4=aZ^4\), \(a\neq 0\), describing the distribution of \(f(C(\overline{k}))\) in \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline{k})\), and so obtaining a different proof of a result of Dem'yanenko. rational points; Jacobian variety; Neron-Tate height function; fixed points of a nontrivial automorphism; curves of genus \(\neq 1\) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Rational points, Jacobians, Prym varieties Rational points of a curve which has a nontrivial automorphism | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper studies the \(1729\) \(K3\) surface associated to Euler's cubic Diophantine equation \(a^3+b^3=c^3+d^3\) considered by Ramanujan. A geometric interpretation of Euler's equation yields the equation \((\star)\quad X^3+Y^3=k(T)\), where \(k(T)=63(3T^2-3T+1)(T^2+T+1)(T^2-3T+3)\).
The main result of this short note is formulated as follows.
{Theorem}: The smooth minimal surface associated to \((\star)\) is an elliptic \(K3\) surface with Picard number \(18\) over \(\overline{\mathbb{Q}}\).
Regarding \((\star)\) as an elliptic curve \(E_{k(T)}\) over the function field \({\mathbb{Q}}(T)\), it is shown that it has rank \(2\), and that for infinitely many \(t\in{\mathbb{Q}}\), \(E_{k(t)}\) has rank \(\geq 2\) over \({\mathbb{Q}}\). Euler's Diophantine equation; \(K3\) surface; elliptic curve over a function field; rank of an elliptic curve; Picard number of a \(K3\) surface Ono, K; Trebat-Leder, S, The 1729 \(K3\) surface, Res. Number Theory, 2, 26, (2016) Rational points, Elliptic curves The 1729 \(K3\) 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 After some of the basic facts about higher dimensional formal groups are reviewed, an explicit construction of the formal group of the Jacobian of a complete nonsingular algebraic curve \(C\) over a field of characteristic zero is given using a basis for the holomorphic differentials on \(C\) at a rational non-Weierstrass point. Let \(l\) be an odd positive prime and
\[
\Gamma_ 0(l) = \left \{ {a\;b \choose c \;d} \in \text{SL}_ 2 (\mathbb{Z}) \mid c \equiv 0 \pmod l \right\}.
\]
Let \(X_ 0 (l)\) be the modular curve associated to \(\Gamma_ 0(l)\). As an example of the above construction, the formal group of the Jacobian of \(X_ 0(l)\) is constructed. The connection between the differentials on \(X_ 0(l)\) and the Fourier expansions of cusp forms of weight 2 on \(\Gamma_ 0(l)\) is reviewed. Using this and a result of \textit{T. Honda}, the author proves that this formal group is \(p\)-integral for all but finitely many primes \(p\). formal group of the Jacobian of a complete nonsingular algebraic curve; holomorphic differentials; rational non-Weierstrass point; modular curve; cusp forms DOI: 10.2140/pjm.1993.157.241 Formal groups, \(p\)-divisible groups, Jacobians, Prym varieties, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials The formal group of the Jacobian of an algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author reports progress on a conjecture of Serge Lang concerning a relationship between the number of integral points and the rank of the group of rational points on an elliptic curve [\textit{S. Lang}, Elliptic curves: Diophantine analysis. Berlin etc.: Springer Verlag (1978; Zbl 0388.10001), page 140]. Thus let \(K/{\mathbb Q}\) be a number field, \(R_ S\) a ring of \(S\)-integers in \(K\), and \(E/K\) an elliptic curve given by a quasi-minimal Weierstrass equation \(y^ 2=x^ 3+Ax+B\), (i.e. \(| \text{Norm}_{K/{\mathbb Q}}(4A^ 3+27B^ 2)|\) is minimized subject to the condition that \(A,B\in K\) are integral). Further, let \(\delta(E/K)\) be the number of primes of \(K\) for which the \(j\)-invariant of \(E\) is non-integral. Then the author proves that there is a constant \(\kappa\), depending only on \(K\), so that
\[
| \{P\in E(K): x(P)\in R_ S\}| \leq \kappa^{(1+rank E(K))(1+\delta (E/K))+| S|}.
\]
The conjecture of Lang referred to above asserts that this estimate should hold using the exponent \(1+\text{rank}\;E(K)+| S|.\)
More generally, let \(C\) be an algebraic family of curves over a base space \(T\). For each \(t\in T(K)\), one can look at the \(S\)-integral points on the fiber \(C_ t\) (relative to a given embedding) and at the group of \(K\)-rational points on the Jacobian variety \(J_ t=\text{Jac}(C_ t)\). The author gives a general estimate
\[
| C_ t(R_ S)| \leq | J_ t(K)_{\text{tors}}| \quad \kappa^{1+| S| +\text{rank}\;J_ t(K)}\quad (h(t)/\mu(t))^{\text{rank}\;J_ t(K)}.
\]
Here \(\kappa\) is a constant depending only on \(K\), \(h(t)\) is the height of the parameter \(t\in T(K)\), and \(\mu(t)\) is a lower bound for the canonical height of non-torsion points on the Jacobian \(J_ t(K)\). This reduces Lang's conjecture to the problem of giving an appropriate lower bound for the canonical height on elliptic curves, which is another conjecture of Lang (loc. cit., page 92). Applying the height lower bound proven by the author [Duke Math. J. 48, 633--648 (1981; Zbl 0475.14033)] gives the result stated above.
As another application of the main theorem, the author proves a similar estimate for the number of integral points on the twisted Catalan curves \(y^ m=x^ n+a\). The proofs are fairly technical due to the necessity of working with non-complete varieties while keeping track of the dependence on the base parameter. The author makes extensive use of his theory of height functions corresponding to arbitrary subschemes, which generalize Weil's theory of heights corresponding to divisors [``Arithmetic distance functions and height functions in diophantine geometry'', Math. Ann. 279, 193--216 (1987; Zbl 0607.14013)]. number of integral points; rank of the group of rational points on an elliptic curve; rational points on the Jacobian variety; lower bound for the canonical height on elliptic curves; twisted Catalan curves ----,A quantitative version of Siegel's theorem, J. reine ang. Math.378 (1987), 60--100 Arithmetic ground fields for curves, Rational points, Elliptic curves over global fields, Heights, Special algebraic curves and curves of low genus, Higher degree equations; Fermat's equation A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan 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 Consider an elliptic curve E with j-invariant \(j=1728\) over \({\mathbb{Q}}\) which has a \({\mathbb{Q}}\)-rational point P of infinite order. For each prime p where the curve has good reduction, the point P defines a cyclic subgroup of the group \(E({\mathbb{F}}_ p)\) of \({\mathbb{F}}_ p\)-rational points on the curve. One can ask how often it actually generates this group. The question is a variant of similar ones in the setting of a unit of infnite order in an algebraic number field. Of course, in certain cases we know a priori that we don't obtain a generator. Whenever \(E({\mathbb{F}}_ p)\) is not cyclic we will not. Also, if the point P is an image \(\phi\) (Q) of some other rational point under an endomorphism \(\phi\) such that \(\phi\) mod p is defined over \({\mathbb{F}}_ p\) and doesn't define an isomorphism on \(E({\mathbb{F}}_ p)\) we also don't have a generator.
The author exhibits two examples where the above obstructions happen for all primes \(p\equiv 1 mod 4\). He proves under the generalized Riemann hypothesis (GRH), that except in these two cases the set of primes \(p\equiv 1 mod 4\) for which P gives a generator, has a positive density. In earlier work he already derived (using GRH) a formula for this density. It involved the degrees over \({\mathbb{Q}}(i)\) of compositions of fields \({\mathbb{Q}}(i,\alpha^{-1}\bar O,\beta^{-1}P)\), for \(\alpha\),\(\beta\in End(E)\cong {\mathbb{Z}}[i]\). The current paper gives explicit formulas for the degree of these fields. This enables the author to derive his density results. generator of set of rational points on a curve; elliptic curve; generalized Riemann hypothesis Gupta, Rajiv, Division fields of \(Y^2 = X^3 - a X\), J. number theory, 34, 335-345, (1990) Rational points, Finite ground fields in algebraic geometry, Elliptic curves Division fields of \(Y^ 2=X^ 3-aX\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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{C. S. Seshadri} has constructed in Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 155-184 (1977; Zbl 0412.14005) a canonical desingularisation \(N\) for the moduli space of semistable vector bundles of rank 2 and degree 0. This therefore provides us with a smooth compactification of the variety of stable bundles of rank 2 and degree 0. The purpose of this paper is to outline a procedure for the computation of the Betti numbers of the above compactification. We go about this by making an in-depth study of this variety \(N\); in particular we give a canonical stratification of \(N\) and determine the normal bundles of these strata. As applications we are able to compute two-thirds of the Betti numbers of \(N\) and determine the intermediate Jacobian of \(N\). Details of proof have appeared elsewhere [cf. the author, Proc. Indian Acad. Sci., Math. Sci. 98, No. 1, 1-24 (1988; Zbl 0687.14014) and Am. J. Math. 112, No. 4, 611-630 (1990; Zbl 0722.14014) and the author and \textit{C. S. Seshadri} in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 87- 120 (1990; Zbl 0726.14012)]. computation of the Betti numbers of a smooth compactification of the variety of stable bundles; intermediate Jacobian Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the cohomology of a moduli space 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 This paper studies the action of the Hecke operators on the height of points on a modular curve. Let \(x\) be a point on \(X=\) the modular curve corresponding to \(\Gamma(N)\), or \(\Gamma_ 1(N)\), or \(\Gamma_ 0(N)\), the usual congruence subgroups. For a point \(x\) in \(X(\overline\mathbb{Q})\), and an integer \(m\), the Hecke correspondance \(T_ m\) associates a formal sum \(\sum y_ i\). The purpose of the paper is to compare the height of \(x\) to the sum \(h_ m(x)\) of the heights of the \(y_ i\). The first step in proving the result is to reduce the computations of heights to the main modular curve corresponding to \(SL_ 2(\mathbb{Z})\). After that the local components of the height are analysed. For the finite prime the situation is quite easy; for the infinite place a comparison to the Eisenstein series is used. The author can deduce from his main result combined with Serre's result on the size of the image of the Galois group on the torsion of an elliptic curve that for \(m\) prime and big enough the \(y_ i\) are non torsion points in the jacobian (unpublished result of Mazur); the proof uses also Raynaud's theorem comparing the image of \(X\) in its jacobian and the torsion part of it. action of the Hecke operators on the height of points on a modular curve; torsion of an elliptic curve; non torsion points in the jacobian J. H. Silverman, Hecke points on modular curves, Duke Math. J. 60 (1990), 401--423. Modular and Shimura varieties, Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic aspects of modular and Shimura varieties Hecke points on modular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0527.14039. surface in projective 4-space; degenerate secant variety; developable ruled surface of a curve; dimension at most 4 Determinantal varieties, Special surfaces, Special varieties Degenerate secant varieties and a problem on 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 (From the author's introduction): There are naturally occuring locally free sheaves called Picard bundles on the Jacobian J of a smooth complete curve C of positive genus \(g\) over \(k=\bar k\). These bundles describe the global variation of the section of invertible sheaves on C with pleasant degree. The inversion problem is to give a description of the Picard bundles globally on J.
As such analytic description is lacking, the author gives two algebraic solutions of this problem. The first solution requires to know the image of some points of C in the Jacobian. This approach uses a method due to R. C. Gunning. The second solution determines the pull-back of the Picard bundle by a multiplication in J in terms of a module over the graded ring of theta sections. Here one uses a form of a theorem of D. Mumford on the equations defining abelian varieties projectively.
[See also part II, ibid. 108, No.1, 59-67 (1990; Zbl 0704.14022).] Picard bundles on the Jacobian; complete curve; section of invertible sheaves; pull-back of the Picard bundle Jacobians, Prym varieties, Analytic theory of abelian varieties; abelian integrals and differentials, Theta functions and abelian varieties Notes of the inversion of integrals. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian variety of an algebraic curve \(C\) is a connected component of the moduli space of line bundles on \(C\). Some subschemes determined by \(h^0(L)\) were classically called the Brill-Noether loci. This article gives a survey of several topics on similar subschemes of the moduli space of vector bundles on \(C\). To be precise, for a line bundle \(\xi\) on \(C\), let \(M_C(2,\xi)\) be the moduli space of stable vector bundles \(E\) of rank two on \(C\) with \(c_1(E)=\xi\). For the canonical bundle \(K\), let \(M_C(2,K,n)\) be the subscheme of \(M_C(2,K)\) consisting of \(E\)'s with \(h^0(E)\geq n+2\). Given a vector bundle \(F\) of rank two, let \(M_C(2,K:\nu F)\) be the subscheme of \(M_C(2,K \otimes\text{det} F)\) consisting of \(E\)'s with \(\dim\Hom (F,E)\geq\nu\).
The author surveys several known results on these vector bundle versions of Brill-Noether loci. For example, if \(g(C)=7\) and if there is no \(g^1_4\) on \(C\), then \(M_C(2,K,3)\) is a smooth Fano threefold of Picard number one and of genus 7; if \(C\) is a smooth plane curve of degree four, \(M_C(2,K:3F)\) is a smooth Fano threefold of Picard number one and of genus 9, if \(F\) is of odd degree and if \(S^2\geq 3\) for any section \(S\) of the \(\mathbb{P}^1\)-bundle \(\mathbb{P}(F)\) over \(C\). Finally, let \(C\) be a curve of genus \(g\geq 11\) with \(g\equiv 3\bmod 4\), and assume that \(C\) can be embedded in some \(K3\) surface and that \(C\) is general among such curves. Then \(T:=M_C(2,K,(g-1)/2)\) is a \(K3\) surface. Moreover, any \(K3\) surface \(X\) containing \(C\) must be isomorphic to a certain subscheme of Brill-Noether type of a certain moduli space of vector bundles on \(T\) (hence \(X\) is uniquely determined by \(C)\). This is based on a duality between \(K3\) surfaces, as certain moduli spaces of vector bundles on each other. line bundles on an algebraic curve; Jacobian variety; Brill-Noether locus; moduli space of stable vector bundles; Fano threefold S. Mukai, Non-abelian Brill-Noether theory and Fano 3-folds [translation of Sūgaku 49 (1997), no. 1, 1-24; MR1478148 (99b:14012)], Sugaku Expositions 14 (2001), no. 2, 125-153. Vector bundles on curves and their moduli, Fano varieties, \(3\)-folds, Algebraic moduli problems, moduli of vector bundles Noncommutativizability of Brill-Noether theory and 3-dimensional Fano variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We present a new technique to handle the study of homogeneous rings of a projective variety endowed with a finite or a generically finite morphism to another variety \(Y\) whose geometry is easier to handle. Under these circumstances it is possible to use the information given by the algebra structure of \({\mathcal O}_X\) over \({\mathcal O}_Y\) to describe the homogeneous ring associated to line bundles which are the pullbacks of line bundles on \(Y\).
In this article we illustrate our technique to study the canonical ring of curves (a well-known ring that we revisit with this new technique) equipped with a suitable finite morphism and homogeneous rings of a certain class of Calabi-Yau threefolds. very ampleness; canonical curve; finite cover; homogeneous rings of a projective variety; finite morphism; Calabi-Yau threefolds F. J. Gallego and B. P. Purnaprajna, Some homogeneous rings associated to finite morphisms, Preprint. To appear in ``Advances in Algebra and Geometry'' (Hyderabad Conference 2001), Hindustan Book Agency (India) Ltd. Rational and birational maps, Coverings in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Coverings of curves, fundamental group Some homogeneous rings associated to finite morphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In Math. Z. 187, 105-123 (1994; Zbl 0543.14014), \textit{E. Kunz} and \textit{R. Waldi} prove that the jacobian algebra \(R/ \widetilde J\) of a reduced algebroid Gorenstein curve \(R\) defined over an algebraically closed field \(k\) of characteristic zero is Gorenstein if and only if \(R\) is quasi- homogeneous. In the article under review the notion of jacobian algebra is extended to a domain \(R\) of CM-type not necessarily one: the ideal \(\widetilde J\) is defined as the annihilator of the cokernel of the fundamental class:
\[
\widetilde J : = \text{Ann}_ R (\text{Coker} \varphi : \Omega^ 1_{R/k} \to \omega_ R).
\]
It is proved that if \(R/ \widetilde J\) is Gorenstein, then \(R\) is Gorenstein. From this result and from the quoted one it follows that \(R/ \widetilde J\) is Gorenstein if and only if \(R\) is Gorenstein and quasi-homogeneous. Jacobian algebra of a curve singularity; reduced algebroid Gorenstein curve Singularities of curves, local rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Modules of differentials, Special algebraic curves and curves of low genus When the Jacobian algebra of a curve singularity is Gorenstein | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Classically, a theorem of \textit{A.~G.~Kushnirenko} [Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)] and \textit{D.~N.~Bernstein} [Funct. Anal. Appl. 9, 183--185 (1976; Zbl 0328.32001)] states that if \(P_1, \ldots, P_n\) are general functions on the complex torus \((\mathbb{C} ^{*})^n\) in a fixed finite-dimensional vector space of functions spanned by monomials, with Newton polytopes \(\Delta_1, \ldots, \Delta_n\) (respectively), then the number of solutions of the simultaneous equations \(P_1= \cdots = P_n=0\) is equal to \(n!V(\Delta_1, \ldots, \Delta_n)\), where \(V\) is the mixed volume of convex bodies in \(\mathbb{R}^n\).
The authors generalize Kushnirenko and Bernstein's theorem by considering the set, denoted by \(\mathbf{K}_ {\mathrm{rat}}(X)\), of all non-zero finite-dimensional vector subspaces (in place of monomial-spanned polynomial spaces) in the space of rational functions on any irreducible \(n\)-dimensional complex algebraic variety \(X\) (in place of \((\mathbb{C}^{*})^n\)). For each \(n\)-tuple \((L_1, \ldots, L_n)\) of elements of \(\mathbf{K}_ {\mathrm{rat}} (X)\), they define their intersection index, denoted by \([L_1, \ldots, L_n]\), to be the number of solutions of the simultaneous equations \(f_1 = \cdots = f_n = 0\), where each \(f_i\) is a function of general form in \(L_i\). In counting the solutions, they neglect those at which all functions in \(L_i\) vanish (for at least one \(i\)) and those at which at least one function in \(L_i\) has a pole. The authors show that the so-defined intersection index share basic properties with the mixed volume of convex bodies: (1) non-negativity, (2) increasing in each variable, (3) multilinearity, and (4) the Alexandrov--Fenchel inequality.
The set \(\mathbf{K}_ {\mathrm{rat}}(X)\) turns out to be a commutative semigroup under the multiplication defined by \(L'L'' =\) the linear span of \(\{fg | f \in L', g\in L''\}\). The core result of the paper under review is that \([L_1, \ldots, L_n]\) is multilinear under multiplication in \(\mathbf{K}_ {\mathrm{rat}}(X)\); for instance, \([L_1'L_1'', \ldots, L_n] = [L_1', \ldots, L_n] + [L_1'', \ldots, L_n]\). Therefore, the intersection index extends to the Grothendieck group of \(\mathbf{K}_ {\mathrm{rat}}(X)\), which provides an extension of the intersection theory of divisors. simultaneous algebraic equations; mixed volume of convex bodies; Kushnirenko and Bernstein's theorem; linear system on a variety; Cartier divisor; intersection index K. Kaveh and A. G. Khovanskii, ''Mixed volume and an extension of intersection theory of divisors,'' Mosc. Math. J., vol. 10, iss. 2, pp. 343-375, 2010. Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Mixed volumes and related topics in convex geometry Mixed volume and an extension of intersection theory of divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In Math. USSR, Izv. 32, No. 3, 523-541 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 3, 522-540 (1988; Zbl 0662.14017), \textit{V. A. Kolyvagin} studied an arbitrary Weil elliptic curve \(E\), whose \(L\)-rank (i.e., the order of the zero of its \(L\)-series at the point \(s=1)\) is equal to 0. The main result of the cited paper is a proof that the Tate-Shafarevich group \(\text{ Ш}(\mathbb{Q},E)\) and the Mordell-Weil group \(E(\mathbb{Q})\) are finite.
