The Local Bourbaki Degree of a Plane Projective Curve

TL;DR

This paper introduces a local version of the Bourbaki degree for plane curves, proves it sums to the global degree, and demonstrates its usefulness in computational and theoretical aspects of curve freeness.

Contribution

It defines the local Bourbaki degree, proves its sum equals the global degree, and applies this to analyze curve freeness and improve computational methods.

Findings

The local Bourbaki degree sums to the global degree.

The local approach simplifies computations of the Bourbaki degree.

Applications to determining (nearly) free curves.

Abstract

The Bourbaki degree of a plane projective curve $F$, denoted by $\mathrm{Bour}(F)$, was introduced in \cite{Marcos} by Jardim, Nejad and Simis. It is defined as the degree of $R/I_\epsilon$, where $R = k[x,y,z]$ is the graded polynomial ring, with $k$ algebraically closed, and $I_\epsilon \subseteq R$ is the Bourbaki ideal associated with a minimal generator $\epsilon$ of the module of first syzygies of the Jacobian ideal $J_F$. In this work, we propose the definition of the local Bourbaki degree at a point $P \in \mathbb{P}^2$, denoted by $\mathrm{Bour}_P(F)$, and prove that $\mathrm{Bour}(F) = \sum_{P \in \mathbb{P}^2}\mathrm{Bour}_P(F).$ Furthermore, we present results that follow from this local definition, which are instrumental in determining the Bourbaki degree and in establishing whether a curve is (nearly) free. In addition, we provide examples of computing the Bourbaki degree via the local formula - an approach that is computationally advantageous, as it, generically, demands fewer calculations.