The Singularity Zq = XYp

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


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].


Singular Point Prime Ideal Local Ring Maximal Ideal Toric Variety 
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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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