An introduction to Hilbert schemes of points on ADE singularities

Alastair Craw

doi:10.2969/aspm/08810119

An introduction to Hilbert schemes of points on ADE singularities

Department of Mathematical Sciences

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

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 A², 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.

Original language	English
Title of host publication	The McKay correspondence, mutation and related topics
Pages	119-157
Number of pages	39
Volume	88
DOIs	https://doi.org/10.2969/aspm/08810119
Publication status	Published - 1 Apr 2023

Publication series

Name	Advanced Studies in Pure Mathematics
Volume	88

Bibliographical note

Publisher Copyright:
© 2023, Mathematical Society of Japan. All rights reserved.

Keywords

cornering
Kleinian orbifold
preprojective algebra
quiver variety
Quot scheme

ASJC Scopus subject areas

General Mathematics

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10.2969/aspm/08810119

Cite this

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AB - 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 A2, 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.

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KW - Kleinian orbifold

KW - preprojective algebra

KW - quiver variety

KW - Quot scheme

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DO - 10.2969/aspm/08810119

M3 - Chapter in a published conference proceeding

AN - SCOPUS:85190649370

VL - 88

T3 - Advanced Studies in Pure Mathematics

SP - 119

EP - 157

BT - The McKay correspondence, mutation and related topics

ER -

An introduction to Hilbert schemes of points on ADE singularities

Abstract

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Bibliographical note

Keywords

ASJC Scopus subject areas

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