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

We prove that the fiber product of the Torelli map $t\text{\hspace{0.17em}⁣}:{\mathcal{M}}_g^{ct}\to {\mathcal{A}}_g$ with any product ${\mathcal{A}}_{g_1}\times \cdots \times {\mathcal{A}}_{g_k}\to {\mathcal{A}}_g$ for $g=g_1+\cdots +g_k$ has a reduced scheme structure. As a consequence, letting $d=\text{codim}(t^{\ast }[{\mathcal{A}}_{g_1}\times \cdots \times {\mathcal{A}}_{g_k}])$, we find that the class $t^{\ast }[{\mathcal{A}}_{g_1}\times \cdots \times {\mathcal{A}}_{g_k}]\in {\mathsf{C}\mathsf{H}}^d({\mathcal{M}}_g^{ct})$ is tautological. In particular, we obtain $t^{\ast }[{\mathcal{A}}_{g_1}\times \cdots \times {\mathcal{A}}_{g_k}]=0$ for $d>2g-3.$