Constructive solution of Zariski's Moduli Problem for Plane Branches

Abstract

In this paper we give an explicit solution to Zariski's moduli problem for plane branches. We compute (in an algorithmic way) the set of Kähler differentials of an irreducible germ of holomorphic plane curve. We show that there is a basis of this set whose main elements correspond to dicritical foliations. Indeed, we discuss several concepts of generation for the semimodule of values of Kähler differentials of the curve and provide basis of Kähler differentials, for every of these concepts, whose geometric properties are described. Moreover, we give an algorithmic construction of the bases.

62 pages. Added Subsection 5.4 on "Hidden Coefficients", which covers in a straightforward way, Zariski's study of the noncompactness for genus greater than 2