Hilbert schemes of points on surfaces

TL;DR

This paper reviews the properties and significance of Hilbert schemes of points on surfaces, highlighting their role as moduli spaces, resolutions of singularities, and their connections to various mathematical and physical theories.

Contribution

It provides an overview of the diverse mathematical and physical connections of Hilbert schemes of points on surfaces, emphasizing their importance as a fundamental model case.

Findings

Hilbert schemes serve as moduli spaces and resolutions of singularities.

They connect to Donaldson invariants and enumerative geometry.

They relate to infinite dimensional Lie algebras and vertex algebras.

Abstract

The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections.