Constructive solution of Zariski's Moduli Problem for Plane Branches

TL;DR

This paper provides an explicit, algorithmic solution to Zariski's moduli problem for plane branches by computing bases of Kähler differentials and analyzing their geometric properties.

Contribution

It introduces an explicit, algorithmic method to determine bases of Kähler differentials for plane branches, addressing Zariski's moduli problem.

Findings

Basis of Kähler differentials corresponds to dicritical foliations

Algorithmic construction of bases for various generation concepts

Detailed geometric properties of the bases

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 K\"{a}hler 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 K\"{a}hler differentials of the curve and provide basis of K\"{a}hler differentials, for every of these concepts, whose geometric properties are described. Moreover, we give an algorithmic construction of the bases.