Irreducibility of the Hilbert schemes of points on surfaces with Kleinian singularities

Zheng, Xudong

doi:10.1080/00927872.2022.2089993

Cited by 2 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our proof is independent of [13], so our approach to Theorem 1.1 also provides a new proof of Fogarty's result, while our proof of irreducibility is independent of the work of Zheng [29]; see Remarks 6.11. We deduce Theorem 1.1 by first proving the following stronger result in the special case where S is isomorphic to A 2 /Γ for a finite subgroup Γ ⊂ SL(2, k).…”

Section: Introductionmentioning

confidence: 85%

“…When S is singular, an example of Miró-Roig-Pons-Llopis [23] shows that Hilb [n] (S) can be reducible. However, for a normal surface S with canonical singularities (also called Du Val, Kleinian or ADE singularities), Zheng [29] proved that Hilb [n] (S) is an irreducible scheme of dimension 2n. Our main geometric result strengthens this statement by generalising Fogarty's famous theorem as follows:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

Craw¹,

Gammelgaard²,

Gyenge³

et al. 2021

Alg. Geom.

View full text Add to dashboard Cite

For a finite subgroup Γ ⊂ SL(2, C) and n 1, we construct the (reduced scheme underlying the) Hilbert scheme of n points on the Kleinian singularity C 2 /Γ as a Nakajima quiver variety for the framed McKay quiver of Γ, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by removing an arrow and then 'cornering', and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of the stability parameter.

show abstract

Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

Craw¹,

Gammelgaard²,

Gyenge³

et al. 2021

Alg. Geom.

View full text Add to dashboard Cite

show abstract

“…See [115, Theorem 2.1(3)] for a sketch of proof. For a normal surface S with all singularities du Val the Hilbert scheme Hilb d (S) is irreducible by [39,135], see also [38]. In general, a normal surface S has only finitely many singular points p 1 , .…”

Section: Open Problemsmentioning

confidence: 99%