Le nombre de modules du germe de courbe planex a +y b =0

Abstract

Etant donn6 un germe d'application analytique f: (~2, 0)~(~E, 0) d6finissant un germe de courbe plane f-1(0) topologiquement 6quivalent fi xa+ yb=0, nous 9 ~{x,y} la dimension de la d6formation semi-universelle de notons z(f) = dlrne~ fr') f-1(0) et Zmin le minimum des entiers z(f) lorsque f parcourt l'ensemble de tels germes.

Moduli spaces for topologically quasi-homogeneous functions

“…This problem is a particular case of an open problem known as the Zariski problem. It has only a very few satisfying answer: Zariski [16] for the very first treatment of some particular cases, Hefez and Hernandez [5,6] for the irreducible curves, Granger [9] in the homogeneous topological class and [2] for some results which are particular case of our present results. Our strategy that we already introduced in a previous work [8], differs from all this works: from our description of the moduli space M, we consider the distribution C on M induced by the equivalence relation ∼ c : two foliations represented by two points in M are in a same orbit of this distribution if and only if they induce the same curve up to analytic conjugacy.…”

On the values of logarithmic residues along curves

“…We then prove that the stratification by the values of logarithmic residues is finite and constructible, and it refines the stratification by the Tjurina number (see Propositions 4.14 and 4.15). We end this section with several algorithms which can be used to compute the set of values of R D , inspired by [BGM88] and [HH07]. For the remainder of this section, we set: • The Milnor number of f is µ…”

“…This algorithm is used to study the equisingular deformation of a quasi-homogeneous polynomial of the form x a − y b , with gcd(a, b) = 1. It is inspired by [BGM88].…”

“…Polynomials of the form (23) are studied in [BGM88]. The authors give an algorithm to compute the "escalier " of the curve, which is by definition the complement in N 2 of the set E = {exp(g); g ∈ (F, F x , F y ) ⊆ C{x, y}}.…”

Moduli of plane curve singularities with a single characteristic exponent

“…Let us call τ min (n, m, s) the minimum of the integers τ (γ) when γ ranges over the set of germs with characteristic exponent {m/n} and fixed invariant s. In sections 2 and 3, we adapt the algorithm of [3] to compute τ min (n, m, s).…”