Singularities of the theta divisor at points of order two

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 θ_null the locus of such ppavs. We describe the locus θ^g-1_null ⊂ θ_null where this singularity is not an ordinary double point. By using theta function methods we first show θ^g-1_null ⊈ θ_null (this was shown in [4], see below for a discussion). We then show that θ^g-1_null is contained in the intersection θ_null ∩ N¹₀ of the two irreducible components of the Andreotti-Mayer N₀ = θ_null+2N ¹₀ , and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in θ^g-1_null . 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.