The 0-th Fitting ideal of the Jacobian module of a plane curve

TL;DR

This paper characterizes the 0-th Fitting ideal of the Jacobian module of a plane curve using Jacobian syzygies, providing new insights into maximal Tjurina curves and their properties.

Contribution

It introduces a novel description of the 0-th Fitting ideal in terms of determinants involving Jacobian syzygies, offering new characterizations of maximal Tjurina curves.

Findings

Provides a determinant-based description of the 0-th Fitting ideal

Characterizes maximal Tjurina curves in terms of Jacobian modules

Establishes connections between syzygies and curve singularities

Abstract

We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global Tjurina number equals an upper bound given by A. du Plessis and C.T.C. Wall.