The tautological ring of the space of pointed genus two curves of compact type

We prove that the tautological ring of $M_{2,n}^{ct}$, the moduli space of n-pointed genus two curves of compact type, does not have Poincar\'e duality for any $n \geq 8$. This result is obtained via a more general study of the cohomology groups of $M_{2,n}^{ct}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^k(M_{2,n}^{ct})$ for any $k$ and $n$ considered both as $S_n$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline M_{2,n}$ is tautological for $n < 20$, and that the tautological ring of $\overline M_{2,n}$ fails to have Poincar\'e duality for all $n \geq 20$. This improves and simplifies results of the author and Orsola Tommasi.