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 work studies the fiber product of the Torelli map $t:\ \mathcal{M}^{ct}_g\to\mathcal{A}_g$ with a product morphism $\mathcal{A}_{g_1}\times\cdots\times\mathcal{A}_{g_k}\to\mathcal{A}_g$ for $g=g_1+\cdots+g_k$ and proves that this fiber product is reduced. Using excess intersection theory, it analyzes the pullback $t^* [\mathcal{A}_{g_1}\times\cdots\times\mathcal{A}_{g_k}]$ in $\mathsf{CH}^*(\mathcal{M}^{ct}_g)$ and identifies it as a tautological class of codimension $d=\mathrm{codim}(t^*[\mathcal{A}_{g_1}\times\cdots\times\mathcal{A}_{g_k}])$. Consequently, $t^* [\mathcal{A}_{g_1}\times\cdots\times\mathcal{A}_{g_k}]$ lies in $\mathsf{CH}^d(\mathcal{M}^{ct}_g)$ and vanishes when $d>2g-3$. These results connect the geometry of decomposable abelian varieties with the tautological structure of the Chow rings on the moduli of compact-type curves, providing concrete constraints on the intersection theory of $\mathcal{A}_g$ via the Torelli map.

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 $d > 2g-3.$