The ideas and methods of the cited paper admit many generalizations. The present paper is a generalization of the method of it to the case of higher-dimensional elliptic curves (i.e., abelian varieties with real multiplication) of \(L\)-rank 0, defined over totally real fields. Such a higher-dimensional elliptic curve \(E\) is a factor of the Jacobian of a Shimura curve \(X\), which, in turn, is a factor of the complex upper half- plane \({\mathfrak H}\) modulo the action of some subgroup (analogous to a congruence subgroup) of the multiplicative group of a quaternion algebra over a totally real field. The finiteness of Ш for such a variety is proved, starting from the conditions that a Heegner point on it is not a torsion point. finiteness of Tate-Shafarevich group; abelian varieties with real multiplication; higher-dimensional elliptic curve; Jacobian of a Shimura curve; Heegner point; Mordell-Weil group Ярощук, В. А., Интегральный инвариант в задаче о качении эллипсоида со специальными распределениями масс по неподвижной поверхности без проскальзывания, Изв. РАН. Мех. тв. тела, 2, 54-57, (1995) Arithmetic ground fields for abelian varieties, Elliptic curves, Totally real fields Finiteness of Ш over totally real 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 For a nonsingular projective curve \(C\) over a field \(K,\) \(H^{0}(C,\Omega^{1})\) is the \(K\)-vector space of holomorphic differentials on \(C\) over \(K.\) Such a differential is said to be exact if it is of the form \(df\) for some function \(f\in K(C)\). If \(f\) is non-constant then \(f\) necessarily has at least one pole. So in characteristic \(0\), there are no non-zero exact holomorphic differentials. But in characteristic \(p > 0\), where \(p\)-th powers differentiate to \(0,\) a pole whose order is a multiple of \(p\) might disappear upon differentiation, and there can be non-zero exact holomorphic differentials. Inside \(H^{0}(C,\Omega^{1}),\) the subspace of exact holomorphic differentials is the kernel of the Cartier operator \(C\). It seems like a natural problem, given a curve \(C/K\) with \(\mathrm{char}(K) = p > 0\), to calculate this subspace.
The authors produce several families of exact holomorphic differentials on a quotient \(X\) of the Ree curve in characteristic 3, defined by \(X : y^{q} - y = x^{q}_{0}(x^{q}- x)/F_{q}\) (where \(q_{0}= 3^{s}, s > 1\) and \(q = 3 q^{2}_{0})\). They conjecture that they span the whole space of exact holomorphic differentials, and prove this in the cases \(s = 1\) and \(s = 2,\) by calculating the kernel of the Cartier operator. Cartier operator; Ree curve; \(a\)-number; the Shafarevich-Tate group; Frobenius morphism; Jacobian of curve Dummigan, N.; Farwa, S., Exact holomorphic differentials on a quotient of the ree curve, J. algebra, 400, 249-272, (2014) Families, moduli of curves (algebraic), Schemes and morphisms, Finite ground fields in algebraic geometry, Arithmetic ground fields for curves Exact holomorphic differentials on a quotient of the Ree 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 \(Z/K\) be a non-singular complete curve over a complete valued field \(K\). The uniformization of the Jacobian variety of \(Z\) is an extension of a principally polarized abelian variety \(A(Z)\), with good reduction, by a torus. Using deformation theory of curves, one shows that \(A(Z)\) is in general not a product of Jacobian varieties. complete curve; complete valued field; uniformization of the Jacobian variety J. Fresnel and M. Van der Put, Uniformisation de variétés de Jacobi et déformations de courbes, Annales de la Faculté des Sciences de Toulouse 3 (1994), 363--386. Jacobians, Prym varieties, Non-Archimedean valued fields Uniformization of Jacobian varieties and deformations 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 In this note one studies the minimal resolution \(\widehat F\) of a quartic \(F\) in \({\mathbb{P}}^3\) with \(16\) nodes. The title of the paper is motivated by the fact that \(\widehat F\) is isomorphic to the Kummer surface of the Jacobian of a certain genus 2 curve. For Picard number of \(\widehat F =16\) (this happens generically) one shows that \(\widehat F\) admits an irreducible elliptic fibration and it admits also a most reducible elliptic fibration. Picard number; elliptic fibration; genus; Kummer surface; K3 surface; Jacobian; node of a surface Jong Hae Keum, Two extremal elliptic fibrations on Jacobian Kummer surfaces, Manuscripta Math. 91 (1996), no. 3, 369 -- 377. , https://doi.org/10.1007/BF02567961 Jonghae Keum, Erratum to: ''Two extremal elliptic fibrations on Jacobian Kummer surfaces'' [Manuscripta Math. 91 (1996), no. 3, 369 -- 377; MR1416718 (97h:14053)], Manuscripta Math. 94 (1997), no. 4, 543. \(K3\) surfaces and Enriques surfaces, Jacobians, Prym varieties, Families, fibrations in algebraic geometry, Singularities in algebraic geometry Two extremal elliptic fibrations on Jacobian 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 a finite set of points \(\mathbb{X}\) in \(\mathbb{P}^2\), the degree of a point \(P\) in \(\mathbb{X}\) is defined as the minimal degree of a curve containing \(\mathbb{X}\setminus\{P\}\), but not \(P\). The idea of the authors is to generalize this definition to sets of points on a projectively normal algebraic surface \(S\subseteq\mathbb{P}^n\) as follows: the degree \(\deg_{\mathbb{X},S}(P)\) of \(P\) in \(\mathbb{X}\) with respect to \(S\) is the least integer \(d\) such that there exists an arithmetically Cohen-Macaulay curve \(C\subset S\) which contains \(\mathbb{X}\setminus\{P\}\), but not \(P\).
The main result of the paper relates \(\deg_{\mathbb{X},S}(P)\) to the usual degree of \(P\) when \(S\) is a surface of minimal degree. The authors also provide an example which shows that their formula does not hold on an arbitrary arithmetically Cohen-Macaulay surface \(S\). degree of a point; arithmetically Cohen-Macaulay curve; variety of minimal degree L. Bazzotti and M. Casanellas, Separators of points on algebraic surfaces,J. Pure Appl. Algebra 207 (2006), 319--326. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Plane and space curves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Separators of points on algebraic surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is devoted to the classification of elliptic fibrations with section on the so-called ``generic Jacobian Kummer surface'': this is the name for \(\mathrm{Km}(\mathrm{Jac}(C))\), where \(C\) is a genus 2 curve whose Jacobian \(\mathrm{Jac}(C)\) has no extra endomorphisms. For this class of surfaces, the author states the following :
Theorem. There are exactly 25 different elliptic fibrations with section on a generic Jacobian Kummer surface \(\mathrm{Km}(\mathrm{Jac}(C))\) over an algebraically closed field of characteristic zero, modulo the action of the automorphism group of the surface and permutations of the Weierstrass points of \(C\). Then the author analyzes in details all the 25 elliptic fibrations in order to show explicitly, for each fibration, the elliptic parameter and the reducible fibers of the fibrations, the Weiestrass equation, torsion and non-torsion sections and a basis of the Mordell-Weil lattice.
The motivation for this paper is due to Problem 5 of [\textit{M. Kuwata} and \textit{T. Shioda}, Adv. Stud. Pure Math. 50, 177--215 (2008; Zbl 1139.14032)], where the authors suggest the interest to study elliptic fibrations on a Kummer surface \(\mathrm{Km}(\mathrm{Jac}(C))\), where \(C\) is a genus 2 curve. elliptic fibrations; Kummer surfaces; Weierstrass equation; , Mordell-Weil lattice; Jacobian of a curve Kumar, A, Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geom., 23, 599-667, (2014) Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces Elliptic fibrations on a generic Jacobian Kummer surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The index \(r=r(X)\) of a nonsingular Fano variety \(X\) is defined as the largest positive integer \(r\) such that \(-K_X\) is divisible by \(r\) in \({\text{Pic}}(X)\). One has always that \(1\leq r(X)\leq \dim (X)+1\). Moreover \(r=\dim (X)+1\) only if \(X={\mathbb P}^n\) and \(r=\dim (X)\) only if \(X\) is a smooth quadric. Fano varieties of index \(r=\dim (X)-1\) were classified by \textit{T. Fujita} [J. Math. Soc. Japan 32, 709-725 (1980); 33, 415-434 (1981) and 36, 75-89 (1984); respectively Zbl 0474.14017; Zbl 0474.14018; Zbl 0541.14036)]. --- \textit{S. Mukai} [Proc. Natl. Acad. Sci. USA 86, No. 9, 3000-3002 (1989; Zbl 0679.14020)] classified Fano varieties of index \(r=\dim (X)-2\) under the assumption of the existence of smooth divisors in the linear system \({}-\frac{1}{r}K_X{}\). In the present paper the author proves a weaker five-dimensional variant of this conjecture: The linear system \({}-\frac{1}{3}K_X{}\) contains an irreducible divisor with at worst canonical singularities. The proof uses Kawamata's technique and the minimal model program in dimension \(\dim (X)-2\).
\medskip\noindent \textit{Remark.} Later the author improved the result of this paper [ see \textit{Yu. G. Prokhorov}, ``On the existence of good divisors on the Fano varieties of coindex three. II'', Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 24 (1995) (Russian, to appear)]. Fano variety; canonical divisor; index; existence of smooth divisors in a linear system; minimal model program Fano varieties, Divisors, linear systems, invertible sheaves The existence of good divisors of Fano varieties of coindex 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 An explicit description is given of the group law on the Jacobian of a curve \({\mathcal C}\) of genus 2. The Kummer surface provides a useful intermediary stage; bilinear forms relating to the Kummer surface imply that the global law may be given projectively by biquadratic forms defined over the same ring as the coefficients of \({\mathcal C}\). It is not assumed that \({\mathcal C}\) has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field. group law on the Jacobian of a curve; Kummer surface E. V. Flynn, The group law on the Jacobian of a curve of genus \(2\) , J. Reine Angew. Math. 439 (1993), 45-69. Jacobians, Prym varieties The group law on the Jacobian of a curve of genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We consider a subgroup of the second Chow group of a surface, which we call \(\Sigma\). The group \(\Sigma\) is interesting because it is analogous to the divisor class group of classical number theory. It is defined as follows. Let \(X\) be a smooth algebraic surface defined over a number field \(K\) and suppose that \(X\) has a smooth proper model \({\mathcal X}\) defined over \({\mathfrak O}[1/N]\), a localization of the ring of integers in \(K\). Let \(j:X\to{\mathcal X}\) be the inclusion of \(X\) into \({\mathcal X}\). Then \(\Sigma\) is the kernel of the flat pull-back map \(j^*\): \(\Sigma:=\ker(j^*:\text{CH}^ 2({\mathcal X})\to\text{CH}^ 2(X))\). -- In this paper we give examples of surfaces with finite \(\Sigma\), but where the Picard number of the special fibre is strictly greater than the Picard number of the generic fibre for infinitely many primes. To do this requires elements in \(H^ 1(X,{\mathcal K}_ 2)\) which do not come from \(K^ \times\bigotimes_ \mathbb{Z}\text{Pic}(X)\). We show \(\Sigma\) is torsion in other similar examples. -- Our main theorem is the following one.
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the whole ring of integers in the quadratic imaginary number field \(K\). Let \(N\) be the discriminant of \(E\), and \({\mathcal E}\) be the smooth curve over \({\mathfrak O}[1/6N]\) defined by \(E\). Then the kernels
\[
\Sigma_ K=\ker(\text{CH}^ 2({\mathcal E}\times{\mathcal E})\to\text{CH}^ 2(E_ K\times E_ K))
\]
and
\[
\Sigma=\ker(\text{CH}^ 2({\mathcal E}_{\mathbb{Z}[1/6N]}\times{\mathcal E}_{\mathbb{Z}[1/6N]})\to\text{CH}^ 2(E\times E))
\]
are finite.
If \(E\) is defined over \(\mathbb{Q}\) and has a modular parametrization but no complex multiplication, then we can prove the following theorem: Let \(E\) be a modular elliptic curve defined over \(\mathbb{Q}\). Then \(\Sigma=\ker(j^*:\text{CH}^ 2{\mathcal X}_{\mathbb{Z}[1/5N]})\to\text{CH}^ 2(X))\) is torsion.
There are several conjectural reasons to expect \(\Sigma\) to be torsion or finite, which we now briefly discuss. Bass conjecture, that \(K_ 0({\mathcal Y})\) is finitely generated for \({\mathcal Y}\) a regular scheme of finite type over \(\mathbb{Z}\), together with the Grothendieck isomorphism \(K_ 0({\mathcal Y})\times\mathbb{Q}\simeq(\bigoplus_ i\text{CH}^ i({\mathcal Y}))\otimes\mathbb{Q}\), would imply that \(\Sigma\) has finite rank. If \(\text{CH}^ 2({\mathcal Y})\) is finitely generated, then \(\Sigma\) would be finitely generated. -- An \(S\)-integral version of Beilinson's conjectures would imply that \(\Sigma\) is torsion. -- Finally, we mention that Bloch- Kato's Tamagawa number conjectures imply that \(\Sigma\) is finite. second Chow group of a surface; Picard number; modular elliptic curve; Bass conjecture; Beilinson conjectures doi:10.1215/S0012-7094-92-06715-9 Parametrization (Chow and Hilbert schemes), Modular and Shimura varieties, Elliptic curves, Algebraic cycles Cycles in a product of elliptic curves, and a group analogous to the class 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 [For the entire collection see Zbl 0741.00047.]
The authors continue the work stated in their joint paper with \textit{V. G. Kac} and \textit{C. Procesi} [Commun. Math. Phys. 117, No. 1, 1-36 (1988; Zbl 0647.17010)]. There the extended moduli space \(\hat{\mathcal M}_ g\) of curves of genus \(g\) and the extended relative Picard variety \(\hat{\mathcal F}_{g,n}\) of degree \(n\) line bundles were studied. The first space consists of isomorphy classes of triples \((C,p,z)\) where \(C\) is a Riemann surface of genus \(g\), \(p\) a point on \(C\) and \(z\) a local parameter at the point \(p\). The second space consists of isomorphy classes of quintuples \((C,p,z,L,\varphi)\) where additionally \(L\) is a degree \(n\) line bundle and \(\varphi\) a local trivialization of this line bundle at \(p\). All these spaces are infinite dimensional.
In a first part of the paper under review the authors recall the basic notions and constructions of the afore mentioned paper. Of special importance is the Krichever map which gives an injection of \(\hat{\mathcal F}_{g,g-1}\) into the infinite dimensional Grassmannian \(Gr_ 0(H)\) \((H\) denotes the algebra of formal Laurent series in one variable). By the Boson-Fermion correspondence (using the \(\tau\)-function introduced by the Sato school) \(Gr_ 0(H)\) can be embedded into the projective Boson space (i.e. the space of formal power series in infinitely many variables). The authors introduce the extended moduli space \(\hat{\mathcal A}_ g\) of principally polarized abelian varieties (p.p.a.v.) and the extended Siegel upper halfspace \(\hat{\mathcal H}_ g\). For example, \(\hat{\mathcal A}_ g\) is the space parametrizing (restricted) isomorphy classes of extensions \(0\to H_ +\to H/K\to X\to 0\) where \(X\) is a p.p.a.v., \(H_ +\) the subspace of \(H\) spanned by positive powers of the variable and \(K\) a suitable subgroup of \(H\). The authors show that \(\hat{\mathcal A}_ g\) is again a quotient of \(\hat{\mathcal H}_ g\) under the action of \(Sp(2g,\mathbb{Z})\). By an analogue of the Torelli period map \(\hat{\mathcal M}_ g\) (respectively \(\hat{\mathcal F}_{g,g-1})\) can be embedded into \(\hat{\mathcal A}_ g\) (respectively \(\hat{\mathcal H}_ g)\). Using the \(\tau\)-function the space \(\hat {\mathcal F}_{g,g-1}\) can be embedded into the projective Boson space.
To prove this the authors show that not only the spaces \(\hat{\mathcal M}_ g\) and \(\hat{\mathcal F}_{g,n}\) are infinitesimally homogeneous spaces (in the sense of Manin) but also \(\hat{\mathcal A}_ g\) and \(\hat{\mathcal H}_ g\). The corresponding infinite dimensional Lie algebras are related to a symplectic form on \(H\). The involved tangent spaces can canonically be identified with quotients of these Lie algebras. The differential of the assignment of the \(\tau\)-function can be described as induced by the (Bosonic) Fock representation. This map is connected with the heat equation of the \(\tau\)-function. Due to the fact that the only theta functions satisfying the KP hierarchy are the theta functions coming from Riemann surfaces the intersection of the embedding of \(\hat{\mathcal H}_ g\) with the embedding of \(Gr_ 0(H)\) into the projective Boson space consists exactly of the elements coming from \(\hat{\mathcal F}_{g,g-1}\). heat equation of the \(\tau\)-function; extended moduli space; extended relative Picard variety; Krichever map; projective Boson space; formal power series in infinitely many variables; Torelli period map; infinitesimally homogeneous spaces; theta functions satisfying the KP hierarchy Algebraic moduli of abelian varieties, classification, Infinite-dimensional Lie (super)algebras, Heat equation, Theta functions and abelian varieties, Infinite-dimensional manifolds Abelian varieties, infinite-dimensional Lie algebras, and the heat equation | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a base field \(k\) of characteristic \(p > 2\), and consider a hyperelliptic curve over \(k\). By a result of \textit{Nygaard}, its Jacobian is a product of supersingular elliptic curves if and only if its Hasse- Witt matrix is the zero matrix. By analyzing Hasse-Witt matrices, the author is able to give an essentially elementary proof of a theorem of \textit{Ekedahl}: If the Jacobian of a hyperelliptic curve is a product of supersingular elliptic curves, then the genus \(g\) of the curve satisfies \(g\leq(p-1)/2\) and, moreover, there is a unique such hyperelliptic curve of genus \((p - 1)/2\) up to isomorphism. The paper concludes with the determination of all hyperelliptic curves with zero Hasse-Witt matrix in two families, namely \(y^2 = x^n - x\) and \(y^2 = x^n - 1\). characteristic \(p\); Jacobian of a hyperelliptic curve; product of supersingular elliptic curves; genus; zero Hasse-Witt matrix Valentini, R., Hyperelliptic curves with zero Hasse-Witt matrix, Manuscr. Math., 86, 2, 185-194, (1995) Elliptic curves, Jacobians, Prym varieties, Finite ground fields in algebraic geometry Hyperelliptic curves with zero Hasse-Witt matrix | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Denote by \(SU_X (r)\) the (coarse) moduli space of semistable rank-\(r\) vector bundles with trivial determinant line bundle over a compact Riemann surface \(X\) of genus \(g \geq 2\). It is known that the Picard group of \(SU_X (r)\) is freely generated by an ample line bundle \({\mathfrak L}\), which is called the determinant bundle on \(SU_X (r)\). For any integer \(k \geq 0\), the global sections of \({\mathfrak L}^{\otimes k}\) are sometimes called the generalized \(k\)-th order theta functions on \(SU_X (r)\), because they may be regarded as non-abelian generalizations of the classical theta functions associated with the Jacobian of a Riemann surface. During the past ten years, determinant bundles and their spaces of generalized theta functions have turned out to be fundamental objects in the construction and study of conformal quantum field theories, and some of the basic geometric properties of these mathematical objects (such as the famous Verlinde formulas) have been predicted by means of physical reasonings.
The present report, delivered by one of the most active contributors to these recent developments, provides a beautiful survey on the fundamental ideas, methods, and most important results in this fascinating area linking algebraic geometry and mathematical physics.
The author reviews, in ten brief sections, the moduli spaces \(SU_X (r)\), their determinant bundles and associated rational maps to projective spaces, the base loci of these rational maps, the particular case of semistable rank-2 vector bundles, the various proofs of the Verlinde formula for rank-2 bundles, the conjectural aspects of the so-called strange duality (which seems to be familiar to physicists), and the (again physically motivated) problem of describing the natural projective representation of the Teichmüller modular group \(\Gamma_g\) in \(\text{PGL} (H^0 (SU_X (r), {\mathfrak L}^{\otimes k})\).
