Hilbert schemes for crepant partial resolutions

Abstract

For $n\ge 1$, we construct the underlying reduced subscheme of 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. 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.

v2: 15 pages, 1 figure, streamlined so that we now present only one proof of the main result. Second author's name updated; v3 includes updated references to a revised paper, and a description of the nef and movable cones