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 theta(null) the locus of such ppavs. We describe the locus theta(g-1)(null) subset of theta(null) where this singularity is not an ordinary double point. By using theta function methods we first show theta(g-1)(null) not subset of theta(null) (this was shown in [4], see below for a discussion). We then show that theta(g-1)(null) is contained in the intersection theta(null) boolean AND N(0)(') of the two irreducible components of the Andreotti-Mayer N(0) = theta(null) + 2N(0)', and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in theta(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.