{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33971",
  "author_total": 1,
  "authors": [
    {
      "name": "Young-Hoon Kiem"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0vy731cadxym2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33971",
  "fulltext_url": null,
  "id": "0vy731cadxym2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33971",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223932050",
  "pdf_url": "https://arxiv.org/pdf/2610.33971",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0vy731cadxym2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33971",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33971"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hodge--Deligne polynomials and $S_n$-representations on the cohomology of $\\overline{\\mathcal M}_{2,n}$",
  "title_text": "Hodge--Deligne polynomials and S_n-representations on the cohomology of ℳ̅_2,n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0vy731cadxym2",
  "venue": "arXiv",
  "versions": []
}