Singularities of the Theta Divisor at Points of Order Two

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 θ<inf>null</inf> the locus of such ppavs. We describe the locus <f>$${\\mathit{\\theta }}_{\\hbox{ null }}^{g-1}\\phantom{\\rule{0.4em}{0ex}}\\subset \\phantom{\\rule{0.4em}{0ex}}{\\mathit{\\theta }}_{\\hbox{ null }}$$</f> where this singularity is not an ordinary double point. By using theta function methods we first show <f>$${\\mathit{\\theta }}_{\\hbox{ null }}^{g-1}\\subsetneq {\\mathit{\\theta }}_{\\hbox{ null }}$$</f> (this was shown in [4], see below for a discussion). We then show that <f>$${\\mathit{\\theta }}_{\\hbox{ null }}^{g-1}$$</f> is contained in the intersection <f>$${\\mathit{\\theta }}_{\\hbox{ null }}\\phantom{\\rule{0.4em}{0ex}}\\cap \\phantom{\\rule{0.4em}{0ex}}{N}_{0}^{\\mathrm{\\prime }}$$</f> of the two irreducible components of the Andreotti-Mayer <f>$${N}_{0}\\phantom{\\rule{0.4em}{0ex}}=\\phantom{\\rule{0.4em}{0ex}}{\\mathit{\\theta }}_{\\hbox{ null }}+2\\phantom{\\rule{0.1em}{0ex}}{N}_{0}^{\\mathrm{\\prime }}$$</f>, and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in <f>$${\\mathit{\\theta }}_{\\hbox{ null }}^{g-1}$$</f>. 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.