The analytic classification of plane branches

Abstract

An effective solution of the problem of analytic classification of plane branches is given. A finite stratification of any given equisingularity class of plane branches is determined and normal forms for each stratum are exhibited in such a way that two branches in normal form are easily recognized to be analytically equivalent or not. In this way, we solve the main problems proposed by O. Zariski (Le problème des modules pour les branches planes.

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.…”

“…The successive translations are t 15 = 2 y − 3 x, t 16 = 2 x − y, t 17 = 2 x − y, t 18 = t 15 etc..., whose components are (0, 1), (1, 1), (1,1). We put the translations on a column on the right side of Figure (6). Now we consider all the parallel paths issued from the integer points (i, 0) on the horizontal axe, under the action of the successive translations t d .…”

“…A block B i in the triangle of moduli is a union of kl consecutive horizontal lines from the line of index d i = ikl + 1 -see Figure (6). We denote by…”

“…The figure (6) in Appendix C shows the procedure in order to construct the normal forms associated to the topological class of…”

“…Therefore, following [11] and [3] its desingularization by blowing up's is the same as a quasi-homogeneous function, that is to say: the exceptional divisor is a chain of components isomorphic to P 1 (C), the strict transform of the cuspidal branches intersect the same component, the principal component, and the strict transform of the axes, if they appear, intersect the end components of this chain (see Appendix A and figure (6)). The previous expression (1) is a topological normal form for such a function.…”