Geometry of Critical Loci |
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Authors: | Trang Le Dung; Maugendre Helene; Weber Claude |
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Institution: | Centre de Mathématiques et d'Informatique, Université de Provence 39 rue F. Joliot-Curie, 13 453 Marseille Cedex 13, France, ledt{at}gyptis.univ-mrs.fr
Centre de Mathématiques et d'Informatique, Université de Provence 39 rue F. Joliot-Curie, 13 453 Marseille Cedex 13, France, maugendr{at}gyptis.univ-mrs.fr
Section de Mathématiques 24 rue du Lièvre, Case Postale 240, CH-1211 Genève 24, Switzerland, Claude.Weber{at}math.unige.ch |
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Abstract: | Let :(Z,z) (U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C2. Let u and v be coordinates of C2 defined on U. Weshall call the triple ( , u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let (![{Delta}](http://jlms.oxfordjournals.org/math/Delta.gif) )![{alpha}](http://jlms.oxfordjournals.org/math/alpha.gif) A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by
where I(, ) denotes the intersection number at0 of complex analytic curves in C2. The set of rational numbersd , for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata ( , u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data ( , u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus. |
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