Resolution of Curve and Surface Singularities pp 205-246 | Cite as

# 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 A^{3} 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 S_{0} = *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].

## Keywords

Singular Point Prime Ideal Local Ring Maximal Ideal Toric Variety## Preview

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