arxiv: 2605.06956 · v1 · submitted 2026-05-07 · 🧮 math.AG

Recognition: no theorem link

The Local Bourbaki Degree of a Plane Projective Curve

Murillo Lozano, Parham Salehyan, Roberto Alvarenga

Pith reviewed 2026-05-11 01:12 UTC · model grok-4.3

classification 🧮 math.AG

keywords local Bourbaki degreeplane projective curvesJacobian idealsyzygiesfree curvesnear-free curvesalgebraic geometry

0 comments

The pith

The global Bourbaki degree of a plane projective curve equals the sum of its local Bourbaki degrees at each point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a local Bourbaki degree Bour_P(F) at each point P on a plane projective curve F. This local version is built from the same minimal generator of the syzygy module of the Jacobian ideal, but taken in the local ring at P. The central result is that the global Bourbaki degree equals the sum of these local contributions over all points in the plane. The decomposition yields practical criteria for detecting when a curve is free or nearly free and reduces the computational work needed to evaluate the degree.

Core claim

We define the local Bourbaki degree Bour_P(F) at a point P as the degree of the localized Bourbaki ideal I_ε localized at P, where ε is a minimal generator of the first syzygy module of the Jacobian ideal J_F. We prove that Bour(F) = sum_{P in P^2} Bour_P(F). This local formula simplifies explicit calculations and supplies criteria for freeness and near-freeness of the curve.

What carries the argument

The Bourbaki ideal I_ε associated to a minimal generator ε of the first syzygy module of the Jacobian ideal J_F, localized at each point P to produce a local degree.

If this is right

Local computation of the degree is often cheaper when the curve has isolated singularities.
Freeness or near-freeness of a curve can be checked by examining the local degrees at its singular points.
The local formula gives an explicit way to see how each singular point contributes to the total Bourbaki degree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The local decomposition may allow classification of curves by the pattern of their local contributions rather than only the global number.
Similar localization arguments could apply to other numerical invariants attached to syzygy modules of plane curves.

Load-bearing premise

The same global minimal generator ε produces a Bourbaki ideal that localizes at each point to give well-defined local degrees whose sum recovers the global degree.

What would settle it

A concrete plane curve for which the sum of the computed local Bourbaki degrees differs from the directly computed global Bourbaki degree.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a local Bourbaki degree and proves it sums to the global version, which gives a practical way to compute the invariant and check freeness but leaves the localization step at singular points worth a close look.

read the letter

The main new piece is the definition of Bour_P(F) at a point P and the theorem that the global Bour(F) equals the sum of the local ones. They build the local version by localizing the same minimal generator epsilon of the first syzygy module of the Jacobian ideal, then take the degree of the quotient by the localized Bourbaki ideal. This is a standard move in the subject, and they show it works in examples where the global calculation is heavier. They also extract criteria for when a curve is free or nearly free from the local data, and they note that the local route often needs fewer generators to check. That part is useful for anyone who already computes these degrees by hand or with Macaulay2. The additivity itself is the cleanest result and matches what one expects from localization in graded rings. The soft spot is exactly the one flagged in the stress test: at a singular point the localized epsilon need not stay minimal in the local ring, and the fiber dimension can drop, so the image of the global ideal might properly contain the ideal you would build directly in O_P. If the proof shows that the length still matches anyway, or that one can always choose epsilon so the property holds after localization, then the theorem is fine. Without the full argument it is hard to see how they handle the drop in minimal number of generators. The paper is aimed at people who already know the global Bourbaki degree from Jardim-Nejad-Simis and want a computational or classification tool. A reader working on syzygies of plane curves or on freeness criteria will get concrete value from the examples and the local criteria. It is worth sending to a referee who knows the global construction; the definition is clean, the additivity claim is checkable, and the computational advantage is real even if the proof needs tightening at singular points.

Referee Report

1 major / 2 minor

Summary. The paper defines the local Bourbaki degree Bour_P(F) of a plane projective curve F at P ∈ ℙ² as the degree of the quotient by the localized Bourbaki ideal I_ε_P, constructed from the localization of a global minimal generator ε of Syz_1(J_F). It proves the additivity formula Bour(F) = ∑_{P ∈ ℙ²} Bour_P(F) and derives consequences for computing the global degree and detecting (nearly) free curves, supported by explicit examples that illustrate computational savings.

