Hilbert schemes of points on canonical surfaces

Abstract

For $n\geq 1$, we investigate the Hilbert scheme of $n$-points on a surface $S$ with canonical singularities. We generalise the well-known theorem of Fogarty by showing that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ is a normal variety of dimension $2n$ with canonical singularities, and for $n\leq 7$, we show that $\operatorname{Hilb}^n(S)$ is reduced. When $S$ has symplectic singularities over $\mathbb{C}$, we show that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ also has symplectic singularities, thereby generalising a result of Beauville. Our results build on work of the first author with Gyenge, Gammelgaard and Szendr\H{o}i that sought to identify the underlying reduced subscheme of the Hilbert scheme of $n$-points on a Kleinian singularity with a Nakajima quiver variety.