Hilbert schemes and cyclic quotient surface singularities

Abstract

Let $G$ be a finite cyclic subgroup of $GL(2, C)$ of order $n$ which contains no reflections. Let ${\mathbf A}^{2}$ be the complex affine plane. We consider a certain subscheme $Hi1b^{G}({\mathbf A}^{2})$ of $Hi1b^{n}({\mathbf{}A}^{2})$ consisting of $G$-invariant zero-dimensional subschemes of length $n$. We describe the structure of $Hi1b^{G}({\mathbf{}A}^{2})$ and prove this is the minimal resolution of the quotient surface singularity ${\mathbf A}^{2}/G$ .