Hilbert schemes for crepant partial resolutions

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. We provide both a short proof involving a combinatorial argument, in which the isomorphism is implicit, and a more satisfying geometric proof where the isomorphic is constructed explicitly. As a corollary, we compute the nef and movable cones of the Hilbert scheme of $n$ points on any crepant partial resolution of a Kleinian singularity in terms of the summands of a tilting bundle on the surface.