Sheaves on nodal singularities I — Draconitic Research

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⊕ 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 𝕃↦ Q𝕃. 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.

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BibTeX RIS

BibTeX

@misc{huang2026sheaves,
  title = {{Sheaves on nodal singularities I}},
  author = {Yifeng Huang and Ruofan Jiang and Alexei Oblomkov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34676},
  eprint = {2610.34676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34676},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Sheaves on nodal singularities I
AU  - Yifeng Huang
AU  - Ruofan Jiang
AU  - Alexei Oblomkov
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34676
UR  - https://arxiv.org/abs/2610.34676
AB  - 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⊕ 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 𝕃↦ Q𝕃. 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.
ER  -

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