The overview of the present state of knowledge is enhanced by the discussion of ten open problems in this context. These problems, among many others, form a crucial obstruction to the complete understanding of the analytic, geometric and physical nature of generalized theta functions. In this regard, the article under review may also be seen as a challenging proposal (or as a guiding program) for further research in this direction. generalized \(k\)-th order theta functions; moduli space of semistable rank-\(r\) vector bundles; determinant bundle; Jacobian of a Riemann surface; conformal quantum field theories; Verlinde formula; strange duality A. BEAUVILLE, \textit{Vector bundles on curves and generalized theta functions: recent results and open} \textit{problems}, Complex Algebraic Geometry, MSRI Publications 28 (1995), 17--33. Vector bundles on curves and their moduli, Theta functions and abelian varieties, Algebraic moduli problems, moduli of vector bundles, Theta functions and curves; Schottky problem, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Vector bundles on curves and generalized theta functions: Recent results and open problems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((X,o)\) be a normal complex surface singularity with rational homology sphere link and \(\tilde X \to X\) a resolution with exceptional set \(E\). A divisor on \(\tilde X\) supported on \(E\) is called a cycle. The authors have studied the theory of Abel maps on effective Cartier divisors on cycles and analyzed the cohomology of invertible sheaves on \(\tilde X\); see [\textit{J. Nagy} and \textit{A. Némethi}, Math. Ann. 375, No. 3--4, 1427--1487 (2019; Zbl 1436.32091)] and [\textit{J. Nagy} and \textit{A. Némethi}, Adv. Math. 371, Article ID 107268, 37 p. (2020; Zbl 1444.32030)]. These methods can apply to investigate analytic invariants and structures of complex surface singularities. The present article provides precise results on Brill-Noether type problem for \(\mathrm{Pic}(\tilde X)\) when \((X,o)\) is an elliptic singularity. It is known that the splice quotient singularities introduced by Neumann-Wahl are characterized by the End Curve Condition (ECC) [\textit{W. D. Neumann} and \textit{J. Wahl}, J. Eur. Math. Soc. (JEMS) 12, No. 2, 471--503 (2010; Zbl 1204.32019)]. The Weak End Curve Condition (WECC) is a weaker condition which enable us to obtain the \(p_g\)-formula similar to that of splice quotients, but a few results are known about this property. In this article, the authors show basic properties of the topology (combinatorial data) of singularities with WECC and prove that an elliptic singularity satisfies the WECC if and only if it satisfies the ECC in terms of the Abel map. This article also introduces a new elliptic sequence for non numerically Gorenstein singularities which connects some invariants to that of numerically Gorenstein sub singularities. normal surface singularity; resolution graph; rational homology sphere; natural line bundle; Poincaré series; Abel map; Brill-Noether theory; effective Cartier divisors; Picard group; Laufer duality; elliptic singularities; elliptic sequence; end curve condition; monomial condition; splice quotient singularities Local theory in algebraic geometry, Local complex singularities, Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) The Abel map for surface singularities. III: Elliptic germs | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the conjecture of \textit{R. Donagi} [Bull. Am. Math. Soc., New Ser. 4, 181-185 (1981; Zbl 0491.14016)] in the ``tetragonal case''. More precisely, let \(\pi: \tilde C\to C\) be a double covering of a tetragonal curve C (where C is non hyperelliptic, non trigonal and non bielliptic) and let the associate Prym variety \(P(\tilde C,\pi,C)\) be isomorphic to some Prym variety \(P(\tilde D,\rho,D)\) (as a principal polarized abelian variety). The author proves that for \(g(C)\geq 13\) the pair \((\tilde D,D)\) can be either \((\tilde C,C)\) or one of the two following pairs \((\tilde C,C')\), \((\tilde C'',C'')\) which correspond to \((\tilde C,C)\) by the tetragonal construction (corollary 4.4). To prove that he applies the methods of \textit{G. E. Welters} [Am. J. Math. 109, 165-182 (1987; Zbl 0639.14026)], especially to the ``tetragonal case''. The first idea in the proof is that the dimension of the locus \(Sing_{ex}\Xi\) of the exceptional singularities of the theta divisor \(\Theta\) of a Prym variety \(P=P(\tilde C,C)\) is equal to dim(P)-6 only if the curve C is tetragonal (theorem 3.1). (Theorem 4.2) gives that Sing(\(\Theta)\) is a union of three components \(S,S'\) and \(S''\) of dimension \(\dim(P)-6.\) The triple \((S,S',S'')\) determines a triple of surfaces \((\Sigma,\Sigma',\Sigma'')\) in the Prym variety P (theorem 3.2). The surfaces \(\Sigma,\Sigma',\Sigma ''\) are singular at \(0\in P\) and the associate projective tangent cones at 0 are actually the semicanonical curves \(C,C',C''\) which determine uniquely some two-sheeted coverings of \(C,C',C''\), namely those from the tetragonal triple (cf. (lemma 4.5). covering of a tetragonal curve; theta divisor; Prym variety O. Debarre , Sur les variétés de Prym des courbes tétragonales , Ann. Sc. Ec. Norm. Sup. 21, (1988), 545-559. Picard schemes, higher Jacobians, Special algebraic curves and curves of low genus, Theta functions and abelian varieties, Jacobians, Prym varieties Sur les variétés de Prym des courbes tétragonales. (On the Prym varieties of tetragonal 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 algebraic curve. We define as gonality of \(C\) the smallest integer \(n\) such that there exists a morphism \(C\to \mathbb{P}^1\) of degree \(n\); i.e. the smallest integer \(n\) such that there exists an invertible sheaf \({\mathcal L}\) on \(C\) of degree \(n\) with \(h^0 ({\mathcal L}) \geq 2\). We define as Clifford index of an invertible sheaf \({\mathcal L}\) on \(C\) the integer \(\deg ({\mathcal L}) -2[h^0 ({\mathcal L})-1]\) and we define as Clifford index of \(C\) the smallest Clifford index of an invertible sheaf \({\mathcal L}\) satisfying \(h^0 ({\mathcal L}) \geq 2\) and \(h^0 ({\mathcal L}^\vee \otimes \omega_C) \geq 2\).
Let \(C\subset \mathbb{P}^3\) be a twisted smooth complete intersection curve and let \(\ell\) be the maximum degree of a collinear positive divisor on \(C\). In the paper under review the author shows that the gonality of \(C\) is \(\deg (C)-\ell\) and that an effective divisor \(\Gamma\) on \(C\) computes this gonality if and only if \(\Gamma\) is the residual of a collinear divisor of degree \(\ell\) on a plane section of \(C\).
Moreover, by results of G. Martens and others [cf., \textit{M. Coppens} and \textit{G. Martens}, Compos. Math. 78, No. 2, 193-212 (1991; Zbl 0741.14035)], the author shows that, if \(\deg (C)\neq 9\), the Clifford index of \(C\) is \(\deg (C)-\ell -2= \text{gon} (C)-2\). divisors on a curve; gonality; Clifford index of an invertible sheaf; complete intersection curve Basili, B, Indice de Clifford des intersections complètes de l'espace, Bull. Soc. Math. France, 124, 61-95, (1996) Plane and space curves, Complete intersections, Projective techniques in algebraic geometry Clifford index of complete intersections of 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 Let \(X\) be a smooth projective variety over \(\mathbb{C}\). We say \(X\) is covered by images of \(\mathbb{P}^1\) if for every point \(p\) of \(X\), we may find a non-constant morphism \(f:\mathbb{P}^1\to X\) such that the point \(p\) belongs to \(f(\mathbb{P}^1)\). Recall one of the principal accomplishments of Mori's program (Miyaoka, Mori, Kawamata, Shokurov and others): Let \(X\) be a smooth projective variety of dimension at most three. Then \(|mK_X|\) is empty for all \(m>0\) iff \(X\) is covered by images of \(\mathbb{P}^1\).
In this paper the authors consider the analogous question for quasi-projective (or more generally log) varieties. If \(U\) is a smooth quasi-projective variety, then, following Iitaka, one picks a smooth compactification \(U\subset X\) such that the complement \(D=X\setminus U\) is a divisor with normal crossings. The linear series \(|m (K_X+D)|\) turn out to depend only on \(U\), not on \(X\) or \(D\). We say \(U\) is dominated by images of a curve \(C\), if there is a dense open subset \(V\) of \(U\) and for every point \(p\) of \(V\), we can find a non-constant morphism \(f:C\to U\) such that \(p\in f(G)\). We say \(U\) is dominated by rational curves, if it is dominated by images of \(\mathbb{P}^1\). -- The main result of the paper is the following:
Let \(U\) be a smooth quasi-projective variety of dimension at most two. Then \(|m(K_X+D) |\) is empty for all \(m>0\) iff \(U\) is dominated by images of \(\mathbb{A}\).
The other main results of the paper are partial classifications of log del Pezzo surfaces, that is projective surfaces \(S\) with quotient singularities and \(-K_S\) ample. variety dominated by images of a curve; Mori's program; smooth quasi-projective variety; log del Pezzo surfaces Keel, Seán; McKernan, James, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc., 140, 669, viii+153 pp., (1999) Fano varieties, Plane and space curves Rational curves on quasi-projective surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A generalized double translation manifold is, by definition, an analytic hypersurface \(S\subset {\mathbb{C}}^{n+1}\) which has two distinct analytic parametrizations:
\[
x_ i=\alpha_ i(t_ 1)+A_ i(t_ 2,...,t_ n)=\beta_ i(u_ 1)+B_ i(u_ 2,...,u_ n)\quad (i-1,...,n+1).
\]
Equivalently, a generalized double translation manifold S can be generated in two different ways by translating an analytic curve (\(\alpha\) or \(\beta)\) along an analytic subvariety of codimension 2 in \({\mathbb{C}}^{n+1}\) (A, or respectively, B).
This paper presents a complete study of generalized double translation 3- folds in \({\mathbb{C}}^ 4\), following some ideas of Sophus Lie. Our major result is that if S is not ''developable'' (that is, the Gauß mapping on S is nondegenerate) and the generating curves and surfaces of a generalized double translation 3-fold S are in suitably general position, then S must in fact be a double translation 3-fold. Indeed, the generating surfaces A and B must be translation surfaces, so S has two other parametrizations of the even more special form:
\[
X_ i=\alpha_ i(t_ 1)+A_{i2}(t_ 2')+A_{i3}(t_ 3')=\beta_ i(u_ 1)+B_{i2}(u_ 2')+B_{i3}(u_ 3')\quad (i=1,...,4),
\]
where \(t_ 2'\) and \(t_ 3'\) depend only on \(t_ 2\) and \(t_ 3\), and similarly for \(u_ 2'\) and \(u_ 3'.\)
This result yields a new characterization of the Jacobian varieties of nonhyperelliptic curves of genus 4 among all 4-dimensional principally poarized abelian varieties. Namely, a principally polarized abelian variety \((A,\theta)\) is the Jacobian of a nonhyperelliptic curve if and only if \(\theta\) is a generalized double translation manifold. The proof consists of combining the main result mentioned above and the Lie- Wirtinger theorem on double translation manifolds.
The results mentioned above may all be extended to generalized double translation manifolds of dimension n for all \(n\geq 3\); details of this will appear subsequently. theta-divisor; generalized double translation manifold; analytic hypersurface; double translation 3-fold; Jacobian varieties of nonhyperelliptic curves; principally polarized abelian variety; Lie- Wirtinger theorem J. B. Little, ''On Lie's approach to the study of translation manifolds'',J. Diff. Geom.,26, No. 2, 253--272 (1987). Picard schemes, higher Jacobians, \(3\)-folds, Analytic spaces, Jacobians, Prym varieties, Theta functions and abelian varieties On Lie's approach to the study of translation manifolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J\) be the Jacobian of a curve \(C\) of genus 2, defined over \(\mathbb{Q}\). Let \(p\) be a prime number. Assume that the reduction of the Néron model of \(J\) over \(\mathbb{Q}_p\) is an extension of an elliptic curve by a torus. We denote by \(\overline\mathbb{Q}\) an algebraic closure of \(\mathbb{Q}\); the Galois group \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) acts on the \(\ell\)-division points of \(J\). We denote by \(\rho_\ell\) the associated representation. Let \(q\) be a prime number where \(J\) has good reduction such that the Galois group over \(\mathbb{Q}\) of the characteristic polynomial of the Frobenius endomorphism associated to \(q\) is the dihedral group with 8 elements (this implies that \(J\) is absolutely simple). Then an infinite set of prime numbers can be found such that the image of \(\rho_\ell\) is \(\text{GSp}(4,\mathbb{F}_\ell)\). Two examples will be given at the end of this article. finite ground field; Jacobian of a curve; Néron model; Galois group; Frobenius endomorphism [15]P. Le Duff, Repr'esentations galoisiennes associ'ees aux points d'ordre l des jacobiennes de certaines courbes de genre 2, Bull. Soc. Math. France 126 (1998), 507--524. Jacobians, Prym varieties, Finite ground fields in algebraic geometry, Rational points, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois representations associated to the points of order \(\ell\) of the Jacobians of special curves of genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies automorphisms of a generic Jacobian Kummer surface. He first analyses the action of classically known automorphisms on the Picard lattice of the surface, and then constructs 192 new automorphisms not generated by the classical ones. These new ones are all conjugate to each other by elements in the symmetry group of the (16,6)-configuration. automorphisms of a generic Jacobian Kummer surface; Picard lattice Keum J H 1997 Automorphisms of Jacobian Kummer surfaces \textit{Compos. Math.}107 269--88 \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties Automorphisms of Jacobian 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 The field of moduli \(k_ A\) for a principally polarized abelian variety A (or a smooth curve) is an inseparable extension of the residue field k(A), A defined over the field k (proposition 1.7). Here the author shows: (1) If \(ch(k)=2\), for the generic point x of the hyperelliptic locus in \(M_ g\), \(g\geq 3\), \(k_ x\) is neither the residue field nor the field of moduli for the canonical polarized jacobian variety of x. (2) If \(ch(k)=3\), for every \(m\geq 2\) there is a non-hyperelliptic curve y of genus 2m with \(k_ y\neq k(y)\). (3) If \(ch(k)=p\), for every \(g\geq p-1\) there is \(z\in A_{g,1}\) with \(k_ z\neq k_ A\). (4) If \(w\in A_{g,1}\) (or \(M_ g)\) \(k_ w=k(w)\) if \(p>2g+1\) or \(p>g+1\), \(p\neq 2g+1\) and p is indecomposable. positive characteristic; coarse moduli space; field of moduli for a principally polarized abelian variety; smooth curve Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Wild ramification of moduli spaces for curves or for Abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complete nonsingular irreducible algebraic curve of genus \(g \) defined over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(\pi_1 (X)\) be its algebraic fundamental group, and \(\Gamma_g\) the topological fundamental group of (any) compact Riemann surface of genus \(g\). It is well known since Grothendieck, that a finite group \(G\) of order prime to \(p\) can be realized as the Galois group of an unramified cover of \(X\) if and only if it is a quotient of \(\Gamma_g\). Other information on the structure of \(\pi_1 (X)\) comes from the Hasse-Witt invariant of \(X\), i.e. the \(\mathbb{F}_p\)-dimension of the \(p\)-torsion subgroup of the Jacobian variety of \(X\).
In this paper the author counts the number of unramified Galois coverings of \(X\) whose Galois group is isomorphic to an extension of a group \(G\) of order prime to \(p\) by a finite group \(H\), which is an irreducible \(\mathbb{F}_p [G]\)-module. This counting depends on some invariants attached to a Galois cover \(Y \to X\) of group \(G\), called generalized Hasse-Witt invariants, arising in the canonical decomposition of the \(p\)-torsion space of the Jacobian variety \(J_Y\) of \(Y\). -- Some particular cases had already been treated by \textit{S. Nakajima} [in: Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 69-88 (1983; Zbl 0529.14016)] and \textit{H. Katsurada} [J. Math. Soc. Japan 31, 101-125 (1979; Zbl 0401.14004)], whose results are generalized in the present paper. algebraic curve; algebraic fundamental group; topological fundamental group; Hasse-Witt invariant; Jacobian variety; number of unramified Galois coverings Pacheco, A., Unramified Galois coverings of algebraic curves, J. number theory, 53, 211-228, (1995) Coverings of curves, fundamental group, Inverse Galois theory, Coverings in algebraic geometry Unramified Galois coverings of algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0519.14021. varieties of special divisors; Jacobian variety; dimension theorem [M3] G. Martens: On dimension theorems of the varieties of special divisors on a curve. Math. Ann.267 (1984), 279--288 Families, moduli of curves (algebraic), Jacobians, Prym varieties, Classification theory of Riemann surfaces, Special algebraic curves and curves of low genus On dimension theorems of the varieties of special divisors on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi: \tilde C\to C\) be an unramified double covering of a smooth curve of genus g. One defines the associated Prym variety as the principally polarized abelian variety (ppav) of dimension \(g-1\): \(P(\tilde C,C)=Ker(Nm_{\pi})^ 0\). Denote by Rg the moduli space for the pairs \((\tilde C,C)\) as above. The map P from Rg to the moduli space of ppav of dimension g-1 sending \((\tilde C,C)\) to \(P(\tilde C,C)\) is called the Prym map. \textit{A. Beauville} extended P to a proper map \(\bar P\) defined on suitable covers of stable curves. This map is known to be generically injective for \(g\geq 7\). On the other hand \textit{R. Donagi} defined a so called tetragonal construction producing counterexamples to the injectivity for all g. The tetragonal conjecture states that, essentially, this construction is the only way to produce counterexamples.
One says that a curve is bi-elliptic if it can be represented as a ramified double covering of an elliptic curve. In this paper, the fibre of the Prym map over an element \(P(\tilde C,C)\), where C is a general bi- elliptic curve of genus \(\geq 10\), is described. In particular, a new construction of non-tetragonal type is exhibited producing counterexamples to the injectivity. This new construction is the only exception to the tetragonal conjecture in the general bi-elliptic case. bi-elliptic curve; double covering of a smooth curve; Prym variety; Prym map; tetragonal conjecture J. C. Naranjo,Prym varieties of bi-elliptic curves, J. reine angew. Math.424 (1992), 47--106. Picard schemes, higher Jacobians, Jacobians, Prym varieties Prym varieties of bi-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 The ring \(G_{m,n}\) of m generic \(n\times n\) matrices over some algebraically closed field k of characteristic 0 is well-known to possess a classical division algebra of fractions \(D_{m,n}\). Since the center \(K_{m,n}\) of \(D_{m,n}\) is known to be purely transcendental over \(K_{2,n}\), the problem of (dis)proving that \(K_{m,n}\) is purely transcendental over k reduces to the case \(m=2.\)
The author presents the following geometric interpretation of \(K_{2,n}\). For any pair of positive integers (n,d), denote by \(Q_{n,d}\) the variety parametrizing the couples (C,D), where \(C\subseteq {\mathbb{P}}^ 2_ k\) is a curve of degree n and where D is a divisor of degree d of C \((Q_{n,d}\) is essentially the Picard scheme over a generic plane curve). The main result of the paper then says that \(K_{2,n}\) is the function field of \(Q_{n,n(n-1)/2}\). As an application, it is then shown that \(K_{2,3}\) is rational, by proving that the variety \(Q_{3,3}\) is rational. (The rationality of \(K_{2,n}\) has been proved by E. Formanek for \(n=2,3,4\).) generic n\(\times n\) matrices; classical division algebra of fractions; center; purely transcendental; Picard scheme; generic plane curve; function field; variety; rationality Van den Bergh M., The center of the generic division algebra, J. Algebra, 1989, 127(1), 106--126 Infinite-dimensional and general division rings, Center, normalizer (invariant elements) (associative rings and algebras), Transcendental field extensions, Algebraic functions and function fields in algebraic geometry, Picard schemes, higher Jacobians The center of the generic division 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 We show that the vanishing order of a non-zero vector field at a generic point of a smooth Fano variety of Picard number 1 cannot exceed the dimension of the Fano variety. Furthermore, if there exist only finitely many rational curves of minimal degree through a generic point of the Fano variety, we show that a non-zero vector field cannot vanish at a generic point of the Fano variety. vanishing order; vector field at a generic point of a smooth Fano variety; Picard number Hwang, J.-M.: On the vanishing orders of vector fields on Fano varieties of Picard number 1. Compos. Math. 125, 255--262 (2001) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fano varieties, Automorphisms of surfaces and higher-dimensional varieties, Vanishing theorems in algebraic geometry On the vanishing orders of vector fields on Fano varieties of Picard number 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 \(f : X \to Y\) be a surjective and projective morphism of smooth quasi-projective varieties over an algebraically closed field of characteristic zero with \(\dim f = 1\). Let \(E\) be a vector bundle of rank \(r\) on \(X\). In this paper, we would like to show that if \(X_y\) is smooth and \(E_y\) is semistable for some \(y \in Y\), then \(f_*\left( 2rc_2(E) - (r-1)c_1(E)^2 \right)\) is weakly positive at \(y\). We apply this result to obtain the following description of the cone of weakly positive \({\mathbb{Q}}\)-Cartier divisors on the moduli space of stable curves. Let \(\overline{\mathcal{M}}_g\) (resp. \(\mathcal{M}_g\)) be the moduli space of stable (resp. smooth) curves of genus \(g \geq 2\). Let \(\lambda\) be the Hodge class, and let the \(\delta_i\)'s (\(i = 0, \ldots, [g/2]\)) be the boundary classes. Then, a \({\mathbb{Q}}\)-Cartier divisor \(x \lambda + \sum_{i=0}^{[g/2]} y_i \delta_i\) on \(\overline{\mathcal{M}}_g\) is weakly positive over \(\mathcal{M}_g\) if and only if \(x \geq 0\), \(g x + (8g + 4) y_0 \geq 0\), and \(i(g-i) x + (2g+1) y_i \geq 0\) for all \(1 \leq i \leq [g/2]\). Bogomolov's inequality; moduli space of stable curve; Chern class; cone of weakly positive Cartier divisors Moriwaki, Atsushi, Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc., 0894-0347, 11, 3, 569\textendash 600 pp., (1998) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Arithmetic varieties and schemes; Arakelov theory; heights, Characteristic classes and numbers in differential topology Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0563.14021. double cover of projective 3-space branched along a sextic; surface; intermediate Jacobian; threefold; Abel-Jacobi map; Albanese variety Ceresa G. and Verra A. (1986). The Abel-Jacobi isomorphism for the sextic double solid. Pac. J. Math. 124(1): 85--105 \(3\)-folds, Picard schemes, higher Jacobians, Projective techniques in algebraic geometry The Abel-Jacobi isomorphism for the sextic double solid | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, the authors define and investigate the \textit{triangulated category of relative singularities} associated to a closed subscheme \(Z\) of a separated Noetherian scheme \(X\) with enough vector bundles, where \(\mathcal{O}_Z\) has finite flat dimension as an \(\mathcal{O}_X\)-module. It is given by the quotient of \(\text{D}^b(Z)\) by the thick subcategory generated by the image of the derived inverse image functor \(\mathbb{L}i^*: \text{D}^b(X) \to \text{D}^b(Z)\), and it is denoted by \(\text{D}^b_{\mathrm{Sing}}(Z/X)\). When \(X\) is regular, \(\text{D}^b_{\mathrm{Sing}}(Z/X)\) is precisely the singularity category of \(Z\) as defined by \textit{R.-O. Buchweitz} in [``Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings'', unpublished manuscript (1987)].
