{
  "abstract": "We study two families of moduli spaces of sheaves attached to the node, namely the punctual Hilbert schemes of the non-reduced plane curve $x^uy^v=0$, and the punctual Quot schemes of the module $k[[x]]^u\\oplus k[[y]]^v$ over the reduced node $xy=0$. We compute their motivic generating functions for all $u,v$ and over every field. Both are specializations of a single rational function whose numerator counts matrices by rank, and they are related by the substitution $\\mathbb{L}\\mapsto Q\\mathbb{L}$. This proves the Hilb--Quot correspondence for the node with independent ranks on the two branches. The main geometric ingredient is an elementary modification of an ideal along a branch of the curve, whose fibres are affine spaces of dimension equal to a contact length. We also construct stratifications of the Hilbert schemes into products of affine spaces and tori, and we formulate a branchwise Hilb--Quot--HOMFLY conjecture, which we test on the colored Hopf link.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34676",
  "author_total": 3,
  "authors": [
    {
      "name": "Yifeng Huang"
    },
    {
      "name": "Ruofan Jiang"
    },
    {
      "name": "Alexei Oblomkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/222crnahkpda4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34676",
  "fulltext_url": null,
  "id": "222crnahkpda4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34676",
  "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": "W7223976720",
  "pdf_url": "https://arxiv.org/pdf/2610.34676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/222crnahkpda4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sheaves on nodal singularities I",
  "title_text": "Sheaves on nodal singularities I",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/222crnahkpda4",
  "venue": "arXiv",
  "versions": []
}