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 Bour(F), measures its freeness and was introduced in \cite{JNS2024}. It is defined as the degree of R/Iε, where R = k[x,y,z] and Iε⊆ R is the Bourbaki ideal associated with a minimal generator ε of the module of first syzygies of the Jacobian ideal JF. 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/JF, describing the possible resolution forms and expressing Bour(F) in terms of the corresponding graded shifts. Finally, we study the realization problem in fixed degree. We prove that for °F ≥ 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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BibTeX
@misc{alvarenga2026bourbaki,
title = {{On the Bourbaki Degree of Plane Projective Curves}},
author = {Roberto Alvarenga and Murillo Lozano and Parham Salehyan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.06956},
eprint = {2605.06956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06956},
}
RIS
TY - UNPB
TI - On the Bourbaki Degree of Plane Projective Curves
AU - Roberto Alvarenga
AU - Murillo Lozano
AU - Parham Salehyan
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2605.06956
UR - https://arxiv.org/abs/2605.06956
AB - Let F be a reduced plane projective curve defined over an algebraically closed field k. The Bourbaki degree of F, denoted by 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⊆ 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 Bour(F) in terms of the corresponding graded shifts. Finally, we study the realization problem in fixed degree. We prove that for °F ≥ 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 $6$.
ER -