© 2001 by London Mathematical Society
© The London Mathematical Society
Geometry of Critical Loci
ng Tráng
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 2–4 rue du Lièvre, Case Postale 240, CH-1211 Genève 24, Switzerland, Claude.Weber{at}math.unige.ch
Received 16 October 1998. Revision received 14 December 1999.
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 surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall 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 (
)
A be the branches of the discriminant locus at O which are not the coordinate axes.
For each A, we define a rational number d by
 |
where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers d, for A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of , otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.
In this paper we prove that the set D is a topological invariant of the initial data (, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.