The fiber product of the Torelli map with any product $\mathcal{A}_{g_1}\times \dots \times \mathcal{A}_{g_k}\to\mathcal{A}_g$ is reduced

TL;DR

This paper proves that the fiber product of the Torelli map with any product of moduli spaces is reduced, leading to new tautological class relations and vanishing results in the Chow ring of the moduli space of curves.

Contribution

It establishes the reducedness of the fiber product of the Torelli map with arbitrary products of moduli spaces, and derives tautological class properties and vanishing results.

Findings

Fiber product of Torelli map with products is reduced.

Tautological class pullbacks are well-defined and vanish in high codimension.

Provides new relations in the Chow ring of the moduli space.

Abstract

We prove that the fiber product of the Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ with any product $\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k} \to \mathcal{A}_g$ for $g=g_1+\dots+g_k$ has a reduced scheme structure. As a consequence, letting $d=\text{codim}(t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}])$, we find that the class $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}]\in \mathsf{CH}^{d}(\mathcal{M}^{ct}_g)$ is tautological. In particular, we obtain $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}] = 0$ for $d > 2g-3.$