Singularities of the Isospectral Hilbert Scheme

TL;DR

This paper analyzes the singularities of the isospectral Hilbert scheme over a smooth surface, establishing their nature for various values of n and providing explicit resolutions for the case n=3.

Contribution

It determines the singularity types of the isospectral Hilbert scheme for different n and constructs explicit resolutions in the case n=3.

Findings

Singularities are canonical for n ≤ 5.

Singularities are log-canonical for n ≤ 7.

Singularities are not log-canonical for n ≥ 9.

Abstract

We study the singularities of the isospectral Hilbert scheme $B^n$ of $n$ points over a smooth algebraic surface and we prove that they are canonical if $n \leq 5$, log-canonical if $n \leq 7$ and not log-canonical if $n \geq 9$. We describe as well two explicit log-resolutions of $B^3$, one crepant and the other $\mathfrak{S}_3$-equivariant.