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.

Full text: dvi.gz 110 k, dvi 264 k, ps.gz 861 k, pdf 497 k.

Home Page of DOCUMENTA MATHEMATICA