Some intersections in the Poincare bundle, and the universal theta divisor on the moduli space of (semi)abelian varieties

Details Some intersections in the Poincare bundle, and the universal theta divisor on the moduli space of (semi)abelian varieties Some intersections in the Poincare bundle, and the universal theta divisor on the moduli space of (semi)abelian varieties

2005-12-22

arxiv.org/abs/math/0512530v3

Int. Math. Res Notices (2008) Vol. 2008, article ID rnm128

Mathematics

Algebraic Geometry

v2: Weaker results: no claim regarding the Kodaira dimension of A_6. The error in version 1 was in the computation of $m$-thet

Scientific paper

10.1093/imrn/rnm128

We compute all the top intersection numbers of divisors on the total space of the Poincare bundle restricted to the product of a curve and the abelian variety. We use these computations to find the class of the universal theta divisor and $m$-theta divisor inside the universal corank 1 semiabelian variety -- the boundary of the partial toroidal compactification of the moduli space of abelian varieties. We give two computational examples: we compute the boundary coefficient of the Andreotti-Mayer divisor (computed by Mumford but in a much harder and ad hoc way), and the analog of this for the universal $m$-theta divisor.

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