A comprehensive approach to the moduli space of quasi-homogeneous singularities

TL;DR

This paper explores the classification of quasi-homogeneous singularities and their associated foliations, providing a detailed description of the moduli space and analytic invariants, with an accessible exposition for non-specialists.

Contribution

It introduces a complete set of invariants for foliations with quasi-homogeneous separatrices and characterizes their moduli space, advancing understanding of singular holomorphic foliations.

Findings

Complete set of analytic invariants for foliations with quasi-homogeneous separatrices

Full description of the moduli space of quasi-homogeneous plane curves

Accessible exposition for non-specialists

Abstract

We study the relationship between singular holomorphic foliations in $(\mathbb{C}^{2},0)$ and their separatrices. Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous separatrices. Further, we give the full moduli space of quasi-homogeneous plane curves. This paper has an expository character in order to make it accessible also to non-specialists.