Résolution des Singularités de Surfaces par Éclatements Normalisés

(Multiplicité, multiplicité polaire, et singularités minimales)
  • Romain Bondil
  • Dũng Tráng Lê
Part of the Trends in Mathematics book series (TM)


A celebrated theorem due to O. Zariski (Ann. of Math. 1939), asserts that if S is an algebraic surface over a field (algebraically closed and of characteristic zero), one can resolve the singularities of S by a finite sequence of point blow-ups followed by normalizations (called normalized blow-ups). We give a new approach to prove this theorem by attaching to a germ (S, O) of a complex normal surface a pair (v, γ) of natural numbers, where v = e(S, O) is the multiplicity of the germ, and γ = e p , O) is the multiplicity of the discriminant Δ p , of a generic projection from a small representative of (S, O) onto an open subset U of ℂ2. We prove that after a finite number N S of normalized blow-ups, all the singular points O i , on the surface obtained, have pairs (v i , γ i ) strictly smaller (for the lexicographic order) than the original pair (v, γ) for (S,O). This implies the theorem by Zariski, since v = 1 or γy = 1 for a germ both yield that the corresponding germ is smooth. Moreover, we have the following bound for the number N S : N S is not greater than the number of point blow-ups necessary to get an embedded resolution of the discriminant curve Δ p U considered above.


Cone Tangent Algebraic Surface Section Hyperplane Section Hyperplane Generale Embed Resolution 
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Authors and Affiliations

  • Romain Bondil
    • 1
  • Dũng Tráng Lê
    • 1
  1. 1.C.M.I.Université de ProvenceMarseille cedex 13France

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