E. Izadi

The Chow Ring of the Moduli Space of Curves of Genus 5

Let M g be the moduli spare of smooth curves of genus g over an algebraically closed field (of characteristic differetd, from 2 and 3) and let \({\bar M_g}\) be its compactification by Deligne-Mumford stable curves.

Subvarieties of Abelian Varieties

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.

Fonctions thêta du second ordre sur la jacobienne d'une courbe lisse

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Correspondences with split polynomial equations

Abstract 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-Tyurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.

The uniformization of the moduli space of principally polarized abelian 6-folds

Abstract Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space 𝒜 6 \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E 6 E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E 6 E_{6} -covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to 𝒜 6 \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

Ampleness of intersections of translates of theta divisors in an abelian fourfold

We prove the ampleness of the cotangent bundle of the intersection of two general translates of a theta divosor of the Jacobian of a general curve of genus 4. From this, we deduce the same result in a general, principally polarized abelian variety of dimension 4.