An introduction to Hilbert schemes of points on ADE singularities

Abstract

This paper is based on a talk at the conference The McKay correspondence, mutation and related topics from July 2020. We provide an introduction to joint work of the author with Søren Gammelgaard, Ádám Gyenge and Balázs Szendrői [CGGS21b] that constructs the reduced scheme underlying the Hilbert scheme of $n$ points on an ADE singularity as a Nakajima quiver variety for a particular stability parameter. After drawing a parallel with two well-known constructions of the Hilbert scheme of $n$ points in $\mathbb{A}^2$, we summarise results of the author and Gwyn Bellamy [BC20] before describing the main result by cornering a noncommutative algebra obtained from the preprojective algebra of the framed McKay graph.