On the Bourbaki Degree of Plane Projective Curves

Abstract

Let $F$ be a reduced plane projective curve defined over an algebraically closed field $k$. The Bourbaki degree of $F$, denoted by $\mathrm{Bour}(F)$, measures its freeness and was introduced in \cite{JNS2024}. It is defined as the degree of $R/I_{\epsilon }$, where $R=k[x,y,z]$ 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 article, we investigate the local Bourbaki degree and apply it to refine the well-known upper bound for the global Bourbaki degree. We then study curves with prescribed Bourbaki degree through the minimal graded free resolution of $R/J_F$, describing the possible resolution forms and expressing $\mathrm{Bour}(F)$ in terms of the corresponding graded shifts. Finally, we study the realization problem in fixed degree. We prove that for $°F\ge 5$, the du Plessis-Wall bounds impose no numerical obstruction to the realization of the Bourbaki degree. Moreover, we show that every possible Bourbaki degree is realized by reduced plane curves of degrees $5$ and $6$.

The previous version has been substantially modified. We now address several problems concerning the Bourbaki degree, extending the scope beyond its local version. The title has been changed accordingly. 25 pages. Comments are welcome