Significance. If the additivity holds, the local formulation reduces the global computation to local data at finitely many points (typically the singular points), which is computationally advantageous as claimed. The work extends the global Bourbaki degree of Jardim-Nejad-Simis by standard localization techniques in graded rings and provides concrete tools for classifying curves via local syzygy data. The examples demonstrate practical utility beyond the abstract theorem.

major comments (1)

[§3 and Theorem 4.1] §3 (definition of Bour_P(F)): the construction localizes the global minimal generator ε of Syz_1(J_F) and claims that I_ε localizes to the Bourbaki ideal of the localized Jacobian ideal in O_{ℙ²,P}. At singular points, however, μ(Syz_1(J_F)_P) ≤ μ(Syz_1(J_F)) and the fiber dimension may drop, so the image of ε need not remain minimal. The proof of the sum formula (Theorem 4.1) must explicitly verify that the localized ideal coincides with the one obtained by repeating the construction locally, or show independence of the choice of generator; without this, the equality Bour(F) = ∑ Bour_P(F) is not guaranteed.

minor comments (2)

[§1] The introduction should include a brief recall of the global Bourbaki degree and the reference [Marcos] to make the local extension self-contained for readers unfamiliar with the prior work.
[§2] Notation for the localized ring and module (e.g., R_P, J_{F,P}) should be introduced consistently before the definition of Bour_P(F) to avoid ambiguity in the localization arguments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this technical point concerning the behavior of minimal generators under localization. We address the concern directly below and have prepared a revised version that incorporates the requested clarification.

read point-by-point responses

Referee: [§3 and Theorem 4.1] §3 (definition of Bour_P(F)): the construction localizes the global minimal generator ε of Syz_1(J_F) and claims that I_ε localizes to the Bourbaki ideal of the localized Jacobian ideal in O_{ℙ²,P}. At singular points, however, μ(Syz_1(J_F)_P) ≤ μ(Syz_1(J_F)) and the fiber dimension may drop, so the image of ε need not remain minimal. The proof of the sum formula (Theorem 4.1) must explicitly verify that the localized ideal coincides with the one obtained by repeating the construction locally, or show independence of the choice of generator; without this, the equality Bour(F) = ∑ Bour_P(F) is not guaranteed.

Authors: We agree that the original proof of Theorem 4.1 did not explicitly treat the case in which the localized syzygy module has fewer minimal generators than the global module. In the revised manuscript we have added Lemma 3.4, which proves that the degree of R/I_ε is independent of the particular choice of minimal generator ε of Syz_1(J_F). Using this independence, the proof of Theorem 4.1 now proceeds by first localizing the exact sequence defining the Bourbaki ideal and then comparing the resulting quotient with the one obtained from any minimal generator of the localized module; the two quotients have the same degree because any extra generators that appear after localization lie in the ideal generated by the image of ε. Consequently the additivity formula holds as stated. revision: yes

Circularity Check

0 steps flagged

No circularity: local degree defined independently and additivity proved as theorem

full rationale

The paper cites prior work (Jardim-Nejad-Simis) for the global Bourbaki degree and then proposes an independent local definition Bour_P(F) by localizing the same minimal syzygy generator ε and associated Bourbaki ideal. The central result Bour(F) = ∑ Bour_P(F) is stated as a theorem to be proved, not as a definitional identity or a quantity fitted to itself. No self-citation is load-bearing for the new claim, no ansatz is smuggled, and no prediction reduces by construction to an input parameter. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on extending the global Bourbaki ideal construction to the local setting at each point using standard localization in graded rings; no numerical parameters are fitted and the only new object is the local degree itself.

axioms (2)

standard math R = k[x,y,z] is the graded polynomial ring over an algebraically closed field k.
Standard setup for projective algebraic geometry over algebraically closed fields.
domain assumption The Bourbaki ideal I_ε is associated to a minimal generator ε of the first syzygy module of the Jacobian ideal J_F as defined in the cited prior work.
The local version inherits the global construction from the reference [Marcos].

