Limites d'espaces tangents � une surface normale

Abstract

Résumé. Nousétudions l'ensemble des hyperplans limites d'hyperplans tangentsà un germe de surface normale. Nous caractérisons ces hyperplans par le fait que le nombre de Milnor de leur section avec la surface n'est pas minimum. Nous donnons ensuite une généralisation des résultats de [14] en termes de résolution simultanée faible de la famille des sections hyperplanes, ce qui nous permet de déterminer avec précision les tangentes exceptionnelles d'une surface normale. Grâceà ces résultats, nous démontrons que "les composantes de Tyurina" d'une désingularisation raisonnable se contractent sur des points fixes du système linéaire des courbes polaires. Abstract.We study the set of limiting tangent hyperplanes of a normal surface germ. We characterize these hyperplanes by the non-minimality of the Milnor number of their section with the surface. Then we generalise the results of [14] in terms of weak simultaneous resolution of the family of hyperplane sections, and hence we precisely determine the exceptional tangents of a normal surface singularity. Applying these results, we prove that the "Tyurina components" of a reasonable desingularisation contract to fixed points of the linear system of polar curves. Mathematics Subject Classification (2000). 32S05, 32S15, 32S25, 32S35, 32S45. Mots

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

“…Thus each limit of tangent hyperplanes at a sequence of points converging to 0 in Z j {0} contains L j . We will show there exist infinitely many such limits, which, by definition, means that L j is an exceptional tangent line (this is one of the two implications of Proposition 6.3 of Snoussi [43], see also Proposition 2.2.1 of [25], but we did not find a clear statement in the literature). Indeed, for any generic (n − 2)-plane H let ℓ H : C n → C 2 be the projection with kernel H. By Lemma 3.8 the strict transform Π * of the polar Π of ℓ H | X intersects ν∈Γj E ν .…”

“…We will take a resolution approach to construct the thick-thin decomposition, but another way of constructing it is as follows. Recall (see [43,25]) that a line L tangent to X at 0 is exceptional if the limit at 0 of tangent planes to X along a curve in X tangent to L at 0 depends on the choice of this curve. Just finitely many tangent lines to X at 0 are exceptional.…”

Linear components of the tangent cone in the Nash modification of a complex surface singularity

“…Remark 3.1. In [10], we proved that a singular point of the normalized blow-up of the origin of a germ of a normal surface singularity is always a base point of the linear system of polar curves. We used to think that the similar behavior of the hyperplane sections in the blow-up of the origin with the polar curves in the Nash modification of normal surfaces would suggest that all singular points of the normalized Nash modification of a normal surface are fixed points of the family of hyperplane sections.…”

“…Indeed, the singular points of the normalized blow-up of the origin of a normal germ of a surface are fixed points of the family of polar curves. Meanwhile a singular point of the normalized Nash modification of a normal germ of surface need not be a fixed point of the family of hyperplane sections ; see [10,11].…”

Pencils and critical locus on normal surfaces

“…In [2] R. Bondil gives an algebraic µ-constant theorem for linear families of plane curves. Other results have been obtained in the case where π is the restriction to (Z, z) of a linear projection of (C n , 0) onto (C 2 , 0) (see [1], [4], [18]). At last, the topology of the morphism π has been studied in [13] and [14].…”