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The Singularity Zq = XYp

  • K. Kiyek
  • J. L. Vicente
Part of the Algebras and Applications book series (AA, volume 4)

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 = kU, 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 = kX,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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • K. Kiyek
    • 1
  • J. L. Vicente
    • 2
  1. 1.Department of MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.Departamento de AlgebraUniversidad de SevillaSevillaSpain

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