In Section 1 of the article, the authors establish some general results regarding \textit{derived categories of the second kind} (cf. [\textit{L. Positselski}, Mem. Am. Math. Soc. 996, i-iii, 133 p. (2011; Zbl 1275.18002)]) associated to curved dg-modules over curved dg-rings. These results are applied later in the article to \textit{matrix factorizations}, which may be considered as curved dg-modules over a certain curved dg algebra.
In Section 2, the authors prove what they refer to as their main result. Let \(X\) be as above, let \(\mathcal{L}\) be a line bundle on \(X\), and let \(w \in \mathcal{L}(X)\) be a section. Let \(X_0 \subseteq X\) denote the closed subscheme given by the zero locus of \(w\). Assume the morphism of sheaves \(w: \mathcal{O}_X \to \mathcal{L}\) is injective. Let \((X, \mathcal{L}, w)-\text{coh}\) denote the category of \textit{coherent matrix factorizations} of \(w\); that is, the pair of \(\mathcal{O}_X\)-modules underlying the matrix factorization is allowed to be a pair of coherent modules, rather than locally free. The authors construct an equivalence
\[
\text{D}^{\text{abs}} ((X, \mathcal{L}, w)-\text{coh})) \to \text{D}^b_{\mathrm{Sing}}(X_0/X),
\]
where \(\text{D}^{\text{abs}}(-)\) denotes a certain derived category of the second kind. When X is regular, this theorem recovers a well-known theorem of \textit{D. Orlov} (Theorem 3.5 of [Math. Ann. 353, No. 1, 95--108 (2012; Zbl 1243.81178)]).
The authors also establish, in this section, what they refer to as covariant and contravariant Serre-Grothendieck duality theorems for matrix factorizations. In Section 3, the authors give some general results on ma- trix factorizations with a support condition, and also pushforwards and pullbacks of matrix factorizations. Hochschild (co)homology of dg categories of matrix factorizations is discussed in an appendix. matrix factorizations; relative singularities of cartier divisors; triangulated categories of singularities; derived categories of the second kind; coderived categories; direct and inverse images; covariant Serre-Grothendieck; localization theory A. I. Efimov & L. Positselski, ``Coherent analogues of matrix factorizations and relative singularity categories'', Algebra Number Theory9 (2015) no. 5, p. 1159-1292 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories and commutative rings, Representation theory of associative rings and algebras Coherent analogues of matrix factorizations and relative singularity categories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0632.00015.]
Those interested in the diverse aspects (not totally divorced from analytic ones) of the arithmetic theory of polynomials ought to be greatly indepted to the author for this very important and interesting survey (with due prominence of conjectures and open questions involved). The starting point is the case of the (simplest) polynomial x and the corresponding theory represented by the classic results (on Hecke L- series) in Tate's dissertation and developments due to Weil.
Next comes the Siegel-Weil formula established by Weil in furtherance of Siegel's fundamental theorem on quadratic forms and subsequent work of Tamagawa; the author then unfolds the present state of the Siegel-Weil formula for forms of higher degree, in the wake of his own important contributions to this area. Modulo Hasse principles, the Siegel-Weil formula is nearly equivalent to a result on Tamagawa numbers (passing down to stabilizer subgroups) together with a generalized Poisson formula. A criterion for the validity of this Poisson formula is then given followed by the relevant local theory - a local zeta function, a formula of Langlands for the principal parts of its meromorphic continuation, etc.
After presenting Bernshtein's theorem concerning this zeta function and Denef's theorem on a conjecture of Serre (on the rationality of a series associated with a finite number of polynomials over \({\mathbb{Z}}_ p\) (in n variables)), the author goes on to sketch recent work (including his own) on zeta distributions attached to invariants for prehomogeneous vector spaces (over local or global fields) and finally an identification, in the 2-variable local case, of the poles of the zeta functions concerned. Poincaré series for p-adic points on a variety; arithmetic theory of polynomials; Siegel-Weil formula; forms of higher degree; generalized Poisson formula; local zeta function; Bernshtein's theorem; Denef's theorem; rationality; zeta distributions; invariants for prehomogeneous vector spaces; poles Analytic theory (Epstein zeta functions; relations with automorphic forms and functions), Research exposition (monographs, survey articles) pertaining to number theory, Zeta functions and \(L\)-functions of number fields, Research exposition (monographs, survey articles) pertaining to field theory, Forms of degree higher than two, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Some aspects of the arithmetic theory of polynomials | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves some results on p-adic K-groups of a smooth variety over a number field F. In particular he shows that for a \(curve\quad X\) of genus \(g\) over F the dimension of \(K_ 2(X,{\mathbb{Q}}_ p/{\mathbb{Z}}_ p)\) is at least g[F:\({\mathbb{Q}}]\). The proof relies on comparison with the corresponding étale K-theory and results of \textit{W. G. Dwyer} and \textit{E. M. Friedlander} [Trans. Am. Math. Soc. 292, 247-280 (1985; Zbl 0581.14012)]. Examples are given for elliptic curves over an imaginary quadratic field such that \(\dim (K_ 2(X,{\mathbb{Q}}_ p/{\mathbb{Z}}_ p))=[F:{\mathbb{Q}}]\) for primes p satisfying certain regularity conditions.
In the final section the author constructs higher p-adic regulator maps for certain elliptic curves and obtains a result on values of a p-adic L- function which can be seen as p-adic analog to results of Beilinson on the generalized conjecture of Birch and Swinnerton-Dyer. p-adic K-groups of a smooth variety; elliptic curves; higher p-adic regulator maps; values of a p-adic L-function; generalized conjecture of Birch and Swinnerton-Dyer Soulé, p-Adic K-theory of elliptic curves, Duke Math. J. 54 pp 249-- (1987) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Global ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic ground fields for curves, Elliptic curves, Special algebraic curves and curves of low genus p-adic K-theory of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves that the only solutions to \(x^p+y^q=z^r\) with gcd\((x,y,z)=1\), \(xyz\neq 0\) and \((p,q,r)\in\{(2,4,6),\;(2,6,4),\;(4,6,2),\;(2,8,3)\}\) are \((\pm 1549034)^2+ (\pm 33)^8=15613^3\). The parametrizations of primitive solutions to \(x^2+y^2=z^2\), \(x^2+y^2=z^3\) and \(x^2+y^4=z^3\) are used to derive curves \(C\) of genus 2 such that the primitive solutions to \(x^p+y^q= z^r\) correspond to ratioal points on these curves. In some cases \(C\) admits a morphism to an elliptic curve \(E\) defined over \(\mathbb{Q}\) with Mordell-Weil rank zero and so the rational points on \(C\) are determined by lifting the torsion points on \(E\) to the cover. In the other cases a version of Chabauty method for curves of genus 2 of a specific type is used to determined the rational points of \(C\). generalised Fermat equation; Jacobian variety; elliptic curve; Chabauty method; curves of genus 2 N. Bruin, The Diophantine equations \(x^{2} \pm y^{4}=\pm z^{6}\) and \(x^{2}+y^8=z^{3}\) , Compos. Math. 118 (1999), 305-321. Higher degree equations; Fermat's equation, Rational points, Elliptic curves over global fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields The diophantine equations \(x^2\pm y^4=\pm z^6\) and \(x^2+y^8=z^3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a given Galois representation in the automorphism group of the pro-\(\ell\) fundamental group of an algebraic curve, we associate a family of linear Galois-representations. These are quotients of \(l\)-adic Tate modules of generalized Jacobian varieties of certain finite coverings of the curve. We prove that the intersection of their kernels coincides with the kernel of the given Galois representation. pro-\(\ell\) fundamental group of an algebraic curve; automorphism group; linear Galois-representations; generalized Jacobian varieties; finite coverings Coverings of curves, fundamental group, Galois cohomology, Jacobians, Prym varieties On a family of linear representations associated with Galois representations in the automorphism groups of pro-\(\ell\) fundamental groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author introduces the notion of curves with singularities of coordinate axes type (TAC). A basic example of TAC singularity is given by the union of the coordinate axes in an affine space. Over an algebraically closed field, any TAC singularity is étale locally isomorphic to this example. Let \(\mathcal{X}\) be a semi-stable curve over a discrete valuation ring. A rational point on the generic fiber \(\mathcal{X}_K\) (assumed to be smooth) defines an immersion \(\mathcal{X}_K \to J_K\), where \(J_K\) is the Jacobian variety of \(\mathcal{X}_K\). However, the induced rational map \(a: \mathcal{X} \cdots \to \mathcal{N}\) (where \(\mathcal{N}\) is the Néron model of \(J_K\)) may contract an irreducible component of the special fiber of \(\mathcal{X}\). Contracting such fibers, one obtains \(a_{\min}: \mathcal{X}_{\min} \cdots \to \mathcal{N}\) through which \(a\) factors. It is proved that \(\mathcal{X}_{\min}\) has only TAC singularities. Moreover, \(a_{\min}\) is defined on the smooth locus, and the restriction of \(a_{\min}\) to the smooth locus of any irreducible component of the special fiber of \(\mathcal{X}_{\min}\) is an immersion. As an application, a result of \textit{C. Deninger} and \textit{A. Werner} [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 553--597 (2005; Zbl 1087.14026)] is proved under a milder assumption. curve singularity; semi-stable curve; Néron model of Jacobian variety Tong J.: Application d'Albanese pour les courbes et contractions. Math. Ann. 338, 405--420 (2007) Singularities of curves, local rings, Local ground fields in algebraic geometry The Albanese map for curves and contractions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 given rational parametrization \({\mathcal P}(\bar{t})\), \(\bar{t}=(t_1,\ldots,t_r)\), of a variety in \({\mathbb K}^n\), the base points are the values of \(\bar{t}\) such that all the numerators and denominators in the parametrization simultaneously vanish. The presence of base points is typically a source of difficulties in a number of geometric problems usually attacked by means of elimination methods; therefore, understanding their behavior is important. Additionally, the \textit{fibre} of a point \(P\in {\mathbb K}^n\) is the set of \(\bar{t}\)'s such that \({\mathcal P}(\bar{t})=P\). Studying the cardinality of the fibre of a generic point is also important, because it is related to the number of times that \({\mathcal P}(\bar{t})\) covers the variety, and therefore with whether or not the parametrization is ``redundant'', i.e. with the optimality, in a certain sense, of the parametrization. Both base points and fibres are addressed in the paper. The content itself is very well described in the abstract of the paper:
Given a rational parametrization \({\mathcal P}(\bar{t})\), \(\bar{t}=(t_1,\ldots,t_r)\), of an \(r\)-dimensional unirational variety, we analyze the behavior of the variety of the base points of \({\mathcal P}(\bar{t})\) in connection to its generic fibre, when successively eliminating the parameters \(t_i\). For this purpose, we introduce a sequence of of generalized resultants whose primitive and content parts contain the different components of the projected variety of the base points and the fibre. In addition, when the dimension of the base points is strictly smaller than one (as in the well-known cases of curves and surfaces), we show that the last element in the sequence of resultants is the univariate polynomial in the corresponding Gröbner basis of the ideal associated to the fibre; assuming that the ideal is in \(t_1\)-general position and radical. rational parametrization; unirational variety; degree of a rational map; fiber of a rational map; base points; generalized resultants Pérez-Díaz, S.; Sendra, J. R.: Behavior of the fiber and the base points of parametrizations under projections, Math. comput. Sci. 7, No. 2, 167-184 (2013) Computational aspects of higher-dimensional varieties, Computational aspects of algebraic surfaces, Symbolic computation and algebraic computation, Rational and birational maps Behavior of the fiber and the base points of parametrizations under projections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the \(SL_2(\mathbb C)\) character variety of the hyperbolic knot in \(S^3\). Furthermore, we prove that the corresponding \(\mathbb C^*\)-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over \(\mathbb C^*\times \mathbb C^*\). Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective. character variety; Chern-Simons invariant; hyperbolic knots; volume conjecture; regulator of a curve W. Li and Q. Wang, On the generalized volume conjecture and regulator, math.GT/0610745 . Invariants of knots and \(3\)-manifolds, Knots and links in the 3-sphere, Plane and space curves, Symbols and arithmetic (\(K\)-theoretic aspects) On the generalized volume conjecture and regulator | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraic number field, \(X\) a smooth projective curve over \(k\) and \(Y\) a regular proper flat model of \(X\) over the ring of integers of \(k\). Take an effective reduced divisor \(D\) on \(X\) and its closure \(\overline D\) on \(Y\). For any real place \(v\) of \(k\) denote by \(\pi_{0}J_{X,D}(k_{v})\) the group of connected components of the generalized Jacobian of \(X\) relative to \(D\). Assume that \((X-D)(k)\) is not empty. Then there is an exact sequence
\[
\coprod_{\text{real} v} \{ \pm \} \times \pi_{0}J_{X,D}(k_{v}) \rightarrow \pi_1^{\text{ab}}(Y-\overline D) \rightarrow CH_{}(Y,\overline D) \rightarrow 0.
\]
Here \(CH_{}(Y,\overline D)\) is the relative Chow group of 0-cycles. It is also proved that \(CH_{}(Y,\overline D)\) is finite. smooth projective curve; generalized Jacobian; relative Chow group Generalized class field theory (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Geometric class field theory Class field theory of arithmetic surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A\) be an abelian variety of dimension \(g\) over a finite field \({\mathbb F}_q\). Suppose that \(A\) is given as a closed subvariety of projective \(n\)-space. The authors exhibit a deterministic algorithm that computes the characteristic polynomial of the Frobenius endomorphism of \(A\) that runs in time \(O((\text{log} q)^c)\), where \(c\) is a polynomial expression in \(g\) as well as \(n\). This improves upon an earlier result of \textit{J. Pila} [Math. Comput. 55, 745-763 (1990; Zbl 0724.11070)], who obtained a similar result but with the constant \(c\) depending exponentially on \(n\).
By applying this to the Jacobian varieties of curves \(X\) over \({\mathbb F}_q\), one obtains a deterministic algorithm to count the number of \({\mathbb F}_q\)-rational points of \(X\) that runs in time \(O((\text{log} q)^c)\), where \(c\) is a polynomial expression in \(n\), as well as the genus \(g\) of \(X\). In the special case of hyperelliptic curves of genus \(g\), the authors show that the number of \({\mathbb F}_q\)-rational points on \(X\) can be counted deterministically in time \((\text{log} q)^{O(g^6)}\). This case is of interest in view of the primality test decribed by the authors in their monograph [Primality testing and abelian varieties over finite fields, Lect. Notes Math. 1512 (Springer-Verlag, 1992; Zbl 0744.11065)]. abelian variety over a finite field; deterministic algorithm; characteristic polynomial of the Frobenius endomorphism; Jacobian varieties; rational points; hyperelliptic curves; primality test; complexity analysis L.M. Adleman and M.-D. Huang, Counting rational points on curves and Abelian varieties over finite fields , %in Algorithmic number theory ANTS-II, Lect. Notes Comp. Sci. 1122 (1996), 1-16%, Berlin (Germany), 1996.%Springer-Verlag.
Number-theoretic algorithms; complexity, Varieties over finite and local fields, Curves over finite and local fields, Arithmetic ground fields for abelian varieties, Rational points, Finite ground fields in algebraic geometry, Computational aspects of algebraic curves, Computational aspects of higher-dimensional varieties, Arithmetic ground fields for curves Counting rational points on curves and abelian varieties over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0674.00007.]
Let X be a smooth projective variety over a finite field. Tate conjectured relations between special values of the zeta function of X and various cohomological invariants. For certain surfaces these conjectures are known to be true.
The authors present a survey on Tate's conjectures with special emphasis on projective surfaces where the Picard number, Brauer group and Néron- Severi group are the related objects. Furthermore they evaluate the corresponding special values of the zeta function for abelian surfaces, Kummer surfaces and Fermat surfaces. variety over a finite field; Tate's conjectures; Picard number; Brauer group; Néron-Severi group; special values of the zeta function Suwa, Noriyuki and Yui, Noriko, Arithmetic of certain algebraic surfaces over finite fields, Number Theory ({N}ew {Y}ork, 1985/1988), Lecture Notes in Math., 1383, 186-256, (1989), Springer, Berlin Finite ground fields in algebraic geometry, Picard groups, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Arithmetic of certain algebraic surfaces over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with some aspects of the theory of algebraic systems of effective divisors on a non singular curve and of one dimensional systems of curves on a smooth surface. In the case of curves, defined over an algebraically closed field, a short proof is given of a remarkable result due to Allen, which extends to non-linear systems of effective divisors the well known formula of De Jonquières. As an application of Allen's theorem a criterion of linear equivalence by Castelnuovo and Torelli is derived; this, in turn, implies a well known inequality by Castelnuovo and Severi concerning correspondences on a curve. Turning to surfaces and assuming the base field of characteristic zero, a formula is proved, giving a bound for the number of curves in a one dimensional system of generically smooth curves having a singular point. This bound was already given, but with incomplete proof, by Torelli. From this result a criterion of linear equivalence, analogous to the aforementioned one of Castelnuovo and Torelli for curves, is deduced. algebraic systems of effective divisors; correspondences on a curve C. Ciliberto--F. Ghione,Serie algebriche di divisori su una curva e su una superficie, Ann. Mat. Pura Appl.,136 (1984), pp. 329--353. Divisors, linear systems, invertible sheaves, Rational and birational maps Algebraic series of divisors on a curve and on a surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This work is devoted to explain the relationship between invariants appearing in different subjects in mathematics. The first field are the modular invariants of the partition function of a rational conformal field theory (RCFT) on a torus, specially in the case of \(su(3)\) models.
In the second section, the authors discuss modular invariance for \(su(3)\) theories. The partition function of an RCFT of height \(n\) on a torus \({\mathbb{C}}/({\mathbb{Z}}+\tau{\mathbb{Z}})\), \({\mathfrak I}\tau>0\), is determined by a matrix of non-negative integers \(N:=\{N_{p,p'}\}_{p,p'\in B_n}\), where
\[
B_n:=\{p:=(r,s,t)\in{\mathbb{Z}}^3\mid r,s,t\geq 1, r+s+t=n\}.
\]
The modular action of \(PSL_2({\mathbb{Z}})\) on \(\{{\mathfrak I}\tau>0\}\) induces two matrices \(S\), \(T\) which reflect the isomorphism of the corresponding elliptic curves. A function determined by a matrix \(N\) is a modular invariant if and only if \(N\) commutes with \(S\) and \(T\). The matrices \(S\) and \(T\) are rational combinations of \(3n\)-roots of unity and Galois theory is involved [see \textit{A. Coste} and \textit{T. Gannon}, Phys. Lett. B 323, 316-321 (1994)]. This theory produces a parity selection rule which makes the computation of modular invariants easier.
In the third section, the authors consider the Jacobian of a Fermat curve \(x^n+y^n=z^n\). The first coincidence with previous results is that a basis of holomorphic differentials is parametrized in a natural way by \(B_n\); in fact the Jacobian is isogenous to a product of Abelian varieties also indexed by \(B_n\). These factors possess complex multiplication (CM) and the general theory for simpleness of abelian varieties with CM may be applied, see the Shimura-Taniyama theorem [\textit{G. Shimura} and \textit{Y. Taniyama}, ``Complex multiplication of abelian varieties and its application to number theory'', Publ. Math. Soc. Jap. 6 (1961; Zbl 0112.03502)].
