Extended McKay correspondence for quotient surface singularities

Let G be a finite subgroup of $GL(2)$ acting on $\mathbf{A}^{2}/\{0\}$ freely. The G-orbit Hilbert scheme $G\text{-}Hilb(\mathbf{A}^{2})$ is a minimal resolution of the quotient $\mathbf{A}^{2}/G$ as given by A. Ishii, On the McKay correspondence for a finite small subgroup of GL(2,C), J. Reine Angew. Math. 549 (2002), 221–233. We determine the generator sheaf of the ideal defining the universal G...