Let
(formula here)
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 (π, u, v) the initial data.
Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the
critical locus Γ of π.
Let (Δα)α∈A be the branches of the discriminant locus
Δ at O which are not the coordinate axes.
For each α ∈ A, we define a rational number dα by
(formula here)
where I(–, –) denotes the intersection number at 0 of complex analytic curves in
C2. The set of rational numbers dα, for α ∈ 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 (π, u, v).
The interesting situation is when one of the two coordinates (u, v) is tangent to some
branch of Δ, 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
(π, 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.