In the fourth section, combinatorial groups for triangulated surfaces are studied, following Grothendieck \textit{dessins d'enfants}. The starting point is the standard triangulation of the Riemann sphere with vertices \(0,1,\infty\); if \(h\) is a meromorphic map from \(\Sigma\) ramified at \(0,1,\infty\), the standard triangulation induces a special one of \(\Sigma\) which may be encoded by the cartographic group. The universal cartographic group is the modular group and any subgroup of finite index of the uniformizing group \(\Gamma_2\) determines up to isomorphism a pair \((\Sigma,h)\). The Kummer projection defines a cartographic group for Fermat curves which is also related with \(B_n\). A rational triangular billiard associated to a triangle of angles \(\pi/r\), \(\pi/s\), \(\pi/t\) with \(r+s+t=n\) is the classical phase space of a particle moving in the corresponding orbifold. The trajectories determine a closed Riemann surface \(C_{r,s,t}\) (called triangular curve) which projects onto the triangle and produce a cartographic group. It may happen that a Fermat curve projects onto a triangular curve.
In the fifth section, the authors study the Riemann surface of a RFCT on a torus. The goal is to show that a compact Riemann surface can be associated with any RCFT. The authors show for example that the curve associated with \(su(3)_1\) is a triangular curve admitting as covering the Fermat curve of degree \(12\).
The paper finishes with some conclusions and questions about the relationships stated in the paper and two appendices. affine Lie algebras; abelian varieties; modular invariant; partition function; rational conformal field theory; Jacobian of a Fermat curve; triangulated surfaces; Riemann surface M. Bauer, A. Coste, C. Itzykson and P. Ruelle, ''Comments on the links between SU(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards,'' J. Geom. Phys. 22 (1997), 134--189. Jacobians, Prym varieties, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Complex multiplication and abelian varieties Comments on the links between \(su(3)\) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an absolutely irreducible, smooth projective curve over a field \(F\); let \(K:=F(X)\) be its function field and \({\mathcal O}_ X\) its structural sheaf. The authors define (for all \(r\geq 1)\) an \(r\)-divisor, \(D\), as a rank \(r\) \({\mathcal O}_ X\)-submodule of the constant sheaf \(K^ r\); a rank \(r\) vector bundle on \(X\) is associated to each such \(D\) and the set of all \(r\)-divisors is partitioned according to the isomorphism classes of the corresponding vector bundles. Counting all the cardinalities related to the partition classes, in this very well written paper the authors give a new proof of the Siegel formula.
In an appendix they sketch the original proof [due independently to the teams \textit{U. V. Desale} and \textit{S. Ramanan}, Invent. Math. 38, 161-185 (1976; Zbl 0323.14012) and \textit{G. Harder} and \textit{M. S. Narasimhan}, Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] of this formula (shown to be equivalent to the classical (i.e. of A. Weil) fact that the Tamagawa number of \(SL(r,K)\) is 1) and to its original application to the computation, via the just proven Weil conjectures, of the Betti numbers of the moduli spaces of stable vector bundles on \(X\). divisors of higher rank on a curve; \(r\)-divisor; Tamagawa number; Betti numbers of the moduli spaces of stable vector bundles Ghione F and Letizia M, Effective divisors of higher rank on a curve and the Siegel formula,Composite Math. 83 (1992) 147--159 Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli Effective divisors of higher rank on a curve and the Siegel 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 Consider a Stein complex space X, a reductive connected complex group G acting holomorphically on X and let X//G denote the categorical quotient (which has a structure of Stein space). The aim of this paper is to prove that the natural map Pic(X//G)\(\to Pic(X)\) is injective in each of the following situations: \((1)\quad G\quad is\) semisimple; and \((2)\quad X\quad is\) smooth and the radical of G has a fixed point on X. Picard group of a complex quotient variety Picard groups, Homogeneous spaces and generalizations, Homogeneous complex manifolds, Group actions on varieties or schemes (quotients) The Picard group of a complex quotient variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper is defined the height of a moduli variety of semi-stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}})\) and a precise description of moduli varieties of height 0 is given. With this description one can prove for example that given such a variety M, there are infinitely many other non trivial moduli varieties isomorphic to M. These results are proved with the help of the generalized Beilinson spectral sequence, built with exceptional bundles. One can find, for every integer d, all the moduli varieties which are of dimension d. height of a moduli variety of semi-stable sheaves; generalized Beilinson spectral sequence; exceptional bundles J.-M. Drézet, Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur \(\mb{P}_2(\mb{C})\), J. Reine Angew. Math. 380 (1987), 14--58. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Fibrés exceptionnels et variétés de modules de faisceaux semi- stables sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Exceptional bundles and moduli varieties of semi-stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author constructs the local Jacobian of a relative formal curve and proves a relative duality formula.
Let \({\mathcal F}\) be the \(S\)-group extension of the completion \(\check W\), of the universal \(S\)-Witt vector group \(W\), by the group of units \({\mathcal O}_S [[T]]^*\).
The author proves the following theorem: Let \(S= \text{Spec} (A)\) be a noetherian affine scheme. Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Let \(B\) be an \(A\)-algebra adic isomorphic to \( A[[T]]\), \({\mathcal X} = \text{Spf} (B)\), \({\mathcal U}= \text{Spec} (A[[T]][T^{-1}])\). In the case then for any section \(\sigma \in G({\mathcal U})\) there exists unique homomorphism \( h: {\mathcal F} \to G\) such that \(\sigma = h_{\text{omb}} \circ f\) where \(f: {\mathcal U} \rightarrow {\mathcal F}_{\text{omb}}\) is an Abel-Jacobi morphism of \(({\mathcal X}, {\mathcal F})\), i.e. the arrow \(\mathrm{Hom}_{S\text{-gr}} ({\mathcal F},G) \rightarrow G({\mathcal U})\) is bijective.
The proof of the theorem is reduced to the proof of the theorem 1.4.4 (see below).
This paper is the next in a sequence of papers by the author in which he is involved with the Grothendieck program concerning global and local dualities with continuous coefficients. The relative generalized Jacobian of the smooth curve \(X - D\) and an Abel-Jacobi morphism \(X - D \rightarrow J\) are constructed in the author's papers [C. R. Acad. Sci., Paris, Sér. A 289, 203--206 (1979; Zbl 0447.14005); Prog. Math. 87, 69--109 (1990; Zbl 0752.14023)].
Let \({\mathfrak X}\) be the formal completion of \(X\) along \(D\), \(\text{omb}({\mathfrak X}) = \text{Spec} (\Gamma({\mathfrak X}, {\mathcal O}_{\mathfrak X}) = \text{Spec} (A[[T]])\).
Let \(\rho: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}^{0}, G) \to F(G)\) and \(\rho^{+}: \mathrm{Hom}_{S\text{-gr}} ({\mathcal F}, G) \to F^{+} (G)\).
Theorem 1.4.4. The notations are taken above. Let \(A\) be a Noetherian ring and \(S = \text{Spec} (A)\). Let \(G\) be any commutative, smooth and separated \(S\)-group scheme. Then \(\rho \) and \(\rho^{+}\) are isomorphisms.
The proof of the theorem is given in sections 2--5.
This interesting article is done (is presented) by the author in the spirit of the algebraic geometry by Grothendieck, Verdier, Artin, Deligne, Saint-Donat [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)], by Grothendieck and Demazure [Schémas en groupes. I: Propriétés générales des schémas en groupes. Exposés I à VIIb. Séminaire de Géométrie Algébrique 1962/64, dirigé par Michel Demazure et Alexander Grothendieck. Revised reprint. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0207.51401)] together with some new developments by \textit{A. Beilinson} [Fields Institute Communications 56, 15--82 (2009; Zbl 1186.14019)], by \textit{K. Rülling} [J. Algebr. Geom. 16, No. 1, 109--169 (2007; Zbl 1122.14006)], by \textit{M. Kapranov} and \textit{É. Vasserot} [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113--133 (2007; Zbl 1129.14022)] and by \textit{A. N. Parshin} [in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 362--392 (2011; Zbl 1266.11118)].
The paper under review closes with some results on the autoduality of \({\mathcal F}\) in the sense of Cartier. local symbol(s); tame symbol; Contou-Carrère symbol; local Abel- Jacobi morphism; universal Witt bivectors; cartier duality; local Jacobian; relative formal curve; Witt residues; local relative class field theory; Rosenlicht Jacobian Contou-Carrère, C., Rend. Semin. Mat. Univ. Padova, 130, 1-106, (2013) Group schemes, Symbols and arithmetic (\(K\)-theoretic aspects) Local Jacobian of a relative formal curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A_{g}\) be the moduli space of principally polarized abelian varieties over \({\mathbb C}\) of dimension \(g,\) and let \(M_{g}\) be the moduli space of smooth algebraically irreducible curves of genus \(g\) over \({\mathbb C}\). There exists a morphism of schemes \(J_{g} : M_{g} \rightarrow A_{g}\) called the Jacobi map which is defined by \(J_{g}(C) = (J(C), W_{g-1}),\) where \(J(C)\) is the Jacobian of the curve \(C\) and \(W_{g-1}\) is its theta divisor. One can define the Jacobian locus \(J_{g}\) in dimension \(g\) as the image of the Jacobi map, and the closed Jacobian locus \(\overline{J_{g}}\) as the closure of \(J_{g}\) in \(A_{g}.\) One has that \(\dim \overline{J_{g}} = 3g - 3\) for \(g > 1\), and \(\dim A_{g} = g(g+1)/2.\) From a general point of view could state the Schottky problem as the problem of characterize \(\overline{J_{g}}\) as a subscheme of \(A_{g}.\) There are several approaches to this problem in terms of geometric properties. Characterizing \(\overline{J_{g}}\) in terms of algebraic equations goes back to the original paper of Schottky-Jung and there have been proved partial results by R. Donagi and B. van Geemen. The problem of characterizing \(\overline{J_{g}}\) in terms of differential equation (the Novikov's conjecture) was solved by T. Shiota and M. Mulase. This approach is related to the geometric characterization Jacobian in terms of the existence of trisecant lines in its Kummer variety (R. Gunning. G. Welters and I. Krichever).
In this paper is to expose the approaches to the Schottky problem more directly related to the recent results proved by I. Krichever and S. Grushevsky. That is authors gave expose the approaches related with the existence of trisecants, the KP equation and the \(\Gamma_{00}\)-conjecture. moduli space of principally polarized abelian varieties; moduli space of smooth algebraically irreducible curves; Jacobi map; Jacobian of the curve; the Novikov's conjecture; trisecant lines; Kummer variety Theta functions and curves; Schottky problem, Theta functions and abelian varieties, Jacobians, Prym varieties Survey on the Schottky problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We construct an admissible pairing of divisors on a curve defined over a non-archimedean field, as an analogue of Arakelov's pairing on a Riemann surface.
For an algebraic curve \(C\) with a given symmetric metric \(|\cdot|_ \Delta\) on \(O(\Delta)\) on \(C\times C\), one can define a pairing \((\;,\;)\) on \(\text{Div}(C)\) such that \((x,y)=-\log| 1|_ \Delta (x,y)\). For a Riemann surface of positive genus, Arakelov constructed a norm \(|\cdot|_ \Delta\) such that the induced pairing extends the Néron local pairing of \(\text{Div}^ 0(C)\), and satisfies certain adjunction formula and certain normalization condition by integration. For a curve defined over a discrete valuation field \(K\), we construct a similar metric \(|\cdot|_ \Delta\) on \(O(\Delta)\). From semistable models, one has a canonical norm \(|\cdot|_ \Delta\) on \(O(\Delta)\) such that \(i(x,y)= -\log |1|_ \Delta (x,y)\) is the normalized intersection number of sections extending \(x\), \(y\) on some semistable model of \(C\). If \(C\) has potentially good reduction, then \(|\cdot|_ \Delta\) satisfies all requirements. In general we need to multiply a term \(\exp(- g_ \mu(x,y))\), where \(\mu\) is certain metric on the reduction graph \(R(C)\) of \(C\) and \(g_ \mu\) is the associated Green's function. In \S1 we define \(i(x,y)\). In \S2 we study intersection theory via \(R(C)\). In \S3 we find the admissible metrics on the metrized graph. In \S4 we define admissible pairings and prove all required properties.
In \S5 we define some applications to curves defined over global fields. For a curve \(C\) defined over a global field \(K\), the local admissible pairings gives a global admissible pairing for divisors on \(C\). We have a relative dualizing sheaf \(\omega_ a\), a Riemann-Roch formula, an adjunction formula, and an index theorem. Let \(\omega_{Ar}\) denote the Arakelov dualizing sheaf, then we have the estimate:
\[
(\omega_{Ar}, \omega_{Ar})\geq (\omega_ a,\omega_ a) \geq 0.
\]
The first equality holds if and only if \(C\) is an elliptic curve, or a curve which has potentially good reduction at all non-archimedean places. The second equality holds if and only if there is a sequence \(\{x_ n\}\) of distinct algebraic points such that the Néron-Tate heights of \((2g-2) x_ n-\omega\) converges to 0. We also prove the Bogomolov conjecture for the embeddings \(j_ D: C\to J(C)\) which takes \(x\) to \(x-D\), where \(D\) is a divisor of \(C\) of degree 1 such that \((2g-2)D-\omega\) is not a torsion divisor.
When \(C\) has potentially good reduction, partial results are obtained by \textit{L. Szpiro} in The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 229-246 (1990; Zbl 0759.14018) and by the author in Ann. Math., II. Ser. 136, No. 3, 569-587 (1992; Zbl 0788.14017). curve over non-archimedean field; curve over a global field; admissible pairing of divisors on a curve; intersection theory; admissible metrics; global admissible pairing; Arakelov dualizing sheaf; Bogomolov conjecture S. Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), 171-193. Arithmetic ground fields for curves, Arithmetic varieties and schemes; Arakelov theory; heights, Local ground fields in algebraic geometry, Global ground fields in algebraic geometry Admissible pairing on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main object of the paper under review is a system of equations
\[
f_0(\xi )=\dots =f_n(\xi)=0
\]
where \(f_i\) are Laurent polynomials in \(n\) variables \(t_1,\dots ,t_n\) with coefficients in \(K[s]\), \(K\) being an algebraically closed field. The authors are interested in the number of isolated solutions \(\xi\in K\times (K^*)^n\) of this system. A classical theorem of \textit{D.~N.~Bernštein} [Funk. Anal. Prilozh. 9, No. 3, 1--4 (1975; Zbl 0328.32001)] and \textit{A.~G.~Kušnirenko} [Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)] provides an upper estimate for the number of those isolated points (counted with multiplicities) lying in the torus \((K^*)^{n+1}\) in terms of the mixed volume of \((n+1)\) Newton polytopes corresponding to the \(f_i\)'s regarded as Laurent polynomials in \((n+1)\) variables \(s, t_1, \dots , t_n\).
The authors' goal is to improve on this estimate. To this end, they introduce refined combinatorial invariants associated to the system under consideration. For each Laurent polynomial \(f\) as above and each \(v\in {\mathbb P}^1\) they define the \(v\)-adic Newton polytope of \(f\) (lying in \({\mathbb R}^{n+1}\)) and the roof function of this polytope above the usual Newton polytope (lying in \({\mathbb R}^n\)); such a roof function also appears in tropical geometry. Furthermore, for any family of \((n+1)\) concave real functions defined on convex sets in \({\mathbb R}^n\) they define the mixed integral using the operation of sup-convolution (this is a generalization of the mixed volume of convex bodies, more details can be found in their earlier paper [J. Inst. Math. Jussieu 7, No. 2, 327--373 (2008; Zbl 1147.11033)]).
For a system given by primitive (i.e. not having constant factors in \(K[s]\)) \(f_i\)'s, the main result of the paper (Theorem 1.2) states that the number of isolated solutions (counted with multiplicities) does not exceed the sum over all \(v\in {\mathbb P}^1\) of the mixed integrals of the roof functions of the \(v\)-adic Newton polytopes corresponding to the \(f_i\)'s. Moreover, the estimate becomes the equality for generic Laurent polynomials.
This result can be extended to nonprimitive Laurent polynomials as well. Moreover, a geometric look at Theorem 1.2 as at a statement on a toric scheme over the projective line allows the authors to generalize it to toric schemes over an arbitrary smooth projective curve \(S\) (Theorem 1.5) which can be applied to systems of equations over a semiabelian variety. Proofs are based on intersection theory in \(S\times (K^*)^n\) which requires thorough study of toric varieties over curves (note that such varieties were earlier studied by \textit{A.~L.~Smirnov} [St. Petersburg Math. J. 8, No. 4, 651--659 (1997; Zbl 0883.14030)]).
The authors provide many concrete examples illustrating the strength of their results. In particular, for each positive integer \(k\) they exhibit polynomials in two variables \(f\) and \(g\) of degree \(2k\) which have exactly one common zero in \(K\times K^*\). Their estimates predicts value 1 for the number of common zeros whereas the standard and bihomogeneous Bézout theorems give \(4k^2\) and \(8k\), respectively, and the Bernštein--Kušnirenko estimate gives the value \(4k+1\). system of polynomial equations; Newton polytope; sup-convolution; mixed integral; toric variety over a curve; mixed degree; Chow weight Philippon, A refinement of the Bernštein-Kušnirenko estimate, Adv. Math. 218 pp 1370-- (2008) Toric varieties, Newton polyhedra, Okounkov bodies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Heights, Arithmetic varieties and schemes; Arakelov theory; heights, General convexity, Power series, series of functions of several complex variables A refinement of the Bernštein-Kušnirenko estimate | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski. The main tool is the set of elements called maximals, especially the absolute and the relative ones. First, we describe the semigroup in terms of the relative maximals and these ones in terms of the absolute maximals by means of a symmetry property which generalizes the well known property of symmetry for the singularities with only one branch. Then the absolute maximals are described in terms of the theory of maximal contact of higher genus developed by Lejeune. semigroup of values of a plane curve singularity with several branches; relative maximals; absolute maximals; maximal contact of higher genus; equisingularity type Félix Delgado de la Mata, ``The semigroup of values of a curve singularity with several branches'', Manuscr. Math.59 (1987) no. 3, p. 347-374 Singularities of curves, local rings, Riemann surfaces; Weierstrass points; gap sequences, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The semigroup of values of a curve singularity with several branches | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0676.00005.]
In this survey article the author reports some explicit and interesting formulae describing the group law on the Jacobian of a curve of genus 2, which are defined over a general ground field of characteristic zero. The author expects that these formulae would be a nice tool to calculate the Mordell-Weil rank of the genus two curve. group law on the Jacobian of a curve of genus 2; Mordell-Weil rank -, Arithmetic of curves of genus \( 2\), Number Theory and Applications , Kluwer, 1989. Jacobians, Prym varieties, Arithmetic varieties and schemes; Arakelov theory; heights Arithmetic of curves of genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The discrete logarithm problem in the group of rational points of an elliptic curve over a finite field is considered to be harder than its traditional multiplicative counterpart. However, there are a few cases where the elliptic problem can can be reduced to the multiplicative one. It was pointed out by \textit{A. I. Menezes}, \textit{T. Okamoto} and \textit{S. A. Vanstone} [IEEE Trans. Inf. Theory 39, 1639--1646 (1993; Zbl 0801.94011)], that this is the case for supersingular curves. Their method exploits the Weil-pairing. Later \textit{G. Frey} and \textit{H.-G. Rück} [Math. Comput. 62, 865--874 (1994; Zbl 0813.14045)] employed the Tate-pairing in certain other cases.
In the present paper the authors discuss practical implementations of both algorithms and compare their running times. elliptic curves; cryptosystems; discrete logarithm; group of rational points; elliptic curve over a finite field; practical implementations; algorithms; running times R. Harasawa, J. Shikata, J. Suzuki, H. Imai, Comparing the MOV and FR reductions in elliptic curve cryptography, in: Advances in Cryptology--Eurocrypt '99, Lecture Notes in Computer Science, Vol. 1592, Springer, Berlin, 1999, pp. 190--205. Cryptography, Applications to coding theory and cryptography of arithmetic geometry Comparing the MOV and FR reductions in elliptic curve cryptography | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(C: y^2=f(x)\) be a curve of genus \(2\) defined over a number field \(F\), and let \(\mathrm{Jac}(C)\) be its Jacobian variety. Consider the quotient \(\mathrm{Jac}(C)/\pm 1\), and the minimal desingularization of the twists of \(\mathrm{Jac}(C)/\pm 1\).
The paper under review studies the twists of \(\mathrm{Jac}(C)/\pm 1\) building on the previous work by the author and \textit{van Luijk} [Math. Comp. 78, 441-483 (2009)]. The main result is formulated for a specific curve \(C\) and twists of \(\mathrm{Jac}(C)/\pm 1\) as follows.
