The Saito module and the moduli of a germ of curve in $\left(\mathbb{C}^{2},0\right)$

TL;DR

This paper investigates the module of vector fields tangent to a germ of a curve in the complex plane, proposing an algorithm to compute the dimension of its moduli space and validating it for certain curve families.

Contribution

It introduces a conjectural algorithm for calculating the moduli space dimension of curve germs, with proofs for its correctness in specific cases.

Findings

Proposed a conjectural algorithm for moduli space dimension

Validated the algorithm for certain families of curves

Enhanced understanding of tangent vector fields to curve germs

Abstract

We study properties of the module of vector fields tangent to a given germ of curve in the complex plane $\mathbb{C}^{2}$. As a consequence, we obtain a conjectural algorithm to compute the generic dimension of its moduli space. For some families of curves, we proved that this algorithm provides the correct dimension.