## Abstract

In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor)—denote by θnull the locus of such ppavs. We describe the locus $θnullg−1⊂θnull$ where this singularity is not an ordinary double point. By using theta function methods we first show $θnullg−1⊊θnull$ (this was shown in [4], see below for a discussion). We then show that $θnullg−1$ is contained in the intersection $θnull∩N0′$ of the two irreducible components of the Andreotti-Mayer $N0=θnull+2N0′$, and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in $θnullg−1$.

Some of the intermediate results obtained along the way of our proof were concurrently obtained independently by C. Ciliberto and G. van der Geer in [3] and by R. de Jong in [5], version 2.