Singularities of theta divisors and the geometry of A_5

Details Singularities of theta divisors and the geometry of A_5 Singularities of theta divisors and the geometry of A_5

2011-12-29

arxiv.org/abs/1112.6285v1

Mathematics

Algebraic Geometry

31 pages

Scientific paper

We study the codimension two locus H in 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 of H in 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 in A_5 has two components, both unirational, which we completely describe. This gives a geometric classification of 5-dimensional ppav whose theta-divisor has a quadratic singularity of non-maximal rank. We then determine the slope of the effective cone of A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is equal to zero.

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