Hodge--Deligne polynomials and $S_n$-representations on the cohomology of $\overline{\mathcal M}_{2,n}$

Abstract

We compute the Hodge-Deligne polynomials of the moduli spaces $\overline{\mathcal M}_{2,n}$ of stable $n$-pointed curves of genus two, together with the action of the symmetric group $S_n$ on their cohomology, for all $n$. Using Hassett's moduli spaces of weighted pointed curves, we first show that the generating function of the $S_n$-equivariant Hodge-Deligne polynomials of $\overline{\mathcal M}_{g,n}$ is obtained from that of the moduli spaces of minimally weighted pointed curves by plethysm with the explicit symmetric function $Z=h_1-\sum_{k\ge3}(x+x^2+\dots+x^{k-2})h_k$; numerically, this is an explicit change of variables. The latter spaces map to $\overline{\mathcal M}_g$, and over each boundary stratum they are built from punctured universal curves and Losev-Manin spaces. Combining the stratification by stable graphs with Burnside's formula and twisted characteristics, we obtain in genus two a closed formula whose only non-elementary input is the cohomology of symplectic local systems on $\mathcal M_{1,1}$ and $\mathscr A_2$, known by work of Eichler-Shimura and Petersen. This determines the $S_n$-representations on all $H^{p,q}(\overline{\mathcal M}_{2,n})$ for $n\le 19$ and on all even degree $H^{p,q}(\overline{\mathcal M}_{2,n})$ for every $n$. We carry out the computation in all degrees for $n\le 18$, where it agrees with an independent computation by point counts, and in even degrees for $n\le 30$. As an application, we find that the even degree cohomology of $\overline{\mathcal M}_{2,n}$ is of Hodge-Tate type if and only if $n\le 20$: the space $H^{23,1}(\overline{\mathcal M}_{2,21})$ is the multiplicity free $S_{21}$-representation with Frobenius characteristic $s_1\cdot h_2[e_{10}]$, of dimension $1939938$, so that $H^{24}(\overline{\mathcal M}_{2,21})$ is not spanned by algebraic cycle classes.