On the Parameterization of Germs of Two-Dimensional Singularities

Abstract

We consider a germ of a two-dimensional complex singularity \((X,x_0)\), irreducible at \(x_0\) and \(F\) the exceptional divisor of a desingularization. We prove that if there exists a normal isolated singularity \((Z,z_0)\) with simply connected link and a surjective holomorphic map \(f:(Z,z_0)\rightarrow (X,x_0)\) then all irreducible components of \(F\) are rational, and if all irreducible components of \(F\) are rational then there exists a surjective holomorphic map \(f:(\mathbb {C}^2,0)\rightarrow (X,x_0)\).