Freeness versus maximal global Tjurina number for plane curves

TL;DR

This paper characterizes nearly free plane curves using their global Tjurina numbers, extending the understanding of free and nearly free curves, and relates symmetries and syzygies to these properties.

Contribution

It introduces new numerical and syzygy-based characterizations of free and nearly free plane curves, expanding existing theoretical frameworks.

Findings

Nearly free curves characterized by Tjurina numbers

Irreducible curves with 1D symmetry are nearly free

New criteria based on syzygies for free and nearly free curves

Abstract

We give a characterization of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterization of free curves as curves with a maximal Tjurina number, due to A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterization of free curves and a simple characterization of nearly free curves in terms of their syzygies conclude this note.