{
  "abstract": "We compute the integral $K_0$-ring of the moduli stack $\\mathscr H_g$ of smooth hyperelliptic curves of genus $g\\ge2$ over a field of characteristic zero. We show that it is a free abelian group of rank $8g+2$ for even $g$ and $8g+4$ for odd $g$. We use the isotypic decomposition of the $K_0$ of a related $μ_{g+1}$-gerbe to obtain presentations for both even and odd $g$'s from the standard quotient presentation of $\\mathscr H_g$. Thus, our technique differs from the $CH^\\bullet(\\mathscr H_g)$ computations of Edidin--Fulghesu, Fulghesu--Viviani, Di Lorenzo, and the $K_0(\\mathscr M_2)$ computation of Edidin--Hu. Using Pagani's description of the inertia stack of $\\mathscr H_g$, we apply Riemann--Roch to interpret the rank of $K_0(\\mathscr H_g)$ as the number of inertia components. Finally, we compare the $K_0$ and Chow rings of $\\mathscr H_g$ through the gamma-filtration and give presentations to the associated graded rings $\\operatorname{gr}_γK_0(\\mathscr H_g)\\otimes\\mathbb F_p$ for every prime $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27267",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Ruofan Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3z6s8kffcdx22.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27267",
  "fulltext_url": null,
  "id": "3z6s8kffcdx22",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27267",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223132584",
  "pdf_url": "https://arxiv.org/pdf/2610.27267",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3z6s8kffcdx22.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27267",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27267"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The integral $K_0$-ring of the stack of smooth hyperelliptic curves",
  "title_text": "The integral K_0-ring of the stack of smooth hyperelliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/3z6s8kffcdx22",
  "venue": "arXiv",
  "versions": []
}