Hodge--Deligne polynomials and Sn-representations on the cohomology of ℳ̅2,n
Abstract
We compute the Hodge-Deligne polynomials of the moduli spaces ℳ̅2,n of stable n-pointed curves of genus two, together with the action of the symmetric group Sn on their cohomology, for all n. Using Hassett's moduli spaces of weighted pointed curves, we first show that the generating function of the Sn-equivariant Hodge-Deligne polynomials of ℳ̅g,n is obtained from that of the moduli spaces of minimally weighted pointed curves by plethysm with the explicit symmetric function Z=h1−∑k≥3(x+x2+…+xk−2)hk; numerically, this is an explicit change of variables. The latter spaces map to ℳ̅g, 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 ℳ1,1 and 𝒜2, known by work of Eichler-Shimura and Petersen. This determines the Sn-representations on all Hp,q(ℳ̅2,n) for n≤ 19 and on all even degree Hp,q(ℳ̅2,n) for every n. We carry out the computation in all degrees for n≤ 18, where it agrees with an independent computation by point counts, and in even degrees for n≤ 30. As an application, we find that the even degree cohomology of ℳ̅2,n is of Hodge-Tate type if and only if n≤ 20: the space H23,1(ℳ̅2,21) is the multiplicity free S21-representation with Frobenius characteristic s1· h2[e10], of dimension $1939938$, so that H24(ℳ̅2,21) is not spanned by algebraic cycle classes.
Related
- Hodge-Deligne and Poincaré polynomials of ℳ̅1,n
- Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks
- Irregular Hodge Bundles for Deligne--Mumford Landau--Ginzburg Families with Fixed Pole Orders
- Products in absolute Hodge cohomology and mixed Hodge formality
- Mixed Hodge structure in the cohomology of moment-angle complexes
BibTeX
@misc{kiem2026hodgedeligne,
title = {{Hodge--Deligne polynomials and $S_n$-representations on the cohomology of $\overline\mathcal M_2,n$}},
author = {Young-Hoon Kiem},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33971},
eprint = {2610.33971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33971},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hodge--Deligne polynomials and S_n-representations on the cohomology of ℳ̅_2,n
AU - Young-Hoon Kiem
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33971
UR - https://arxiv.org/abs/2610.33971
AB - We compute the Hodge-Deligne polynomials of the moduli spaces ℳ̅_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 ℳ̅_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−∑_k≥3(x+x^2+…+x^k−2)h_k; numerically, this is an explicit change of variables. The latter spaces map to ℳ̅_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 ℳ_1,1 and 𝒜_2, known by work of Eichler-Shimura and Petersen. This determines the S_n-representations on all H^p,q(ℳ̅_2,n) for n≤ 19 and on all even degree H^p,q(ℳ̅_2,n) for every n. We carry out the computation in all degrees for n≤ 18, where it agrees with an independent computation by point counts, and in even degrees for n≤ 30. As an application, we find that the even degree cohomology of ℳ̅_2,n is of Hodge-Tate type if and only if n≤ 20: the space H^23,1(ℳ̅_2,21) is the multiplicity free S_21-representation with Frobenius characteristic s_1· h_2[e_10], of dimension $1939938$, so that H^24(ℳ̅_2,21) is not spanned by algebraic cycle classes.
ER -