arXiv subjects
Publications and source records attributed to E. Izadi.
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. We survey some of the results known about this primitive cohomology, prove a few general facts and mention some interesting open problems.
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.
We give examples of curves of arbitrarily high Clifford index such that the Prym map is not injective at any of their étale double covers.
We show that the coarse moduli space $\cR_5$ of étale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational.
We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tjurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.
We describe an inductive approach most appropriate for abelian varieties with an action of an imaginary quadratic field.
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.
We prove that the cotangent bundle of a complete intersection of two general translates of the theta divisor of the jacobian of a general curve of genus 4 is ample. From this the same result for a general principally polarized abelian variety of dimension 4 follows.
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 symmetric powers $C^{(e)}$ of $C$ where $3\leq e\leq g-3$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing divisors of degree $e$ in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair $(C, g^1_d)$, we prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=2e$, dim$H^0 (g^1_d) = e$ or $d=2e+1$, dim$H^0 (g^1_d) = e+1$. The analogous result in the case $e=2$ without genericity assumptions was proved earlier.
We discuss various constructions which allow one to embed a principally polarized abelian variety in the jacobian of a curve. Each of these gives representatives of multiples of the minimal cohomology class for curves which in turn produce subvarieties of higher dimension representing multiples of the minimal class. We then discuss the problem of producing curves representing multiples of the minimal class via deformation-theoretic methods.
Let $P \cup P'$ be the two component Prym variety associated to an étale double cover $\tilde{C} \to C$ of a non-hyperelliptic curve of genus $g \geq 6$ and let $|2Ξ_0|$ and $|2Ξ_0'|$ be the linear systems of second order theta divisors on $P$ and $P'$ respectively. The component $P'$ contains canonically the Prym curve $\tilde{C}$. We show that the base locus of the subseries of divisors containing $\tilde{C} \subset P'$ is scheme-theoretically the curve $\tilde{C}$. We also prove canonical isomorphisms between some subseries of $|2Ξ_0|$ and $|2Ξ_0'|$ and some subseries of second order theta divisors on the Jacobian of $C$.
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.
Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed characterizations of the locus of jacobians which use the linear system of $2Θ$-divisors. We give new evidence for these conjectures in the case of Prym varieties.
Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined étale 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 $P^n$.
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 $g$.