Journal of the European Mathematical Society
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Published online: 2014-10-22
Singularities of theta divisors and the geometry of $\mathcal A_5$
Gavril Farkas[1], Samuel Grushevsky[2], R. Salvati Manni[3] and Alessandro Verra (1) Humboldt-Universität zu Berlin, Germany(2) Stony Brook University, USA
(3) Università di Roma La Sapienza, Italy
We study the codimension two locus $H$ in $\mathcal A_g$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $[H]\in CH^2(\mathcal A_g)$ for every $g$. For $g=4$, this turns out to be the locus of Jacobians with a vanishing theta-null. For $g=5$, via the Prym map we show that $H\subset \mathcal A_5$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\overline{\mathcal A_5}$ and show that the component $\overline{N_0'}$ of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension $\kappa(\overline{\mathcal A_5}, \overline{N_0'})$ is equal to zero.
Keywords: Theta divisor, moduli space of principally polarized abelian varieties, effective cone, Prym variety
Farkas Gavril, Grushevsky Samuel, Salvati Manni R., Verra Alessandro: Singularities of theta divisors and the geometry of $\mathcal A_5$. J. Eur. Math. Soc. 16 (2014), 1817-1848. doi: 10.4171/JEMS/476