G-Equivariant Hilbert Schemes and Quadratic Forms

Abstract

For a finite abelian group G acting linearly on the plane let 𝒳 be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of 𝒳 whose corresponding subschemes of the plane have a given representation type, denoted by 𝒳^[ρ]. By means of a Grothendieck-Riemann-Roch theorem of Toën we give explicit representation theoretic formulas of dim(𝒳^[ρ]) for all ρ∈ 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 [𝔸²/μ_n] given by the cyclic action ζ· (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.

Non-reducedness of the Hilbert schemes of few points
Michał Szachniewicz
Mathematische Zeitschrift1 Oct 2026Peer-reviewedLicence not statedFull text16 citedGeometry & topologyOpenAlex
Hochschild cohomology of Hilbert schemes of points on surfaces
Pieter Belmans, Lie Fu, Andreas Krug
Journal für die reine und angewandte Mathematik18 Sep 2026Peer-reviewedCC BY 4.0Full textGeometry & topologyHAL
Asymptotics of holomorphic torsion on orbifolds and an equivariant arithmetic Hilbert-Samuel theorem
Kai Köhler
arXiv2 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv
Disconnectedness of the Hilbert Schemes of E₆/P₆
Minyoung Jeon
arXiv5 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Stability conditions on products of curves and Hilbert schemes of surfaces
Chunyi Li, Emanuele Macrì, Alexander Perry +2
Geometry & Topology12 Oct 2026Peer-reviewedCC BYFull text14 citedGeometry & topologyOpenAlex

Cite

BibTeX RIS

BibTeX

@misc{bachhuber2026gequivariant,
  title = {{$G$-Equivariant Hilbert Schemes and Quadratic Forms}},
  author = {Samuel Jay Bachhuber and Amin Gholampour},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12736},
  eprint = {2610.12736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12736},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - G-Equivariant Hilbert Schemes and Quadratic Forms
AU  - Samuel Jay Bachhuber
AU  - Amin Gholampour
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12736
UR  - https://arxiv.org/abs/2610.12736
AB  - For a finite abelian group G acting linearly on the plane let 𝒳 be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of 𝒳 whose corresponding subschemes of the plane have a given representation type, denoted by 𝒳^[ρ]. By means of a Grothendieck-Riemann-Roch theorem of Toën we give explicit representation theoretic formulas of dim(𝒳^[ρ]) for all ρ∈ 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 [𝔸^2/μ_n] given by the cyclic action ζ· (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.
ER  -

Abstract

Related

Cite