A characterization of quasi-homogeneous singularities of free and nearly free plane curves

TL;DR

This paper introduces a new way to identify quasi-homogeneous isolated singularities in free and nearly free plane curves using a criterion based on the first syzygy matrix of the Jacobian ideal.

Contribution

It provides a novel characterization of quasi-homogeneous singularities for free and nearly free plane curves via syzygy matrices.

Findings

New criterion for quasi-homogeneity based on syzygy matrices

Characterization applies to free and nearly free curves

Enhances understanding of singularity structure in plane curves

Abstract

The goal of this paper is to establish a new characterization of quasi-homogeneous isolated singularities of free curves and nearly free curves $C$ in $\mathbb{P}_\mathbb{C}^2$. The criterion will be in terms of a first syzygy matrix associated with the Jacobian ideal $J_f$ of $f$, where $f=0$ is the equation of the plane curve $C$.