Description
Title: The Taylor expansion of the Torelli map and applications to intersection theory
Abstract: Using results of Hu, Norton '18 and Yamada '80, we can deduce the full Taylor expansions of the Torelli map and the Prym map at the boundary in terms of plumbing coordinates. We use the Taylor expansion to determine the scheme structure for the fiber product of the Torelli map $t: M_g^{ct} \rightarrow A_g$ and the product map $A_{g_1} \times ... \times A_{g_k} \rightarrow A_g$, where $g_1 + ... + g_k = g$. This allows us to compute the class $t^*[A_{g_1} \times ... \times A_{g_k}]$ using excess intersection theory in arxiv:2601.04353. The Taylor expansions can also be used for studying the scheme structure for the self-fiber product of the Torelli map, as well as for the fiber product of the product map and Beauville's extended Prym map.