Hilbert schemes of points on canonical surfaces

For n≥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 Hilbn(S) is a normal variety of dimension 2n with canonical singularities, and for n≤7, we show that Hilbn(S) is reduced. When S has symplectic singularities over C, we show that the underlying reduced subscheme of Hilbn(S) also has symplectic singularities, thereby generalising a result of Beauville. Our results build on work of the first author with Gyenge, Gammelgaard and Szendrői that sought to identify the underlying reduced subscheme of the Hilbert scheme of n-points on a Kleinian singularity with a Nakajima quiver variety.