Self-intersection of the Torelli map

Abstract

The Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ is far from an immersion for $g\geq 3$: the self-fiber product of the Torelli map for $g\geq 3$ has several components with nontrivial intersections. We give a stratification of the self-fiber product for arbitrary genus and describe how components in the fiber product intersect. In genus $4$, the Torelli fiber product is nonreduced, which we prove by analyzing the expansion of the period map near a nodal curve. We use the geometry of the Torelli fiber product to: Calculate the class of the pullback to $\mathcal{M}^{ct}_4$ of the Torelli cycle $t_*[\mathcal{M}^{ct}_4]$ on $\mathcal{A}_4$; Find the class $t_*[\overline{\mathcal{M}}_4]$ for suitable toroidal compactifications $\overline{\mathcal{A}}_4$; Calculate the class $t^*t_*[\mathcal{M}^{ct}_5]|_{\mathcal{M}_5}$. In the first appendix, we write down a calculation for finding the Chern classes of $\overline{\mathcal{M}}_{g,n}$. In the second, we give a formula for a coefficient occurring in an intersection of excess dimension.