Theorem. Let \(f(x)=x^6-3x^5+3x^4-3x^2-x-1\), let \(A_f={\mathbb{Q}}[x]/(f(x))\), let \(C\) be the curve \(y^2=f(x)\), and let \(\delta\in A_f\) be the image of \(x^5-7x^4+4x^3+3x^2+x\). Then the Brauer-Manin obstruction blocks the existence of rational points on the surface \(V_{f,\delta}\), a twist of the Kummer surface of \(C\). Here \(V_{f,\delta}\) is defined explicitly by three quadrics in \({\mathbb{P}}^5\).
The result has no direct relevance to \(2\)-descent. However, the theorem provides an example of a Brauer-Manin obstruction in an arithmetically ``generic'' situation in which the degree of the field of definition of generators of the Néron-Severi group is as large as possible. The author conjectures that there are infinitely many twists for which the Brauer-Manin obstruction blocks the existence of rational points on the twists. Kummer surface; Jacobian of genus 2 curve; twist; Brauer-Manin obstruction \(K3\) surfaces and Enriques surfaces, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points The two faces of the twisted Kummer 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 First we describe the Weierstrass semigroups on a plane curve of degree \(\leqslant 6\). Using this description we determine the Weierstrass semigroups at a ramification point and a branch point on a double covering from a plane curve of degree \(\leqslant 6\). In the case of a double covering from a plane curve of degree 7 we determine all the Weierstrass semigroups at branch points. Weierstrass point; Weierstrass semigroup; smooth plane curve; double covering of a curve Komeda, J.; Kim, S.J., The Weierstrass semigroups on the quotient curve of a plane curve of degree \(###\)7 by an involution, J. Algebra, 322, 137-152, (2009) Special divisors on curves (gonality, Brill-Noether theory) The Weierstrass semigroups on the quotient curve of a plane curve of degree \(\leqslant 7\) by an involution | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complex projective manifold that is uniruled, i.e., there exists a family of rational curves \(f: \mathbb P^1 \rightarrow X\) that dominates \(X\). For a given polarization a minimal rational curve is a rational curve that belongs to such a dominating family and whose degree with respect to the polarization is minimal. If we fix a general point \(x \in X\), the minimal rational curves passing through \(x\) are parametrized by a closed subset \(\mathcal K_x\) of the Hilbert scheme. There is a natural rational map \(\mathcal K_x \dashrightarrow \mathbb P(T_{X,x})\) which associates to a curve its tangent direction at the marked point \(x\). The variety of minimal rational tangents \(\mathcal C_x\) is defined as the strict transform of \(\mathcal K_x\) under this rational map. The geometric study of varieties of minimal rational tangents (VMRTs) is guided by the idea that one should be able to recover the geometry of \(X\) from \(\mathcal C_x\), at least when \(X\) is a Fano manifold with Picard number one.
The main result of this paper confirms this idea for certain model spaces. More precisely let \(S\) be a rational homogeneous manifold with Picard number one which is either a Hermitian symmetric space or a Fano contact manifold, and denote by \(\mathcal C_0 \subset \mathbb P(T_{S,0})\) its VRMT at a point \(0 \in S\). Let \(X\) be a Fano manifold with Picard number one, and denote by \(\mathcal C_x \subset \mathbb P(T_{X,x})\) its VMRT at a \textit{general} point \(x \in X\). If \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic as projective subvarieties, then \(S\) is isomorphic to \(X\). This result generalizes a famous characterization of the projective space by \textit{K. Cho, Y. Miyaoka} and \textit{N. Shepherd-Barron} [Adv. Stud. Pure Math. 35, 1--88 (2002; Zbl 1063.14065)]. It also generalizes an earlier result due to \textit{J.-M. Hwang} and the author [J. Reine Angew. Math. 490, 55--64 (1997; Zbl 0882.22007)] where they made the stronger assumption that \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic for \textit{every} point \(x \in X\). The main difficulty of the proof is that a priori there might exist a divisor \(H \subset X\) such that for \(x \in H\), the VMRT \(\mathcal C_x\) is not isomorphic to \(\mathcal C_0\). In order to exclude this possibility the author uses a notion of parallel transport along the tautological lifting of a standard minimal rational curve, a concept that already appeared implicitly in \textit{N. Mok} [Trans. Am. Math. Soc. 354, No. 7, 2639--2658 (2002; Zbl 0998.32013)]. minimal rational curve; variety of minimal rational tangents; analytic continuation Mok, N.: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. In: \textit{Third International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt. 1, vol. 2, A. Math. Soc., Providence, RI}, pp. 41-61 (2008) Fano varieties, Homogeneous spaces and generalizations, Rational and unirational varieties, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is well known that the Prym variety of an étale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree \(n\) of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. It is a natural question whether this is true for arbitrary degrees. We show that this is not the case by computing the degree of the isogeny for \(n\) a power of 2. Furthermore, we compute the degree of a closely related isogeny for arbitrary \(n\). cyclic covering of a hyperelliptic curve; isogenous to the product of two Jacobians Jacobians, Prym varieties, Coverings of curves, fundamental group Prym varieties of étale covers of hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We construct natural relative compactifications for the relative Jacobian over a family \(X/S\) of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf \(F\) of simple, torsion-free, rank-1 sheaves on \(X/S\), and showing that certain open subsheaves of \(F\) have the completeness property. Strictly speaking, the functor \(F\) is only representable by an algebraic space, but we show that \(F\) is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones. compactifications for the relative Jacobian; Poincaré sheaf; étale base change; theta functions; family of reduced curves Busonero, S.: Compactified Picard schemes and Abel maps for singular curves, PhD thesis, Sapienza Università di Roma (2008) Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties, Vector bundles on curves and their moduli Compactifying the relative Jacobian over families of reduced curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We compute the multiplicity of the discriminant of a line bundle \({\mathcal L}\) over a nonsingular variety \(S\) at a given section \(X\), in terms of the Chern classes of \({\mathcal L}\) and of the cotangent bundle of \(S\), and the Segre classes of the jacobian scheme of \(X\) in \(S\). For \(S\) a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities of \(X_{\text{red}}\). Also, under certain hypotheses, we provide formulas for the ``higher discriminants'' that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be ``small''. higher discriminants; multiplicity of the discriminant of a line bundle; Chern classes; cotangent bundle; Segre classes of the jacobian scheme P. Aluffi and F. Cukierman, Manuscripta Math., 78, 245--258 (1993); M. Chardin, J. Pure Appl. Algebra, 101, 129--138 (1995); L. Ducos, J. Pure Appl. Algebra, 145, 149--163 (2000); L. Busé and C. D'Andrea, C. R. Math. Acad. Sci. Paris, 338, 287--290 (2004); C. D' Andrea, T. Krick, and A. Szanto, J. Algebra, 302, 16--36 (2006); arXiv:math/0501281v3 (2005). Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Picard schemes, higher Jacobians Multiplicities of discriminants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 gives an answer to the following question posed by \textit{W. Fulton}, \textit{S. Kleiman} and \textit{R. MacPherson} [Algebraic Geometry --- Open Problems, Proc. Conf., Ravello 1982, Lect. Notes Math. 997, 156-196 (1983; Zbl 0529.14030)]: Are there integral varieties \(A\), \(A'\subset\mathbb{P}^ n\) (possibly \(A=A')\) of the same dimension, and an irreducible subset \(E\subset A\times A'\) of dimension \(2\dim A-1\), such that for all \((x,y)\in E\) with \(T_ xA\neq T_ yA'\), the intersection \(T_ xA\cap T_ yA'\) contains the line \([x;y]\), but \(A\cup A'\) does not contain \([x;y]\)?
Over an algebraically closed field of characteristic two the author exhibits examples with \(\dim A\) even and \(A=A'\) resp. \(A\neq A'\). He also shows that for ordinary varieties \(A\), \(A'\), those examples are essentially unique. Finally, he obtains a qualitative bound for a result of \textit{A. Hefez} and \textit{S. L. Kleiman} [Geometry Today, Int. Conf., Roma 1984 Prog. Math. 60, 143-183 (1985; Zbl 0579.14047)] about the non- vanishing of the ranks of a projective variety which improves the bound given by \textit{A. Holme} [Manuscr. Math. 61, No. 2, 145-162 (1988; Zbl 0657.14033)]. bitangent; dual of projective varieties; characteristic 2; ordinary varieties; rank of a projective variety Ballico E.: On the dual of projective varieties. Canad. Math. Bull. 34, 433--439 (1991) Projective techniques in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Finite ground fields in algebraic geometry On the dual of projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give generalizations of various classical results on the normal generation and presentation of line bundles on a smooth curve X of genus g. Concerning normal generation, the main result is the following characterization of hyperelliptic curves. Suppose that no line bundle L on X of degree 2g or 2g-1 defines a projectively normal embedding. Then under mild assumptions on the genus g, X is hyperelliptic. Conversely, if X is hyperelliptic then no line bundle of degree \(\leq 2g\) is normally generated. The authors also obtain somewhat less precise results for line bundles of lower degree. As to normal presentation, the authors show that if \(d\geq (3g+4+\sqrt{8g+1})/2\) then a general line bundle of degree d on a general curve of genus g defines an embedding in which the homogeneous ideal of X is generated by quadrics. For the most part the techniques used are those introduced by \textit{D. Mumford} in C.I.M.E. \(3\circ\) Ciclo Varenna 1969, Quest. algebr. Varieties, 29-100 (1970; Zbl 0198.258)].
[We remark that \textit{M. Green} and the reviewer have recently obtained a more complete picture for normal generation [Invent. Math. 83, 73--90 (1986; Zbl 0594.14010)].] normal generation of line bundles; normal presentation of line bundles on a smooth curve; hyperelliptic curves; degree; embedding Lange H., Martens G.: Normal generation and presentation of line bundles of low degree on curves. J. Reine. Angew 356, 1--18 (1985) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus, Embeddings in algebraic geometry Normal generation and presentation of line bundles of low degree on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Grothendieck ring \(K(x)\) for complete, nonsingular toric varieties (Demazure models) \(X\) is studied. At the beginning of the work, an explicit description of \(K(x)\) is given in combinatorial terms of the corresponding fan \(\Sigma(x)\) over an algebraically closed ground field. After that, the author studies the action of the Galois group on the ring \(K(x)\), if \(X\) is a smooth complete variety over a nonclosed field, which is a form of some Demazure model. Demazure models; fan; action of the Galois group; complete variety over a nonclosed field Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients), \(K\)-theory of schemes, Other nonalgebraically closed ground fields in algebraic geometry \(K\)-theory of Demazure models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the following result: Suppose \(X \subseteq \mathbb{P}^ n\) is an integral nondegenerate local complete intersection variety such that \(\deg X > 2\) and \(h^ 0(X, {\mathcal N}_ X (-1)) \leq n + 1\). If \(X\) is a hyperplane section of a local complete intersection variety \(W \subseteq \mathbb{P}^{n + 1}\), then \(W\) is a cone with base \(X\). The author mentions that the assumption on the normal bundle \({\mathcal N}_ X\) of \(X\) seems to be extremely restrictive, and that finding a nontrivial class of examples would be nice. Then he proceeds to give a weak version of his theorem in the case in which the ambient variety is a Grassmannian. hyperplane section of a local complete intersection variety; cone; normal bundle; Grassmannian Projective techniques in algebraic geometry, Complete intersections, Grassmannians, Schubert varieties, flag manifolds, Surfaces and higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On varieties as hyperplane sections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper contains some of the intermediate results from Hrushovski's model theoretic proof of the Manin-Mumford Conjecture which are used in Pillay's presentation of the proof in the same volume. Some general properties of groups definable in a model of the theory ACFA, the model companion of the theory of fields with an automorphism, are discussed. One of the main results is as follows. Let \((K,\sigma)\) be a model of ACFA of characteristic 0. Let \(A\) be an abelian variety defined in \((K,\sigma)\) over a field \(k\), and the automorphism \(\sigma\) fixes \(k\). If \(p[T]\) is a nonzero polynomial over the integers then \(\text{ ker}(p(\sigma))\), the kernel of the endomorphism \(p(\sigma)\) of the group \(A\), is stable and 1-based iff \(p[T]\) has no roots of unity among its roots. difference field; abelian variety; theory ACFA; group definable in a model; model theoretic stability; 1-basedness; Manin-Mumford conjecture; model companion of the theory of fields with an automorphism Z. Chatzidakis, ''Groups definable in ACFA,'' in Algebraic Model Theory, Dordrecht: Kluwer Acad. Publ., 1997, vol. 496, pp. 25-52. Model-theoretic algebra, Applications of logic to group theory, Model theory of fields, Difference algebra, Classification theory, stability, and related concepts in model theory, Abelian varieties of dimension \(> 1\), Rational points Groups definable in ACFA | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 extend the theory of \textit{G. P. Pirola} and \textit{F. Zucconi} [J. Algebr. Geom. 12, No. 3, 535--572 (2003; Zbl 1083.14515)]. We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree \(>2\) and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics. extension class of a vector bundle; torsion freeness; infinitesimal Torelli problem; canonical map; holomorphic forms; Albanese variety; families of varieties; generic Torelli problem Torelli problem, Variation of Hodge structures (algebro-geometric aspects), Families, moduli, classification: algebraic theory, \(n\)-folds (\(n>4\)) Differential forms and quadrics of the canonical image | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main aim of the paper is to prove (in Section 3) that if \(K(C)\) is the function field of a smooth projective curve \(C\) over a local non-dyadic field \(K\) then the \(u\)-invariant \(u(K(C))\) is finite. In fact the result is more general. The authors show that if \(F\) is a field of characteristic not 2, \(I^4F=0\) and to each form \(\varphi\in I^2F\) there exist \(a,b\in F^*\) such that \(\varphi_{F(\sqrt{a},\sqrt{b})}\) is hyperbolic, then \(u(F)\leq 22\).
The recent result of \textit{D. Saltman} [J. Ramanujan Math. Soc. 12, 25-47 (1997; Zbl 0902.16021)] on a common splitting field of a finite subset of \(_nBr(F)\), \(F\) being a function field in one variable over a local field, guarantees that \(K(C), K\) and \(C\) being as above, satisfies the assumptions of the main theorem. Saltman's paper, however, contains a gap completed later by \textit{O. Gabber} [cf. J. Ramanujan Math. Soc. 13, No. 2, 125-129 (1998)]. Moreover the authors need only Saltman's result for \(n=2\). To make the paper self-contained the authors discuss (in Section 2) Saltman's result restricting themselves to elements of \(_2Br(F)\) and taking into account Gabber's corrections.
The paper sums up with an application of the main result to a problem raised by \textit{J.-L. Colliot-Thélène} and \textit{A. N. Skorobogatov} [K-Theory 7, 477-500 (1993; Zbl 0837.14002)]. Let \(\varphi\) be a quadratic form over \(K(C)\), \(K\) and \(C\) as above, and let \(N_{\varphi}(K(C))\) be the norm group of \(\varphi\). The question is whether \(K^*N_{\varphi}(K(C))=K(C)^*\) for all forms \(\varphi\) over \(K(C)\) of dimension \(\geq 5\). The authors of the paper under review yield a positive answer for all forms \(\varphi\) of dimension \(\geq 7\). \(u\)-invariant; \(p\)-adic field; smooth projective curve; Brauer group; norm group of a quadric D. W. Hoffmann andJ. Van Geel, Zeroes and norm groups of quadratic forms over function fields in one variable over a local non-dyadic field,J. Ramanujan Math. Soc. 13 (1998), 85--110. Quadratic forms over general fields, Curves over finite and local fields, Algebraic theory of quadratic forms; Witt groups and rings, Brauer groups of schemes Zeros and norm groups of quadratic forms over function fields in one variable over a local non-dyadic 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 The main object of the interesting paper is the \(L\)-function of the family of superelliptic curves representing by the affine equation \(y^r = (x - f_1(t)) \cdots (x - f_d(t)) \), where \(f_i(t)\) are pairwise distinct polynomials in \({\mathbb F}_q[t]\) and \(r > 1\) is a natural number dividing \(q - 1\).
Let \(E/K\) be a superelliptic curve over \(K = {\mathbb F}_q(t)\) and \({\mathcal E} \to {\mathbb P}^1\) its model. The author assumes that all singularities of \({\mathcal E}\) are rational and gives conditions ensuring this.
Let \(K\) be a field. It is well known by Grothendieck and others that the Jacobian of the algebraic curve \(C/K\) is nice motivic object: \(J_{C/K} = \mathrm{Ext}^1_{\mathrm{Mot}}({\mathbb Z},H^1(C,{\mathbb Z}(1)))\).
The author explores this motivic interpretation, develops and applies a new general technique for constructing the \(L\)-function and obtaining upper bounds on the analytic rank of the corresponding Jacobian.
He extends results of \textit{C. Hall} [J. Number Theory 119, No. 1, 128--147 (2006; Zbl 1144.11051)] to families of superelliptic curves.
The approach is based on the resolution of singularities of models of relative curves by \textit{J. Lipman} [Publ. Math., Inst. Hautes Étud. Sci. 36, 195--279 (1969; Zbl 0181.48903)] such that the model has only rational singularities, on equivariant \(L\)-functions and on the cohomology of trivial modules, a technique developed by the author and \textit{C. D. Popescu} [Int. Math. Res. Not. 2012, No. 5, 986--1036 (2012; Zbl 1254.11063)].
The paper ends with consequences on the degree of \(L\)-functions and the ramification divisor. Five interesting examples classes of curves and their \(L\)-functions also are given. Picard 1-motives; class groups; models of relative curves; Galois coverings; equivariant \(L\)-functions; Galois module structure; Fitting ideals Zeta and \(L\)-functions in characteristic \(p\), Curves over finite and local fields, Varieties over finite and local fields, Geometric class field theory, \(p\)-adic cohomology, crystalline cohomology, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Families of curves with Galois action and their \(L\)-functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves two results, which give negative answers to a problem of Kleiman about the plane geometry in characteristic \(p>0\) [\textit{S. L. Kleiman} in Algebraic Geometry, Proc. Conf., Sundance 1988, Contemp. Math. 116, 71-84 (1991; Zbl 0764.14020)]. The question was whether every curve of degree at least two was the dual of a smooth curve whose Gauss map had any assigned inseparable degree. Note that if one does not care about the smoothness of the curve in question, then the answer is affirmative [see \textit{A. H. Wallace}, Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956; Zbl 0072.160) and \textit{S. L. Kleiman} in Algebraic Geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 163-225 (1986; Zbl 0601.14046)].
The first result is: For two given distinct smooth curves in the projective plane over an algebraically closed field of characteristic \(p\), their duals in the dual plane coincide if and only if they are conics in characteristic 2 with the same center. -- This result implies that for any given curve (which may be singular) of degree \(\geq 3\) in the dual plane, the number of smooth plane curves whose duals coincide with the assigned one is at most 1.
The second one is: If the degree \(d^*\) of the dual of a smooth curve \(C\) is a prime number, then either (1) \(d^* = 2\), \(p \neq 2\), and \(C\) is a conic; (2) \(d^* = 3\), \(p=2\), and \(C\) is cubic; or (3) \(d^*\) is of form \(2^ e + 1\), \(p=2\), and \(C\) is projectively equivalent to the curve \(X^{2^ e+1} + Y^{2^ e+1} + Z^{2^ e+1} = 0\). -- This result implies that there are many curves which are not dual of any smooth plane curves.
The first result has been generalized by \textit{H. Kaji} [Manuscr. Math. 80, No. 3, 249-258 (1993) and J. Reine Angew. Math. 437, 1-11 (1993; Zbl 0764.14019)]. plane geometry in characteristic \(p\); dual of a smooth curve; Gauss map Homma M. (1993). On duals of smooth plane curves. Proc. Amer. Math. Soc. 118(3): 785--790 Rational and birational maps, Projective techniques in algebraic geometry, Finite ground fields in algebraic geometry On duals of smooth plane curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0635.00008.]
A reduced projective curve is semistable if it is connected and each singular point is a node. It is stable if its automorphism group is finite; more concretely, if each smooth rational component meets the union of the other components in at least 3 distinct points. The dual graph of a semistable curve C has as vertices the irreducible components of C, and as edges the nodes of C. (When two different irreducible components meet at a node, the node contributes an edge which joins two distinct vertices; otherwise the node contributes a loop at one vertex.)
