The Singularity Z^q = XY^p

Abstract

In this chapter we describe a resolution process for the singularity of the normalization of the surface in A³ defined by the equation Z^P = XY^q over an algebraically closed field k of characteristic zero; here 0 < p < q are integers and gcd(p, q) = 1. These singularities arise in a natural way: In section 1 we show in (1.6) that if L is a finite extension of Q = k((U, V)), S is the integral closure of R = k 〚U, V〛 in L, and the only prime ideals of R which are ramified in S are at most RU or RV, then L = k((X,Y))[Z]/(Z^P − XY^q), and that S is the integral closure of S₀ = k 〚X,Y 〛[Z]/(Z^p − XY^q). Also, they arise as the only singular point of the normalization of a toric variety associated with a not nonsingular strongly convex rational polyhedral cone of dimension 2 [cf. section 4].