Tautological rings of spaces of pointed genus two curves of compact type (Q2816794)

Let \(\mathcal M_{2,n}^{\text{ct}}\) be the moduli space of \(n\)-pointed genus two curves of compact type. The main result of the paper shows that \(\mathcal M_{2,n}^{\text{ct}}\) does not satisfy Poincaré duality for any \(n\geq 8\). In order to prove the result, the author conducts a general study of the cohomology of \(\mathcal M_{2,n}^{\text{ct}}\), decomposing it into pieces corresponding to local systems and identifying it with the tautological cohomology. The method developed in the paper allows general computation of \(H^k(\mathcal M_{2,n}^{\text{ct}})\) for any \(k\) and \(n\). As a consequence, the author shows that all even cohomology of the moduli space of stable genus two curves \(\overline{\mathcal M}_{2,n}\) is tautological for \(n < 20\) and that the tautological ring of \(\overline{\mathcal M}_{2,n}\) fails to satisfy Poincaré duality for any \(n\geq 20\). This paper generalizes and simplifies earlier results of the author and \textit{O. Tommasi} [Invent. Math. 196, No. 1, 139--161 (2014; Zbl 1295.14030)].