Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

TL;DR

This paper presents an algorithm to compute the generic component of the analytic moduli of plane branches, focusing on Kähler differentials, and explores its behavior under blow-up, with applications to semimodules and moduli dimensions.

Contribution

It introduces an algorithm for computing the generic semimodule and provides an alternative proof for the moduli dimension formula of plane curve singularities.

Findings

Algorithm for generic semimodule based on multiplicity and Puiseux exponents

Explicit basis of Kähler differentials for generic curves

Alternative proof for Genzmer's moduli dimension formula

Abstract

Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of K\"ahler differentials for generic $\Gamma \in {\mathcal C}$, algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of $\Gamma$.