Aitor Iribar López | Searcharxiv

arXiv2026

The fifteenth cohomology group of moduli spaces of stable curves

The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let $S^*_ω(\overline{\mathcal M}_{g,n}) \subset H^*(\overline{\mathcal M}_{g,n})$ denote the semi-tautological extension generated by $H^{11}(\overline{\mathcal M}_{1,11})$, that is, the smallest system of subrings of $H^*(\overline{\mathcal M}_{g,n})$ containing $H^{11}(\overline{\mathcal M}_{1,11})$ and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that $H^{11}(\overline{\mathcal M}_{g,n}) = S^{11}_ω(\overline{\mathcal M}_{g,n})$ and $H^{13}(\overline{\mathcal M}_{g,n}) = S^{13}_ω(\overline{\mathcal M}_{g,n})$ for all $g$ and $n$. Meanwhile, $H^{15}(\overline{\mathcal M}_{1,n}) \neq S_ω^{15}(\overline{\mathcal M}_{1,n})$ for all $n \geq 15$. Nevertheless, we prove that $H^{15}(\overline{\mathcal M}_{2,n}) = S^{15}_ω(\overline{\mathcal M}_{2,n})$ for all $n$. Moreover, we determine all relations in $S_ω^{15}(\overline{\mathcal M}_{g,n})$ for all $g \geq 2$ and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of $H^{15}(\overline{\mathcal M}_{3,14})$, we also prove that $H^{15}(\overline{\mathcal M}_{3,n}) = S^{15}_ω(\overline{\mathcal M}_{3,n})$ for all $n$. As an application, we prove $H^*(\overline{\mathcal M}_{g,n}) = RH^*(\overline{\mathcal M}_{g,n})$ for several new small pairs $(g,n)$.

math.AG↗

arXiv2026

Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$

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 $\mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}})$.

math.AG↗