The Bourbaki Degree of Plane Projective Curves

TL;DR

This paper introduces the Bourbaki degree, a new numerical invariant for plane projective curves, derived from Bourbaki ideals, aiming to classify curves based on their singularities and bounds of this invariant.

Contribution

It defines the Bourbaki degree for graded modules and applies it to the study of plane curve singularities, providing a new tool for curve classification.

Findings

Introduces the Bourbaki degree as a new invariant.

Analyzes bounds of the Bourbaki degree for plane curves.

Connects the invariant to singularity classification.

Abstract

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due to their nature, no numerical invariant was immediately available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the {\em Bourbaki number} in the category of graded modules as the shifted degree of a Bourbaki ideal corresponding to submodules generated in degree at least the maximal degree of a minimal generator of the given module. The present work introduces the{\em Bourbaki degree} as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree. The main intent is a study of plane curve singularities via this new numerical invariant. Accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form $f\in k[x,y,z]$ -- i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds.