Quelques calculs utiles pour la résolution des singularités

Abstract

This text is the redaction of lectures given in 1971-72 by Monique Lejeune-Jalabert and Bernard Teissier at the Centre de Mathématiques de l'Ecole Polytechnique. The rédaction is due to Lê Dung Tráng and Jean-Jacques Risler. The aim was to give a complete proof of the continuity of maximal contact along the strata of a Samuel stratification in complex analytic geometry. This result is a part of Hironaka's strategy to prove resolution of singularities in complex analytic geometry. The main ingredients are generalizations of the concept of normal cone of a singular space along a subspace and of the concept of Newton polygon, In particular, we study a generalization of the maximal contact exponent at a singular point of a singular plane curve with a non singular curve through that point. For plane curves, this invariant appears as the inclination of a compact face of the Newton polygon in suitable coordinates, and also as a " critical tropism " for a family of weighted tangent cones