On the Resolution Diagrams of the Brieskorn Singularities (2,q,r) of Type II

Abstract

Let p, q, r be pairwise coprime integers with 2 ≦ p < q < r. The equation defines a complex hypersurface V(p, q, r) ⊂ C3 which has an isolated singular point at the origin. We call the singularity the Brieskorn singularity (p, q, r). An algorithm of resolving this singularity is known [1]. According to the algorithm, the resolution diagram which describes the configuration of the pre-image of the singular point in the resolved surface is a star-shaped tree Γp, q, r with three branches:1.1The positive integers (weights) ai, bj, ck are given as follows: Let x, y, z, b be integers satisfying1.2