An introduction to Hilbert schemes of points on ADE singularities

TL;DR

This paper introduces a construction of the Hilbert scheme of points on ADE singularities as a Nakajima quiver variety, linking it to noncommutative algebra and the McKay correspondence.

Contribution

It presents a novel geometric realization of Hilbert schemes on ADE singularities via Nakajima quiver varieties and relates it to noncommutative algebraic structures.

Findings

Constructed the reduced scheme of points on ADE singularities as a Nakajima quiver variety.

Connected the Hilbert scheme construction to the preprojective algebra of the McKay graph.

Summarized previous results and established new links with noncommutative algebra.

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 S{\o}ren Gammelgaard, \'{A}d\'{a}m Gyenge and Bal\'{a}zs Szendr\H{o}i 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 before describing the main result by cornering a noncommutative algebra obtained from the preprojective algebra of the framed McKay graph.