Some Intersections in the Poincaré Bundle and the Universal Theta Divisor on $\overline{\mathcal{A}}_{g}$

We compute all the top intersection numbers of divisors on the total space of the Poincaré bundle restricted to B×C (where B is an abelian variety, and C⊂B is any test curve). 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.