Second order theta divisors on Pryms

Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed characterizations of the locus of jacobians which use the linear system of $2\Theta$-divisors. We give new evidence for these conjectures in the case of Prym varieties.Comment: AMS-Latex, 11 pages, the exposition has been modified, Proposition 5 has been modifie

A Prym construction for the cohomology of a cubic hypersurface

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined \'etale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in $P^n$.Comment: AMS-Latex, 39 page

A new Homological Invariant for Modules

Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $\zeta$-dimension, denoted by $\zeta-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $\zeta$-dimension. Our definition of $\zeta$-dimension offer a new homological perspective on the projective dimension, complete intersection dimension of Avramov et al. and $G$-dimension of Auslander and Bridger

Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space \cA_g of principally polarized abelian varieties of dimension $g$ or of the moduli space \tM_g of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if V \subset \cA_g is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V$ if V \subset \tM_g (or \cA_g) is complete of codimension $g$.Comment: AMS-LaTeX, 15 page

Deforming curves in jacobians to non-jacobians I: curves in C(2)C^{(2)}

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in the second symmetric power $C^{(2)}$ of $C$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing pairs of points in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). We prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=4$ or $d=5$, dim$H^0 (g^1_5) = 3$ and $C$ has genus 4.Comment: amslatex, 25 page

On Rectification of Circles and an Extension of Beltrami's Theorem

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of \r^3, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in \r^3 passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in \r^3 in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circles.}Comment: 18 page

Congruent Numbers Via the Pell Equation and its Analogous Counterpart

The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections between Pythagorean triples and the Pell equation $x^2-dy^2=1$ plus its analogous counterpart $x^2-dy^2=-1$ which give rise to congruent numbers n with arbitrarily many prime factors.Comment: 8 page

An inductive approach to the Hodge conjecture for abelian varieties

We describe an inductive approach most appropriate for abelian varieties with an action of an imaginary quadratic field.Comment: amslatex, 11 page

On the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of A^k+B^k+C^k=D^k+E^4; k=1,3.Comment: 7 pag

Diophantine Equation X4+Y4=2(U4+V4)X^4+Y^4=2(U^4+V^4)

In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation $X^4+Y^4=2(U^4+V^4)$ Keywords: Diophantine equation, Elliptic curve, Congruent numberComment: 5 page