Prym varieties and the Schottky problem

be the moduli space of principally polarized abelian varieties of dimension g, Jg c ~q/g the locus of Jacobians. The problem is to find explicit equations for Jg (or rather its closure Jg) in s/g. In their beautiful paper [A-M], Andreotti and Mayer prove that Jg is an irreducible component of the locus N~_ 4 of principally polarized abelian varieties (A, O) with dim Sing O > g 4 . Then they give a procedure to write "explicit" equations for N~_ 4. There is no hope that Jg be equal to Ng_ 4: already in genus 4, there is at least one other component, namely the divisor 0,un of principally polarized abelian varieties with one vanishing theta-null (i.e. such that Sing O contains a point of order 2). Our aim is to prove the following: