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arXiv · 2605.06956

On the Bourbaki Degree of Plane Projective Curves

Roberto Alvarenga·Murillo Lozano·Parham Salehyan
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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 $R/I_ε$, where $R = k[x,y,z]$ and $I_ε\subseteq R$ is the Bourbaki ideal associated with a minimal generator $ε$ 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 $°F \geq 5$, 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 $6$.

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Roberto Alvarenga, Murillo Lozano, Parham Salehyan. 2026-09-21. On the Bourbaki Degree of Plane Projective Curves. https://arxiv.org/abs/2605.06956

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