The present article surveys several results where the dual graph plays an important supporting role: first J. Carlson's Torelli theorem for irreducible stable curves [\textit{J. A. Carlson}, Journées de géométrie algébrique, Angers/France 1979, 107-127 (1980; Zbl 0471.14003)], then R. Friedman's generic Torelli theorem for curves of positive genus [\textit{R. Friedman}, ``Hodge theory, degenerations, and the global Torelli problem'' (Thesis, Harvard 1981)], and finally a proof of the irrationality of the general cubic 3-fold, based on ideas of \textit{A. Collino} [Boll. Unione Mat. Ital., V. Ser. B 16, 451-465 (1979; Zbl 0425.14011)]. dual graph of a semistable curve; generic Torelli theorem; irrationality of the general cubic 3-fold Special algebraic curves and curves of low genus Semistable curves, their dual graphs and some applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to find character formulae for every finite-dimensional simple module over the basic classical complex Lie superalgebra \(\mathfrak{g} = \mathfrak{osp}(m,2n)\). The results, without proofs and with a mistake in one statement, were enunciated in a previous paper by the second author [\textit{V. Serganova}, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 583--593 (1998; Zbl 0898.17002)]. Here, a different Borel subgroup and the language of weight diagrams are used in order to provide the needed proofs, which exploit geometrical methods adapted from Borel-Weil-Bott theory along the following lines:
1) Let \(G\) be the orthosymplectic Lie supergroup \(\text{OSP}(m,2n)\) and let \(P\) be a parabolic subgroup of \(G\). The multiplicities of simple \(\mathfrak{g}\)-submodules of the sheaf cohomology groups of a generalized supergrassmannian \(G/P\) are determined for certain \(P\).
2) These multiplicities are employed to express the characters of finite-dimensional simple \(\mathfrak{g}\)-modules as linear combinations of characters of the Euler characteristic of some invertible sheaves, characters which are determined by means of Borel-Weil-Bott theory. character formula; basic classical Lie superalgebra; cohomology of a generalized supergrassmannian; Borel-Weil-Bott theory; Euler characteristic; weight diagram C. Gruson, V. Serganova, \textit{Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras}, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852-892. Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Sheaf cohomology in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Superalgebras Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 investigates a conjecture of Beilinson and Bloch which relates the rank of the Griffiths group of a smooth projective variety over a number field to the order of vanishing of an L-function at the center of the critical strip. Numerical data is gathered for 1-cycles on \(E^3\) for 76 different elliptic curves \(E/\mathbb{Q}\). The empirical order of vanishing of the L-function is at least as large as a lower bound on the rank of the Griffiths group coming from the existence of certain genus three curves on \(E^3\). For 11 of the cases considered, the rank of \(\text{Griff}^2(E^3_{\mathbb{Q}})\) is shown to be at least 2. conjecture of Beilinson and Bloch; rank of the Griffiths group; smooth projective variety over a number field; order of vanishing of an L-function; elliptic curves J. Buhler, C. Schoen, and J. Top, ''Cycles, \(L\)-functions and triple products of elliptic curves,'' J. reine angew. Math., vol. 492, pp. 93-133, 1997. \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over global fields, Algebraic cycles, Global ground fields in algebraic geometry Cycles, \(L\)-functions and triple products of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0752.00024.]
Let \(\text{Sh} (G,X)\) be the Shimura variety associated to a reductive group \(G\) over \(\mathbb{Q}\), and let \(\text{Sh}_ p (G,X) = \text{Sh} (G,X)/K_ p\), where \(K_ p\) is a compact open subgroup of \(G(\mathbb{Q}_ p)\). Let \(E\) be a reflex field of \(\text{Sh} (G,X)\), and let \(v\) be a prime of \(E\) lying over \(p\). If \(\mathbb{Q}^{\text{al}}_ p\) is the algebraic closure of \(E_ v\) and \(\mathbb{Q}^{\text{un}}_ p\) is the maximal unramified extension of \(\mathbb{Q}_ p\) contained in \(\mathbb{Q}^{\text{al}}_ p\), then \(\mathbb{F}\) denotes the residue field of \(\mathbb{Q}^{\text{un}}_ p \subset \mathbb{Q}^{\text{al}}_ p\). Let \({\mathfrak P}\) be the pseudomotivic groupoid associated with the Tannakian category of motives over \(\mathbb{F}\), and let \({\mathfrak G}_ G\) be the neutral groupoid defined by \(G\). Then a homomorphism \(\varphi : {\mathfrak G}_ G \to {\mathfrak P}\) defines a triple \((S (\varphi), \Phi (\varphi), \times (\varphi))\), where \(S(\varphi)\) is a set of the form \(I_ \varphi (\mathbb{Q})^ - \backslash X^ p (\varphi) \times X_ p (\varphi)\), \(\Phi (\varphi)\) is a Frobenius operator, and \(X(\varphi)\) is an action of \(G(\mathbb{A}^ p_ f)\) on \(S(\varphi)\) commuting with the action of \(\Phi (\varphi)\). Then the conjecture of \textit{R. P. Langlands} and \textit{M. Rapoport} [J. Reine Angew. Math. 378, 113-220 (1987; Zbl 0615.14014)] can be stated as
\[
\bigl( \text{Sh}_ p (\mathbb{F}), \Phi, \times \bigr) \approx \coprod_ \varphi \bigl( S (\varphi), \Phi (\varphi), \times (\varphi) \bigr),
\]
where the disjoint union is over a certain set of isomorphism classes of \(\varphi\). Let \({\mathcal V} (\xi)\) be the local system on \(\text{Sh} (X)\) defined by a representation \(\xi : G \to GL (V)\) of \(G\). Let \({\mathcal T} (g)\) be the Hecke operator defined by \(g \in G (\mathbb{A}_ f^ p)\), and let \({\mathcal T} (g)^{(r)}\) be the composite of \({\mathcal T} (g)\) with the \(r\)-th power of the Frobenius correspondence. In this paper the author derives from the conjecture of Langlands and Rapoport the formula for the sum
\[
\sum_{t'} \text{Tr} \bigl( {\mathcal T} (g)^{(r)} | {\mathcal V}_ t (g) \bigr),
\]
where \(t'\) runs over the elements of \(\text{Sh}_{K \cap gKg^{-1}} (G,X) (\mathbb{F})\) such that \({\mathcal T} (g)(t') = t\), as a sum of products of certain orbital integrals. He also introduces the notion of an integral canonical model for a Shimura variety, extends the conjecture of Langlands and Rapoport to Shimura varieties defined by groups whose derived group is not simply connected, and reviews results of R. Kottwitz concerning the stabilization. zeta functions; Picard modular surfaces; \(L\)-functions; automorphic representations; Shimura variety; Tannakian category of motives; Frobenius operator; conjecture of Langlands and Rapoport J. S. Milne, The points on a Shimura variety modulo a prime of good reduction, The Zeta Functions of Picard Modular Surfaces, University Montréal, Montreal (1992), 151-253. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Modular and Shimura varieties, Representation-theoretic methods; automorphic representations over local and global fields, Finite ground fields in algebraic geometry The points on a Shimura variety modulo a prime of good reduction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((X,{\mathcal X})\) be a polarized abelian variety of dimension \(n\) over an algebraically closed field \(k\) and \(C\) an irreducible curve that generates \(X\). The author proves the inequality: \(C\cdot {\mathcal X} \geq n(\deg {\mathcal X})^{1/n}\).
If \({\mathcal X}\) is separable, this is an equality if and only if \(C\) is smooth, \(X\) is the Jacobian of \(C\) and \({\mathcal X} = \delta \cdot \Theta_C\), where \(\delta\) is an integer prime to \(\text{char} (k)\). -- The cases \(C \cdot {\mathcal X} = n+1\) and \(C \cdot {\mathcal X}=n+2\) are studied in detail.
As a corollary of the above results the author finds an upper bound for the geometric genus of \(C\) in terms of its degree. endomorphisms; polarized abelian variety; irreducible curve; Jacobian; geometric genus Debarre O. (1994). Degrees of curves in Abelian varieties. Bull. Soc. Math. France 122(3):343--361 Algebraic theory of abelian varieties, Jacobians, Prym varieties Degrees of curves in abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order Frank Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110 (English, with English and French summaries). Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pieri's formula for flag manifolds and Schubert polynomials | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0624.00007.]
The author reviews Dwork's preprint ``On the Tate constant'' [see \textit{B. Dwork}, Compos. Math. 61, 43-59 (1987; Zbl 0622.14016) and Groupe Étud. Anal. Ultramétrique, 11e Année 1983/84, Exposé No.11 (1985; Zbl 0614.14009)] describing the Picard-Fuchs equation and the Hasse-Witt matrix of a family of elliptic curves by considering the formal group of the Tate curve. The author states generalizations to the case of curves of higher genus; the details will appear in the author's thesis. Tate constant; hypergeometric differential equation; Legendre family of elliptic curves; Tate curve; Picard-Fuchs equation; Hasse-Witt matrix; formal group Local ground fields in algebraic geometry, Families, moduli of curves (algebraic), Algebraic functions and function fields in algebraic geometry, Formal groups, \(p\)-divisible groups, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) On the Tate-matrix | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0685.00007.]
From the authors' abstract: Several facts about \(SK_ 0\) and \(SK_ 1\) are presented both for commutative rings and schemes. If A is the homogeneous coordinate ring of a projective variety over a field k, then Pic(A), \(SK_ 0(A)\) and \(SK_ 1(A)\) are naturally modules over the ring W(k) of Witt vectors over k. If A is any commutative ring, NPic(A), \(NSK_ 0(A)\) and \(NSK_ 1(A)\) are naturally modules over W(A). The K- theory transfer map defined when B is an A-algebra which is a finite projective A-module, sends \(SK_ 0(B)\) to \(SK_ 0(A)\) and \(SK_ 1(B)\) to \(SK_ 1(A)\). \(SK_ 0\); \(SK_ 1\); coordinate ring of a projective variety; Pic; Witt vectors; K-theory transfer map B.H. Dayton and C.A. Weibel, On the naturality of Pic, SK0 and SK1, to appear. Grothendieck groups, \(K\)-theory and commutative rings, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Picard groups On the naturality of Pic, \(SK_ 0\) and \(SK_ 1\) | 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 The study of the extrinsic geometry of the Jacobian locus \(\overline{J_g}\) in the moduli space \(A_g\) of principally polarized abelian varieties, namely of the closure of the locus of Jacobian varieties in \(A_g\), has been stimulated by different many problems throughout its history (two of the best known are the Torelli and Schottky problems). In particular, the theory of generalized Prym varieties as developed by [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)] highlighted a natural inclusion of the Jacobian locus inside boundary of the Prym locus \(\overline{P_{g+1}}\), which is the closure in \(A_g\) of the locus of Prym varieties associated to étale coverings of degree \(2\) onto smooth projective curves of genus \(g+1\). In terms of the study of the extrinsic geometry of \(\overline{J_g}\subset A_g\) such a theory suggested a refined study of \(\overline{J_g}\subset \overline{P_{g+1}}\) by using the parametrization of Jacobian varieties as generalized Prym varieties.
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(R_{g+1}\) be the moduli space of degree \(2\) étale coverings of curves of genus \(g+1\). For any \([C]\in M_g\), let \(JC\) be the Jacobian variety and let \(\mathrm{Cliff }C\) denote the Clifford index of \(C\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]). For any \(\zeta \in T_{[JC]}J_g\), let \(\xi \in H^1(C, T_C)\) such that \(\zeta \) can be identified with the cup product map \(\cup \xi: H^0(C, \omega_C)\to H^0(C, \omega_C)^\vee\), under the isomorphism \(T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\). We define the rank of \(\zeta\) as the rank of \(\cup \xi\). The following is one of the main results of the paper.
Theorem 1. Let \(JC\) be a general Jacobian variety of dimension \(g\geq 7\). Then for any \(\zeta\in T_{[JC]}J_g\) of rank \(k=rk\zeta <\mathrm{Cliff }C-3\), the local geodesic in \(A_g\) at \([JC]\) with direction \(\zeta\) (defined with respect to the Siegel metric) is not contained in the Prym locus \(\overline{P_{g+1}}\) (in particular, also in \(\overline{J_g}\subset \overline{P_{g+1}}\)).
Assume \(g\geq 7\).
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(A_g\) be the moduli space of principally polarized abelian varieties.
The Torelli morphism
\[
j: M_g\to A_g , \quad [C]\mapsto [JC]
\]
sends a genus \(g\) smooth projective curve \(C\) to its Jacobian variety \(JC\) (up to isomorphism). Its image \(J_g=j(M_g)\subset A_g\) defines a proper locus for \(g>3\) and its closure \(\overline{J_g}\subset A_g\) is called the Jacobian locus. By using the isomorphisms
\[
T_{[C]}M_g\simeq H^1(C,T_C), \quad T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},
\]
the differential of the Torelli map
\[
dj:H^1(C,T_C)\to \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},\quad \xi \mapsto \zeta
\]
is given by \(\zeta=\cup\xi:H^0(C, \omega_C)\to H^0(C,\omega_C)^\vee,\) the cup product map with \(\xi.\)
Let \(R_{g+1}\) be the moduli space parametrizing \(2\)-sheeted étale coverings \(\pi: \tilde{C}\to C'\) between smooth projective curves of genus \(\tilde{g}=g(\tilde{C})=2g(C')-1\) and \(g'=g(C')=g+1\), respectively (modulo isomorphism). A point in \(R_{g+1}\) is an isomorphism class assigned equivalently by
\begin{itemize}
\item[(i)] a pair \((\tilde{C}, i)\), where \(i:\tilde{C}\to \tilde{C}\) is an involution such that \(\tilde{C}/(i)=C'\),
\item[(ii)] a pair \((C', \eta)\), where \(\eta \in \mathrm{Pic}^0(C')\setminus \{\mathcal O_{C'}\}\) is a \(2\)-torsion point and \(\tilde{C}= \mathrm{spec} (\mathcal O_C\oplus \eta)\).
\end{itemize}
One can reconstruct the data \((i)\) from \((ii)\) and conversely (see [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)], [\textit{D. Mumford}, in: Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers, 325--350 (1974; Zbl 0299.14018)]) and so with a little abuse of notations we will describe a point of \(R_{g+1}\) in both ways depending on the setting.
The Prym morphism
\[
Pr: R_{g+1}\to A_g, \quad [(C', \eta)]\mapsto [P(C',\eta)]
\]
maps a pair \((C', \eta)\) to its Prym variety \(P(C',\eta)\)(up to isomorphisms).
By definition, the Prym variety is \(P(C',\eta)=\ker Nm^0,\) namely the connected component containing zero in the kernel of the norm map \(Nm: J\tilde{C}\to JC'\), and it is a principally polarized abelian variety of dimension \(g\), with the principal polarization given by one half of the restriction of that on \(J\tilde{C}\).
The image \(P_{g+1}=Pr(R_{g+1})\subset A_g\) of the Prym morphism defines a proper locus for \(g\geq 6\) and its closure \(\overline{P_{g+1}}\) is called the Prym locus.
By construction, the forgetful functor \(\pi_{R_{g+1}}: R_{g+1}\to M_{g+1},\) sending \([(C',\eta)]\mapsto [C']\), is an étale finite covering and so we can identify \(T_{[(C', \eta)]}R_{g+1}\simeq T_{[C']}M_g.\) Using the isomorphisms
\[
T_{[C', \eta]}R_{g+1}\simeq H^1(C', T_{C'}), \quad T_{[P(C, \eta)]}A_g\simeq \mathrm{Sym}^2H^1(C',\eta)\simeq \mathrm{Sym}^2 H^0(C', \omega_{C'}\otimes \eta)^{\vee},
\]
the differential of the Prym map
\[
d Pr:H^1(C', T_{C'})\to \mathrm{Sym}^2H^1(C',\eta), \quad \xi' \longmapsto \zeta',
\]
is given by \(\zeta'=\cup\xi':H^0(C', \omega_{C'}\otimes \eta)\to H^0(C', \omega_{C'}\otimes \eta)^{\vee}\), the cup product with \(\xi'.\)
Let \(C\) be a smooth projective curve. The gonality and the Clifford index of \(C\) are defined as
\begin{align*}
&\operatorname{gon} C= \min\{n\in\mathbb N \,| \, C \mbox{ has a } g^1_n\} ;\\
&\mathrm{Cliff }C = \min\{\mathrm{Cliff }D=\deg D- 2 (h^0(D)-1)\, | \, D\subset C \mbox{ divisor}, h^0(D)\geq 2, h^1(D)\geq 2 \}.
\end{align*}
By [\textit{M. Coppens} and \textit{G. Martens}, Compos. Math. 78, No. 2, 193--212 (1991; Zbl 0741.14035)], these are related by
\[
\operatorname{gon} C -3 \leq\mathrm{Cliff }C \leq \operatorname{gon} C- 2 ,
\]
where the second inequality is an equality on a general \(C\) in \(M_g\), while the first one is conjectured to be extremely rare [\textit{D. Eisenbud} et al., Compos. Math. 72, No. 2, 173--204 (1989; Zbl 0703.14020)]. The followings are well known.
\begin{itemize}
\item[1.] \(\operatorname{gon} C \leq \left\lfloor \frac{g+3}{2}\right \rfloor\) (and so \(\mathrm{Cliff} C \leq \left\lfloor \frac{g-1}{2}\right \rfloor\)) and the equality holds for a general curve \([C]\in M_g\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]);
\item[2.] let \(f:\mathcal C\to B\) be a fibration of projective curves over a complex curve \(B\) and such that the general fibre is smooth. The invariants \(\operatorname{gon} C_b \) and \(\mathrm{Cliff} C_b\) are maximal on a general fibre \(C_b\) of the set of smooth fibres ([\textit{K. Konno}, J. Algebr. Geom. 8, No. 2, 207--220 (1999; Zbl 0979.14004)]).
\end{itemize}
An \textit{admissible covering} of degree \(k\) with \(m\) ramification points is the data of
\begin{itemize}
\item[1.] a stable \(m\)-pointed reduced connected curve \((E, x_1, \dots ,x_m)\) of arithmetic genus \(0\) (i.e. a curve with ordinary double points where any rational component is smooth and stable and the dual graph is connected);
\item[2.] a reduced connected curve \(X\) with ordinary double points and a morphism \(\pi : X \to E\) of degree \(k\) (everywhere) such that over any marked point \(x_i\) of \(E\), \(X\) is smooth and \(f\) has a unique simple ramification point \(y_i\), on any smooth point of \(E\), \(f\) is étale and over double points \(p\) of \(E\), \(X\) has an ordinary double point \(q\) and \(f\) is locally described as
\[
X\,:\, xy=0; \quad E\,:\, uv=0;\quad \, f\,:\, u=x^{k'}\quad ,\, v= y^{k'},
\]
for some \(k'\leq k\).
\end{itemize}
We have the following
Lemma. Let \(C'\) be a stable nodal curve of genus \(g'\), let \(\nu': C^{\nu'}\to C'\) be its normalization and let \(C\subset C^{\nu'}\) be a smooth connected curve of genus \(g\). Consider \(f':\mathcal C'\to \Delta\), a family of smooth projective curves of genus \(g'\) over \(\Delta \setminus \{0\}\), the complex disk minus zero, such that \(C'=f^{-1}(0)\). Then any family of pencils \(g^1_k(t)\) over \( \Delta\setminus\{0\}\) compatible with \(f\) (i.e. \(g^1_k(t)\) is a pencil on \(C'_t=f^{-1}(t)\)) defines a pencil \(g^1_{k'}\) on \(C\) for some \(k'\leq k\). In particular, \(\operatorname{gon} C\leq \operatorname{gon} C'_t\) and \(\mathrm{Cliff }C\leq \mathrm{Cliff }C'_t+1\), for a general fibre \(C'_t=f^{-1}(t)\).
Let \(Y\) be a smooth complex variety and let
\((\mathbb H_{\mathbb Z}, \mathcal H^{1,0}, \mathcal Q) \) be a polarized variation of Hodge structures (pvhs, in short) of weight 1 on \(Y\). Namely, \(\mathbb H_Z\) is a local system of lattices, \(\mathcal H^{1,0}\) is a Hodge bundle of type \((1,0)\) (equivalently, the Hodge filtration in this case) and \(\mathcal Q\) is a polarization. Let \(\mathbb H_{\mathbb C}=\mathbb H_{\mathbb Z}\otimes_{\mathbb Z}\mathbb C\) and let \(\mathcal H=\mathbb H_{\mathbb C} \otimes \mathcal O_Y\) be the holomorphic flat bundle with the holomorphic flat connection \(\nabla\) defined by \(\ker \nabla \simeq \mathbb H_{\mathbb C}\), the Gauss-Manin connection. The holomorphic inclusion \(\mathcal H^{1,0}\subset\mathcal H\) of vector bundles induces the short exact sequence
\[
\begin{tikzcd}
0 \arrow{r} & \mathcal H^{1,0} \arrow{r} & \mathcal H \arrow{r}{\pi^{{0,1}}} &
\mathcal H/\mathcal H^{1,0} \arrow{r}& 0.
\end{tikzcd}
\]
Let \(\pi^{{0,1}'}: \mathcal H\otimes \Omega^1_Y\to\mathcal H/\mathcal H^{1,0}\otimes \Omega^1_Y\) be the map induced by \(\pi^{0,1}\) and \(\sigma =\pi^{{0,1}'}\circ \nabla: \mathcal H^{1,0}\to\mathcal H/\mathcal H^{1,0}\otimes \Omega_Y^1\) the second fundamental form of \(\mathcal H^{1,0}\subset\mathcal H\) with respect to \(\nabla\).
