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Equivariant Hilbert schemes of points

Let $G$ be a finite subgroup of $\mathrm{SL}(2, \mathbb{C})$ and let $X$ be the quotient surface $X = \mathrm{Spec}(\mathbb{C}[x, y]^{G})$. Denote by $\mathrm{Hilb}^r([X])$ the equivariant Hilbert sch …