We study the codimension two locus in consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class for every . For , this turns out to be the locus of Jacobians with a vanishing theta-null. For , via the Prym map we show that has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of and show that the component of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension is equal to zero.
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Gavril Farkas, Samuel Grushevsky, R. Salvati Manni, Alessandro Verra, Singularities of theta divisors and the geometry of . J. Eur. Math. Soc. 16 (2014), no. 9, pp. 1817–1848DOI 10.4171/JEMS/476