Dihedral groups and G-Hilbert schemes

Abstract

Let G ⊂ GL(2,C) be a finite subgroup acting on the complex plane C2, and consider the following diagram C2 -> X <- π:Y where π is the minimal resolution of singularities. Since Du Val in the 1930s the explicit calculation of Y was made from X by blowing up the singularity at the origin, where we lose any information about the group G in the process. But, is there a direct relation between the resolution Y and the group G? McKay [McK80] in the late 1970s was the first to realise the link between the group action and the resolution Y , thus giving birth to the so called McKay correspondence. This beautiful correspondence establishes an equivalence between the geometry of the minimal resolution Y of the quotient singularity C2/G, and the G-equivariant geometry of C2