Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities. I

Abstract

Let (X, p) be a germ of a normal isolated singularity of dimension two. (1.1) Definition. (X,p) is a hyperbolic unimodular singularity if the minimal resolution of (X, p) has either a cycle of nonsingular rational curves or a rational curve with a node as exceptional set. For brevity we call a hyperbolic unimodular singularity (X,p) a cusp singularity. (1.2) Theorem (Karras [5]). A cusp singu.!arity (X,p) is a complete intersection singularity iff (X, p) is one of the following: Tp, q,,: xP+y~+z'-xyz=O with 1 /p+l /q+l / r< l ,