Following [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)], let \(\mathbb U=\ker \nabla_{|\mathcal H^{1,0}}\) and define
\[
\mathcal U:=\mathbb U\otimes
\mathcal O_Y,\quad \quad
\mathcal K:=\ker ( \sigma : \mathcal H^{1,0}\longrightarrow\mathcal H /\mathcal H^{1,0} \otimes
\Omega_Y^1).
\]
Then \(\mathcal U\) is a holomorphic vector bundle and \(\mathcal K\) is a coherent sheaf which is a vector bundle when \(\sigma\) is of constant rank.
Definition 1. We call \(\\mathcal U\) and \(\mathcal K\) as defined in (1), \textit{the unitary flat bundle} and \textit{the kernel sheaf} of the variation, respectively
Proposition 1. We have \( \mathcal U\subset \mathcal K\) and if \(\tau\equiv 0\), then \(\mathcal U=\mathcal K\).
For any \(B\subset A_g\) smooth complex curve, let \(f: \mathcal A\to B\) be the family of abelian varieties (defined up to finite base change) and let \(A_b\) be the fibre over \(b\in B\). Then the p.v.h.s. is defined by
\[
\mathbb H_{\mathbb Z}\simeq R^1f_\ast \mathbb Z, \quad \mathcal H^{1,0}= f_\ast \Omega^1_{\mathcal A/B}\subset \mathcal H= R^1f_\ast \mathbb C\otimes \mathcal O_{B};
\]
\[
\mbox{where }\quad (R^1f_\ast \mathbb Z)_b\simeq H^1(A_b,\mathbb Z), (f_\ast\Omega^1_{\mathcal A/B })_b\simeq H^0(A_b, \Omega^1_{A_b}), {(R^1f_\ast \mathbb C\otimes \mathcal O_{B})}_b\simeq H^1(A_b, \mathbb C).
\]
Let \(H_g\) denote the Siegel upper half space. As a symmetric space of non-compact type it is endowed by a symmetric metric \(h^s\), called the Siegel metric, defining a metric connection \(\nabla^{LC}\) on the tangent bundle \(TH_g\). As a parametrizing space of weight \(1\) p.v.h.s., it carries a universal p.v.h.s. \((\mathbb H_{\mathbb Z},\mathcal H^{1,0}, \mathcal Q)\) with its Hodge metric defined by \(Q\), inducing a metric \(h\) together with a metric connection \(\nabla^{hdg}\) on \(\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0})\). There is a natural inclusion
\[
(TH_g, \nabla^{LC})\subset (\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0}), \nabla^{hdg})
\]
compatible with the metric structure (see e.g. a classical reference [\textit{P. A. Griffiths}, in: Actes Congr. internat. Math. 1970, 1, 113--119 (1971; Zbl 0227.14008)] or some more recent references [\textit{A. Ghigi}, Boll. Unione Mat. Ital. 12, No. 1--2, 133--144 (2019; Zbl 1444.14028)], [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)]).
Consider the universal covering \(\psi: H_g\to A_g\) and the metric properties introduced before on \(H_g\). Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\).
Take \(\tilde{A}\in \psi^{-1}([A])\) and consider \(\zeta\) as \(\zeta \in T_{\tilde{A}}H_g\simeq T_{[A]}A_g\). Then in \(H_g\) a (local) geodesic at \((\tilde{A}, \zeta)\) is simply a curve \(\gamma:(-\epsilon, \epsilon)\to H_g\) such that \(\gamma(0)=\tilde{A}\) and \(\gamma'(0)=\zeta\) satisfying \(\nabla^{LC}_{\gamma'}\gamma'=0\). Working locally, we can assume w.l.o.g that \(\gamma((-\epsilon, \epsilon))\) is contained in one sheet of \(\psi\).
Definition 2. Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\)
A (local) geodesic associated to \(([A], \zeta)\) is a map \(\psi\circ \gamma: (-\epsilon, \epsilon)\to A_g\), where \(\gamma\) is a local geodesic in \(H_g\) defined as above.
We are interested in points of \(J_g\subset A_g\) and directions in \(T_{[A]}J_g\subset T_{[A]}A_g\). If \([A]=[JC]\), namely the Jacobian of some \([C]\in M_g\), and \(\zeta = \cup \xi\), for some \(\xi \in H^1(C, T_C)\), under the isomorphisms \(T_{[C]}M_g\simeq H^1(C, T_C)\) and \(T_{[J_g]}A_g\simeq \mathrm{Sym}^2H^1(C, \mathcal O_C)\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\), we will also refer to the geodesic at \((JC, \zeta)\) as the geodesic at \((C, \xi)\). This is admitted since the Torelli map is an immersion outside the Hyperelliptic locus and we are not considering hyperelliptic curves.
We have the following (see [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013), Lemma 3.3 and Lemma 3.4] for the proof)
Lemma 2. Let \(\gamma:(-\epsilon, \epsilon)\to A_g\) be a local geodesic associated to \(([A], \zeta)\). Then there exists a complex curve \(B\subset H_g\) containing the geodesic and such that \(\mathcal K =\mathcal U\).
We can shrink \(B\) around \(\gamma((-\epsilon, \epsilon))\) in such a way that it is contained in one sheet on \(\psi\) and so we can then look at it as a curve in \(A_g\).
Let \(C\) be a smooth projective curve, let \(\mathcal F\) be a rank \(2\) vector bundle over \(C\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\) be a linear map. A subspace \(W\subset H^0(C,\mathcal F) \) is called \textit{isoptropic} with respect to \(\alpha\) if \(\alpha_{|\bigwedge^2 W}\equiv 0\).
Let \(\mathcal L, \mathcal L'\) be two line bundles on \(C\), let \(0\to\mathcal L \to\mathcal F \to\mathcal L'\to 0\) be a s.e.s. associated to \(\xi\in \mathrm{Ext}^1_{\mathcal O_C}(\mathcal L', \mathcal L)\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\simeq H^0(C,\mathcal L\otimes\mathcal L')\). A subspace \(V\subset H^0(C,\mathcal L')\) is called \textit{isotropic} with respect to \(\alpha\) if it lifts to a subspace \(W\subset H^0(C,\mathcal F)\) isotropic with respect to \(\alpha\).
Note that a subspace \(V \subset H^0(C,\mathcal L')\) lifts to \(W\subset H^0(C,\mathcal F)\) if and only if it lies in the kernel of the coboundary morphism \(\delta: H^0(C,\mathcal L')\to H^1(C,\mathcal L)\) on the long exact sequence in cohomology.
Theorem 2. Let \(f: \mathcal C\to B\) be a fibration of smooth projective curves over a smooth complex curve \(B\) and let \(\mathcal U\subset f_\ast \omega_{\mathcal C/B}\) be the associated unitary flat bundle (Definition 1). Assume that the fibres \(U_b\subset H^0(\omega_{C_b})\) are isotropic subspaces with respect to \(\alpha_b\) for any \(b\in B\). Then (up to a finite base change) there exists a smooth projective curve \(\Sigma\) of genus \(g'=rk \mathcal U\) and a non constant fibre-preserving map \(\varphi: \mathcal C \to \Sigma\) such that \(U_b\simeq \varphi^\ast H^0(\Sigma,\omega_{\Sigma})\). moduli space of curves and abelian varieties; geodesics; Prym locus; Jacobian locus; generalized Prym varieties; admissible coverings Jacobians, Prym varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Period matrices, variation of Hodge structure; degenerations, Subvarieties of abelian varieties On the Jacobian locus in the Prym locus and geodesics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 consider morphisms of algebraic stacks \(\mathcal X \to \mathcal Y\) which are torsors under a group stack \(\mathcal G\). We show that line bundles on \(\mathcal Y\) correspond exactly with \(\mathcal G\)-linearized line bundles on \(\mathcal X\) (with a suitable definition of a \(\mathcal G\)-linearization). We use this fact to determine the precise structure of the Picard group of the moduli stack of \(G\)-bundles on an algebraic curve when \(G\) is any group of type \(A_n\). linearization; morphisms of algebraic stacks; Picard group; torsors; moduli stack on an algebraic curve Laszlo Y., Linearization of group stack actions and the Picard group of the moduli of SLr/{\(\mu\)}s-bundles on a curve, Bull. Soc. Math. France, 1997, 125(4), 529--545 Vector bundles on curves and their moduli, Picard groups, Families, moduli of curves (algebraic), Group actions on varieties or schemes (quotients) Linearizaton of group stack actions and the Picard group of the moduli of \(SL_r/\mu_s\)-bundles on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(L\) be a finite distributive lattice and \(k\) an algebraically closed field. Let \({\mathbf X}=\{x_\alpha|\alpha\in L\}\) be commuting indeterminates over \(k\). Let \(I(L)\) be the ideal in \(k[{\mathbf X}]\) generated by the quadratic forms \(\{x_\alpha x_\beta- x_{\alpha\wedge\beta} x_{\alpha\vee \beta}\mid\alpha, \beta\in L\}\). Set \(k[L]=k [{\mathbf X}]/I(L)\). Since \(I(L)\) is generated by homogeneous polynomials, \(k[L]\) is a graded \(k\)-algebra and \(\text{Proj} k[L] =V(L)\) is a projective variety. It is known that \(V(L)\) is the toric variety associated to a complete fan.
In this article, we compute the Brauer group of the variety \(V(L)\). The Brauer group is the second étale cohomology group of \(V(L)\) with coefficients in the sheaf of units. The Brauer group functor on toric varieties is an interesting invariant, intricately tied to the arithmetic and topology of the underlying fan. Our main result (theorem 4.3) is that the Brauer group of \(V(L)\) is always trivial. Brauer group of a toric variety; finite distributive lattice; Brauer group functor Brauer groups of schemes, Toric varieties, Newton polyhedra, Okounkov bodies, Étale and other Grothendieck topologies and (co)homologies, Structure and representation theory of distributive lattices The Brauer group of a toric variety associated to a finite distributive lattice | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth irreducible curve of genus \(g\) and let \(C_ d\) be its \(d\)-fold symmetric product. For a curve \(C\) with general moduli its Néron-Severi group is generated by the classes of two divisors. If \(\vartheta\) and \(x\) stand for those classes, the slope of a divisor on \(C_ d\), whose class is \(a\vartheta-bx\), is defined by \(b/a\). The author looks for lower and upper bounds for the slope of effective and ample divisors on \(C_ d\). curve; genus; symmetric product; Néron-Severi group; slope of a divisor Kouvidakis, A.: Divisors on symmetric products of curves. Trans. Am. Math. Soc. 337, 117--128 (1993) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Divisors on 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 The deformation theory of a two-dimensional singularity, which is isomorphic to the affine cone over a curve, is intimately linked with the (extrinsic) geometry of this curve. In recent times various authors have studied one-parameter deformations, partly under the guise of extensions of curves to surfaces. In this paper we consider the versal deformation of cones, in the simplest case: cones over hyperelliptic curves of high degree. In particular, we show that for degree \(4g + 4\), the highest degree for which interesting deformations exist, the number of smoothing components is \(2^{2g + 1}\) (except for \(g = 3)\). Powerful methods exist to compute \(T^1\) for surface singularities, without using explicit equations. In the homogeneous case the graded part \(T^1 (-1)\) is the hardest. We review in a general setting the relation with Wahl's Gaussian map [cf. \textit{J. Wahl}, J. Differ. Geom. 32, No. 1, 77-98 (1990; Zbl 0724.14022)]. We prove that \(T^1 (-1)\) vanishes for a cone over a general curve, embedded with an arbitrary linear system of degree at least \(2g + 11\). We find the dimension of \(T^2\) for hyperelliptic cones with the main lemma of a paper by \textit{K. Behnke} and \textit{J. A. Christophersen} [Compos. Math. 77, No. 3, 233-268 (1991; Zbl 0728.14034)], which connects the number of generators of \(T^2\) with the codimension of smoothing components in the versal base of a general hyperplane section. Therefore we compute \(T^1\) for the cone over \(d\) points on a rational normal curve of degree \(d - g - 1\). We use explicit equations for the curve singularity. Actually, the equations for the cone over a hyperelliptic curve have a nice structure. We give an interpretation of \(T^2 (-2)\) in terms of this structure.
Smoothing components are related to surfaces with \(C\) as hyperplane section. According to Castelnuovo, these are rational ruled. We get all from a given one by elementary transformations. An explicit description of the corresponding infinitesimal deformations enables us to conclude that the base space is a complete intersection of degree \(2^{2g + 1}\). We also consider smoothing data in the sense of \textit{E. Looijenga} and \textit{J. Wahl} [Topology 25, 261-291 (1986; Zbl 0615.32014)]. affine cone over a curve; versal deformation of cones; Wahl's Gaussian map; smoothing components; infinitesimal deformations Stevens, J.: Deformations of cones over hyperelliptic curves. J. reine angew. Math. 473, 87-120 (1996) Formal methods and deformations in algebraic geometry, Curves in algebraic geometry, Infinitesimal methods in algebraic geometry, Surfaces and higher-dimensional varieties Deformations of cones over hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0684.00004.]
In this expository paper the author reviews recent results on the cohomology classes defined by automorphic forms. Moreover the author tries to find evidence for the statement that if for two motives \(M_ 1\) and \(M_ 2\) occurring in the cohomology of a Shimura variety one has an isomorphism of \(\ell\)-adic cohomology \(H_{\ell}(M_ 1)\cong H_{\ell}(M_ 2)\) as Galois modules for all \(\ell\) then the arithmetic Hodge structures defined by \(M_ 1\) and \(M_ 2\) are isomorphic. (Here the arithmetic Hodge structure is the triple: \(H_{DR},H_ B,H_ B(M)\otimes {\mathbb{C}}\cong H_{DR}(M,{\mathbb{C}})\).) cohomology classes defined by automorphic forms; motives; cohomology of a Shimura variety; arithmetic Hodge structures M. Harris, Automorphic forms and the cohomology of vector bundles on Shimura varieties , Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. II (Ann Arbor, MI, 1988) eds. L. Clozel and J. S. Miline, Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 41-91. Modular and Shimura varieties, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Generalizations (algebraic spaces, stacks), Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Applications of methods of algebraic \(K\)-theory in algebraic geometry Automorphic forms and the cohomology of vector bundles on Shimura 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 aim of this article is to study the structure of \(H^2 (X,K_2)\) of a surface \(X\) with prescribed singularities.
Let \(X'\) be a smooth projective surface over \(\mathbb{C}\), \(m\) an effective divisor on \(X'\) and \(\pi : X' \to X\) such that \(\pi : m \to S\) is a finite surjective map between reduced curves and \(U = X' \backslash m \simeq X \backslash S\). The author studies the relations between the group \(H^2 (X,K_2)\), the 1-motive \(J^2 (X)\) defined by \(J^2 (X) = H^3 (X,\mathbb{C})/F^2 H^3 (X, \mathbb{C}) + H^3 (X, \mathbb{Z})\) and \(G_{um}\) the generalized albanese variety of \(X'\) with modulus \(m : G_{um} (\mathbb{C}) \simeq H^0 (X', \Omega_{X'} (m))^*_{d = 0}/H_2 (U, \mathbb{Z})\); the generalized albanese map \(\alpha_{um} : U \to G_{um} (\mathbb{C})\) is given by \(\alpha_{um} (x)=(\int_\gamma \omega_1, \ldots, \int_\gamma \omega_n)\) modulo periods, for any path \(\gamma\) joining a fixed point \(x_0\) to \(x\) and where \((\omega_i)\) is a basis of \(H^0 (X', \Omega_{X'} (m))_{d = 0}\). The author gets the following results:
(i) There is a surjective homomorphism \(G_{um} (\mathbb{C}) \to J^2 (X)\) which is an isomorphism if \(S\) is integral.
(ii) If \(H^2 (X', {\mathcal O}_{X'}) = 0\) then \(J^2 (X)\) is an extension of \(\text{Alb}_{X'}\) by a torus of dimension \(d = \text{rank(NS}(m)/(\text{NS}(S) +\text{NS}(X')))\) where NS\(( )\) is the Néron-Severi group.
(iii) \(\alpha_{um}\) induces a surjective homomorphism \(H^2 (X, K_2)_0 \to J^2 (X)\) where \(H^2 (X,K_2)_0\) is the group of zero cycles of degree zero.
To prove the first result, the author considers the following diagram:
\[
\begin{matrix} H^2 (X, \mathbb{C}) & \longrightarrow & H^2 (S, \mathbb{C}) & \longrightarrow & H^3_c (U, \mathbb{C}) & \longrightarrow & H^3 (X, \mathbb{C}) & \longrightarrow & 0 \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \pi^* \\ H^2 (X', \mathbb{C}) & \longrightarrow & H^2 (m, \mathbb{C}) & \longrightarrow & H^3_c (U, \mathbb{C}) & \longrightarrow & H^3 (X', \mathbb{C}) & \longrightarrow & 0 \end{matrix}
\]
By Hodge theory, we have \(F^2 H^3 (X, \mathbb{C}) \cap W^2 H^3 (X, \mathbb{C}) = F^2 H^3 (X, \mathbb{C}) \cap \text{Ker} (\pi^*) = 0\), and we get \(F^2 H^3 (X, \mathbb{C}) = F^2 H^3 (X', \mathbb{C}) = H^1 (X', \Omega^2)\). Then we deduce from the isomorphism \(H^1 (U, \mathbb{C}) \simeq \Omega^{\text{inv}}_{G_{um}} \oplus H^1 (X', \Omega^2)\) a surjective map \(\Omega^{\text{inv} *}_{G_{um}} \to H^3 (X, \mathbb{C})/F^2 H^3 (X, \mathbb{C})\), and taking quotients respectively by \(H^3_c (U, \mathbb{Z}) \simeq H_1 (U, \mathbb{Z})\) and \(H^3 (X, \mathbb{Z})\) we obtain the surjective map \(G_{um} (\mathbb{C}) \to J^2 (X)\). If \(S\) is integral \(H^2 (S, \mathbb{C}) \simeq \mathbb{C}\), then \(H^3_c (U, \mathbb{C}) \simeq H^3 (X, \mathbb{C})\), and the second part of the first statement follows. We obtain the second statement from the exact sequence:
\[
0 \to {H^2 (m, \mathbb{C}) \over H^2 (S, \mathbb{C}) + H^2 (X', \mathbb{C})} \to {H^3_c (U, \mathbb{C}) \over H^2 (S, \mathbb{C}) + H^1 (X', {\mathcal O}_{X'})} \to {H^3 (X', \mathbb{C}) \over F^2 H^3 (X', \mathbb{C})} \to 0
\]
and from the isomorphism \(H^3_c (U, \mathbb{C})/(H^2 (S, \mathbb{C}) + H^1 (X', {\mathcal O}_{X'})) \simeq H^3 (X', \mathbb{C})/F^2 H^3 (X', \mathbb{C})\). To obtain the last statement, we have to look at the decomposition in irreducible components of the divisors \(S\) and \(m\) and to study the map \(H^2 (S, \mathbb{C}) \to H^3_c (U, \mathbb{C})\). Albanese variety; algebraic cycles; second cohomology group of a surface with prescribed singularities; curves on surfaces; effective divisor; 1- motive; periods (Co)homology theory in algebraic geometry, Generalizations (algebraic spaces, stacks), Algebraic cycles, Singularities of surfaces or higher-dimensional varieties, Curves in algebraic geometry On glueing curves on surfaces and zero cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The monad of a curve \(C\) in \(\mathbb{P}^3\) is a complex of decomposable vector bundles \(0\to F_2\to F_1\to F_0\to 0\) which is exact except in the middle where the homology is \(I_C\). The monad may be computed from the minimal free resolution of the normalized coordinate ring of the curve. A curve is said to have the expected monad if some cohomological conditions are satisfied by \(I_C\) twisted by the dualizing sheaves.
In this paper the author studies the monads of curves of given degree with very low genus, the so called ``range \(A\)'' curves. In his main theorem he proves that for given \((d, g)\) there exists a curve of degree \(d\) in \(\mathbb{P}^3\) and of genus \(g\) with the expected monad if and only if \((d, g)\) satisfy the degree and genus inequality of ``range \(A\)'' curves. The main tool in this paper is the smoothing theory developed by \textit{C. Walter} [``Normal bundles and smoothing of algebraic space curves'' (preprint)]. The author also constructs examples of curves with the expected monad. monad of a curve; resolution of the normalized coordinate ring; degree; low genus; smoothing Walter, CH, Curves in \(\mathbb{P}^r\) with the expected monad, J. Algebr. Geom., 4, 301-320, (1995) Plane and space curves, Special algebraic curves and curves of low genus, Vector bundles on curves and their moduli Curves in \(\mathbb{P}^ 3\) with the expected monad | 0 |
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