Inventiones math Inventione matbematicae by Springer Verlag

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Inventiones math. 41, 149 - 196 (1977) Inventione$ matbematicae 9 by Springer-Verlag 1977 Prym Varieties and the Schottky Problem Arnaud Beauville D6partment de Math6matiques, Universit6 d'Angers, F-49000 Angers, France O. Introduction The Schottky problem is the problem of characterizing Jacobian varieties among all abelian varieties. More precisely, let: stg = HJSp(Z, 2g) 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: Theorem.