Self-intersection of the Torelli map

Abstract

The Torelli map $t:{\mathcal{M}}_g^{ct}\to {\mathcal{A}}_g$ is far from an immersion for $g\ge 3$: the self-fiber product of the Torelli map for $g\ge 3$ 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 ${\mathcal{M}}_4^{ct}$ of the Torelli cycle $t_{\ast }[{\mathcal{M}}_4^{ct}]$ on ${\mathcal{A}}_4$; Find the class $t_{\ast }[{\overline{\mathcal{M}}}_4]$ for suitable toroidal compactifications ${\overline{\mathcal{A}}}_4$; Calculate the class $t^{\ast }{t}_{\ast }[{\mathcal{M}}_5^{ct}]{|}_{{\mathcal{M}}_5}$. 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.

58 pages