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 θnullg1θnull where this singularity is not an ordinary double point. By using theta function methods we first show θnullg1θnull (this was shown in [4], see below for a discussion). We then show that θnullg1 is contained in the intersection θnullN0 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 θnullg1.

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.

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