Hilbert schemes for crepant partial resolutions

TL;DR

This paper constructs Hilbert schemes of points on crepant partial resolutions of Kleinian singularities using Nakajima quiver varieties, unifying previous constructions for minimal resolutions and singularities.

Contribution

It provides a unified framework for constructing Hilbert schemes on crepant partial resolutions via explicit GIT stability parameters.

Findings

Constructs Hilbert schemes as Nakajima quiver varieties for crepant partial resolutions.

Generalizes existing quiver variety constructions to broader classes of resolutions.

Unifies previous approaches for minimal resolutions and Kleinian singularities.

Abstract

For $n\geq 1$, we construct the Hilbert scheme of $n$ points on any crepant partial resolution of a Kleinian singularity as a Nakajima quiver variety for an explicit GIT stability parameter. This generalises and unifies existing quiver variety constructions of the Hilbert scheme of points on the minimal resolution of a Kleinian singularity, and on the Kleinian singularity itself.