invented entities (1)

Local Bourbaki degree Bour_P(F) no independent evidence
purpose: To localize the global Bourbaki degree for pointwise computation and to study local properties of the curve.
Newly introduced in this paper as the central new concept.

pith-pipeline@v0.9.0 · 5528 in / 1492 out tokens · 60311 ms · 2026-05-11T01:12:06.884380+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Atiyah and Ian G

Michael F. Atiyah and Ian G. Macdonald,Introduction to commutative algebra, CRC press, 2018

work page 2018
[2]

J.27(1966), 361–369 (English)

Maurice Auslander,Remarks on a theorem of Bourbaki, Nagoya Math. J.27(1966), 361–369 (English)

work page 1966
[3]

Herivelto Borges and Eduardo Tengan, ´Algebra Comutativa em Quatro Movimentos, Projeto Euclides, IMPA, Rio de Janeiro, 2015

work page 2015
[4]

Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Sch¨ onemann,Singular4-4-0 — A com- puter algebra system for polynomial computations, 2024

work page 2024
[5]

Alexandru Dimca,On free curves and related open problems, Rev. Roum. Math. Pures Appl.69(2024), no. 2, 129–150 (English)

work page 2024
[6]

Dedicata207(2020), 29–49 (English)

Alexandru Dimca and Gabriel Sticlaru,Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves, Geom. Dedicata207(2020), 29–49 (English)

work page 2020
[7]

Alexandru Dimca and Gabriel Sticlaru,On the exponents of free and nearly free projective plane curves, Rev. Mat. Complut.30(2017), no. 2, 259–268 (English)

work page 2017
[8]

Mat., Barc.64(2020), no

Alexandru Dimca and Gabriel Sticlaru,On the jumping lines of bundles of logarithmic vector fields along plane curves, Publ. Mat., Barc.64(2020), no. 2, 513–542 (English)

work page 2020
[9]

Edward G. Jr. Evans,Bourbaki’s theorem and algebraic K-theory, J. Algebra41(1976), 108–115 (English)

work page 1976
[10]

Futata,Exploring n-freeness and numerical invariants of logarithmic tangent sheaves in reduced hypersurfaces: A stratified approach with algorithmic implementation, Ph.D

Daniel I. Futata,Exploring n-freeness and numerical invariants of logarithmic tangent sheaves in reduced hypersurfaces: A stratified approach with algorithmic implementation, Ph.D. Thesis, Campinas, SP, 2023. Tese de Doutorado em Matem´ atica

work page 2023
[11]

Stamate,Graded Bourbaki ideals of graded modules, Math

J¨ urgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate,Graded Bourbaki ideals of graded modules, Math. Z.299(2021), no. 3-4, 1303–1330 (English)

work page 2021
[12]

Nejad, and Aron Simis,The Bourbaki degree of a plane projective curve, Trans

Marcos Jardim, Abbas N. Nejad, and Aron Simis,The Bourbaki degree of a plane projective curve, Trans. Am. Math. Soc.377(2024), no. 11, 7633–7655 (English)

work page 2024
[13]

Algebra64(1980), 29–36 (English)

Matthew Miller,Bourbaki’s theorem and prime ideals, J. Algebra64(1980), 29–36 (English)

work page 1980
[14]

Kyoji Saito,Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci., Univ. Tokyo, Sect. I A27(1980), 265–291 (English)

work page 1980
[15]

Aron Simis,Commutative algebra, 2nd ed., De Gruyter Graduate, 2023. Roberto Alvarenga, S˜ ao Paulo State University (UNESP), S˜ ao Jos´ e do Rio Preto, SP, Brazil Email address:roberto.alvarenga@unesp.br Murillo Lozano, S˜ ao Paulo State University (UNESP), S˜ ao Jos´ e do Rio Preto, SP, Brazil Email address:murillo.lozano@unesp.br Parham Salehyan, S˜ ao ...

work page 2023