Seminar in Number Theory and Algebraic Geometry

Abstract: The Torelli map t: M_g^(ct) -> A_g is a useful tool for studying algebraic cycles on the moduli space of abelian varieties. Using excess intersection theory, we can compute the Torelli pullback of any cycle [A_(g_1) x ... x A_(g_k)] on A_g where g_1 + ... + g_k = g. However, this requires a full understanding of the local scheme structure for the pullback of A_(g_1) x ... x A_(g_k) along the Torelli map.

In the first part of my talk, I will give a background on the Torelli map and indicate how it can be used to study cycles on A_g. In the second part, I will describe a strategy for finding the precise local structure of the fiber product of the Torelli map and the product map A_(g_1) x ... x A_(g_k) -> A_g. This involves studying series expansions of period integrals in terms of local coordinates on the moduli space of curves.