{
  "abstract": "For a finite abelian group $G$ acting linearly on the plane let $\\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\\mathcal{X}^{[ρ]}$. By means of a Grothendieck-Riemann-Roch theorem of Toën we give explicit representation theoretic formulas of $\\dim(\\mathcal{X}^{[ρ]})$ for all $ρ\\in \\operatorname{Rep}(G)$ which are similar to the McKay correspondence. We also give a classification of all such quotient stacks according to an associated quadratic form called the Euler form. For $[\\mathbb{A}^2/μ_n]$ given by the cyclic action $ζ\\cdot (x,y) = (ζx, ζ^{-1} y)$, we give an interpretation of the corresponding Euler form in terms of border strips which may be removed by a diagram, an equational characterization of $n$-core partitions, and the number of significant arrows of a monomial ideal as defined in \"Smooth and Irreducible Multigraded Hilbert Schemes\" by D. Maclagan and G. Smith.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12736",
  "author_total": 2,
  "authors": [
    {
      "name": "Samuel Jay Bachhuber"
    },
    {
      "name": "Amin Gholampour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4acjwm06hnqak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12736",
  "fulltext_url": null,
  "id": "4acjwm06hnqak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12736",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221538494",
  "pdf_url": "https://arxiv.org/pdf/2610.12736",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4acjwm06hnqak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12736",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12736"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$G$-Equivariant Hilbert Schemes and Quadratic Forms",
  "title_text": "G-Equivariant Hilbert Schemes and Quadratic Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/4acjwm06hnqak",
  "venue": "arXiv",
  "versions": []
}