The tautological \mathbb{Q}-subalgebra \mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g) of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical \mathbb{Q}-linear projection operator \mathsf{taut}: \mathsf{CH}^*(\mathcal{A}_g) \rightarrow \mathsf{R}^*(\mathcal{A}_g). We present here new calculations of intersection products of the Torelli locus in \mathcal{A}_g with the product loci \mathcal{A}_{r}\times \mathcal{A}_{g-r} \rightarrow \mathcal{A}_g for r\leq 3. The results suggest that \mathsf{taut} is a \mathbb{Q}-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property.
Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps.
We define tautological projections of cycles on the fiber products \mathcal X_g^s \to \mathcal A_g of the universal family. We compute these projections for a class of product cycles on \mathcal X_g^s in terms of a determinant involving the universal theta divisors and Poincaré classes. Using Abel-Jacobi pullbacks of product cycles on \mathcal X_g^s and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings \mathsf{R}^*(\mathcal M^{\mathrm{ct}}_{g,n}). In particular, we construct nontrivial elements of the Gorenstein kernels of \mathsf{R}^5(\mathcal{M}_{5,2}^{\mathrm{ct}}) and \mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}}).
