The rational double points of surfaces in characteristic zero are related to the finite subgroups G of SL2 [6, 7]. Namely, if V denotes the affine plane with its linear G-action, then the variety X = V/G has a singularity at the origin, which is the one corresponding to G. Let p be a prime integer. If p divides the order of G, this subgroup will degenerate when reduced modulo p, and the smooth reduction of V will usually not be compatible with an equisingular reduction of X. Nevertheless, it turns out that every rational double point in characteristic p has a finite (possibly ramified) covering by a smooth scheme. In this paper we prove the existence of such a covering by direct calculation, and we compute the local fundamental groups of the singularities.

Generalities on Coverings

We are interested in the local behavior of singularities and so we work with a scheme of the form X=Spec A, where A is the henselization of the local ring of a normal algebraic surface over an algebraically closed field k. We could also work with complete local rings.

In general, U will denote the complement of the closed point of X : U = X – x0. By fundamental group of X, we mean π=π1(U). This is the group which classifies finite étale coverings of U, or equivalently, normal, pure 2-dimensional schemes Y, finite over X, which are étale except above x0.