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 , 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  and by R. de Jong in , version 2.
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|Titolo:||Singularities of the Theta Divisor at Points of Order Two|
|Data di pubblicazione:||2007|
|Appartiene alla tipologia:||01a Articolo in rivista|