JournalsjemsVol. 16, No. 9pp. 1817–1848

Singularities of theta divisors and the geometry of A5\mathcal A_5

  • Gavril Farkas

    Humboldt-Universität zu Berlin, Germany
  • Samuel Grushevsky

    Stony Brook University, USA
  • R. Salvati Manni

    Università di Roma La Sapienza, Italy
  • Alessandro Verra

    Università degli studi Roma Tre, Italy
Singularities of theta divisors and the geometry of $\mathcal A_5$ cover
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Abstract

We study the codimension two locus HH in Ag\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]CH2(Ag)[H]\in CH^2(\mathcal A_g) for every gg. For g=4g=4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5g=5, via the Prym map we show that HA5H\subset \mathcal A_5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of A5\overline{\mathcal A_5} and show that the component N0\overline{N_0'} of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension κ(A5,N0)\kappa(\overline{\mathcal A_5}, \overline{N_0'}) is equal to zero.

Cite this article

Gavril Farkas, Samuel Grushevsky, R. Salvati Manni, Alessandro Verra, Singularities of theta divisors and the geometry of A5\mathcal A_5. J. Eur. Math. Soc. 16 (2014), no. 9, pp. 1817–1848

DOI 10.4171/JEMS/476