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

TL;DR

This paper computes intersection numbers on the Poincare bundle to determine the classes of universal theta divisors on the moduli space of semiabelian varieties, advancing understanding of their geometric structure.

Contribution

It provides explicit intersection number calculations and new formulas for universal theta divisor classes on the boundary of the moduli space.

Findings

Computed all top intersection numbers of divisors on the Poincare bundle.

Derived the class of the universal theta divisor and m-theta divisor in semiabelian varieties.

Calculated boundary coefficients for the Andreotti-Mayer divisor and the universal m-theta divisor.

Abstract

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.