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Résolution des Singularités de Surfaces par Éclatements Normalisés

(Multiplicité, multiplicité polaire, et singularités minimales)

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Trends in Singularities

Part of the book series: Trends in Mathematics ((TM))

Abstract

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(Δ 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.

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  1. C.M.I., Université de Provence, 39, rue F. Joliot-Curie, 13453, Marseille cedex 13, France

    Romain Bondil & Dũng Tráng Lê

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  1. Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL, 60607, USA

    Anatoly Libgober

  2. Mathématiques, UMR-CNRS 8524, Université de Lille 1, 59655, Villeneueve d’Ascq Cedex, France

    Mihai Tibăr

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Bondil, R., Lê, D.T. (2002). Résolution des Singularités de Surfaces par Éclatements Normalisés. In: Libgober, A., Tibăr, M. (eds) Trends in Singularities. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8161-6_2

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