Punctual Hilbert schemes for Kleinian singularities as quiver varieties

Craw, Alastair and Gammelgaard, Soren and Gyenge, Ádám and Szendroi, Balazs (2021) Punctual Hilbert schemes for Kleinian singularities as quiver varieties. ALGEBRAIC GEOMETRY, 8 (6). pp. 680-704. ISSN 2313-1691

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Abstract

For a finite subgroup Gamma subset of SL(2, C) and n >= 1, we construct the (reduced scheme underlying the) Hilbert scheme of n points on the Kleinian singularity C-2/Gamma as a Nakajima quiver variety for the framed McKay quiver of Gamma, 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.

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