DOCUMENTA MATHEMATICA, Vol. 13 (2008), 453-504

Ciro Ciliberto, Gerard van der Geer

Andreotti--Mayer Loci and the Schottky Problem

We prove a lower bound for the codimension of the Andreotti-Mayer locus $N_{g,1}$ and show that the lower bound is reached only for the hyperelliptic locus in genus $4$ and the Jacobian locus in genus $5$. In relation with the intersection of the Andreotti-Mayer loci with the boundary of the moduli space ${\Acal}_g$ we study subvarieties of principally polarized abelian varieties $(B,\Xi)$ parametrizing points $b$ such that $\Xi$ and the translate $\Xi_b$ are tangentially degenerate along a variety of a given dimension.

2000 Mathematics Subject Classification: 14K10

Keywords and Phrases: Abelian variety, theta divisor, Andreotti-Mayer loci, Schottky problem.

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