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arxiv: 2511.00372 · v4 · submitted 2025-11-01 · 🧮 math.AG · math.AC

Bourbaki degree of pairs of projective surfaces

Felipe Monteiro This is my paper

Pith reviewed 2026-05-18 02:11 UTC · model grok-4.3

classification 🧮 math.AG math.AC MSC 14J60
keywords logarithmic tangent sheafBourbaki degreecodimension-one foliationprojective three-spacesyzygiesnearly free sheavessheaf stabilityTjurina number
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The pith

A pair of homogeneous polynomials in four variables can induce an unstable non-split tangent sheaf for a degree-three foliation on projective three-space.

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

The paper studies the logarithmic tangent sheaf attached to two homogeneous polynomials in four variables and introduces two discrete invariants: the number m, which behaves like a Tjurina number, and the Bourbaki degree of the sequence. These invariants are shown to satisfy a quadratic upper bound in the degrees of the polynomials and to govern the minimal degree of syzygies on the Jacobian matrix. The authors relate the sheaf to foliation theory by observing that it is, up to twist, the tangent sheaf of a codimension-one foliation in projective three-space. They classify the possible invariants for pencils of cubics and for pairs consisting of one quadric and one cubic, and they exhibit one nearly-free case whose associated sheaf is unstable and non-split.

Core claim

By defining the invariant m and the Bourbaki degree for a sequence of two homogeneous polynomials, the authors obtain bounds relating these numbers to the syzygy degrees of the Jacobian matrix. In the special case of a nearly-free example, the logarithmic sheaf becomes an unstable non-split tangent sheaf of a codimension-one foliation of degree three in projective three-space, furnishing a counterexample to the conjecture that every such sheaf must be stable or split.

What carries the argument

The Bourbaki degree of the sequence of two polynomials, which measures the algebraic complexity of the pair and controls both the syzygy relations and the stability properties of the induced foliation tangent sheaf.

If this is right

  • The minimal syzygy degree is bounded above by a quadratic expression involving m and the Bourbaki degree.
  • For pencils of cubics only finitely many pairs of values for the two invariants can occur.
  • For a quadric and a cubic the possible nearly-free configurations are limited and include at least one unstable case.
  • Equality can hold in the quadratic bound on m for certain explicit pairs.

Where Pith is reading between the lines

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

  • The same invariants could be defined and bounded for sequences of three or more homogeneous polynomials.
  • Stability questions for foliation sheaves in higher-dimensional projective spaces might be addressed by analogous discrete invariants.
  • Random sampling of polynomial pairs could reveal how often the resulting sheaves are unstable.

Load-bearing premise

The logarithmic sheaf associated to the pair is, up to a twist, the tangent sheaf of a codimension-one foliation in projective three-space.

What would settle it

Explicit computation of the stability and splitting type of the tangent sheaf for the specific nearly-free example of degree three would confirm or refute the claimed instability and non-splitting.

read the original abstract

The present work focuses on studying the logarithmic tangent sheaf associated with sequences of two homogeneous polynomials in four variables. We introduce two positive discrete invariants: the invariant m and the Bourbaki degree of a sequence, inspired by the framework of the Bourbaki degree recently developed for projective plane curves by Jardim-Nejad-Simis. The invariant m plays the role of the Tjurina number of plane projective curves and is bounded by a quadratic relation of the degrees. We establish results concerning the interplay of minimal degree for syzygies of the Jacobian matrix and the introduced discrete invariants. Our approach uses tools from foliation theory, taking advantage of the fact that the logarithmic sheaf is, up to a twist, the tangent sheaf of a codimension-one foliation in projective three-space. We provide examples and classification results for pencils of cubics and for pairs of a quadric and a cubic. In particular, one of the nearly-free examples induces an unstable, non-split tangent sheaf for a codimension-one foliation of degree 3, answering, in the negative, a conjecture of Calvo-Andrade, Correa and Jardim from 2018.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces two new discrete invariants, m and the Bourbaki degree, for sequences of two homogeneous polynomials in four variables. It establishes a quadratic bound on m, studies the minimal degree of syzygies of the Jacobian matrix in relation to these invariants, and employs tools from foliation theory by identifying the logarithmic sheaf (up to twist) as the tangent sheaf of a codimension-one foliation in P^3. The work includes examples and classification results for pencils of cubics and for pairs consisting of a quadric and a cubic. Notably, one nearly-free example is used to construct an unstable, non-split tangent sheaf for a codimension-one foliation of degree 3, which negatively answers a conjecture by Calvo-Andrade, Correa, and Jardim from 2018.

Significance. If the central claims hold, this work advances the study of logarithmic sheaves and foliations by introducing new discrete invariants analogous to those for plane curves, provides concrete classifications for low-degree cases, and resolves an open conjecture in the negative. The explicit examples and use of foliation tools to transfer stability results constitute a solid contribution to algebraic geometry.

major comments (1)
  1. Abstract and the section introducing the foliation identification: The claim that the logarithmic tangent sheaf is, up to a twist, the tangent sheaf of a codimension-one foliation in P^3 is load-bearing for the refutation of the 2018 conjecture. For the specific nearly-free degree-3 example, explicit verification is required that the defining 1-form satisfies the integrability condition and that the sheaf properties (stability and non-splitting) transfer directly; without this check the counterexample does not necessarily address the conjecture about foliation tangent sheaves.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive feedback on the foliation identification. We address the major comment below and will revise the manuscript to include the requested explicit verification for the degree-3 example.

read point-by-point responses

  1. Referee: Abstract and the section introducing the foliation identification: The claim that the logarithmic tangent sheaf is, up to a twist, the tangent sheaf of a codimension-one foliation in P^3 is load-bearing for the refutation of the 2018 conjecture. For the specific nearly-free degree-3 example, explicit verification is required that the defining 1-form satisfies the integrability condition and that the sheaf properties (stability and non-splitting) transfer directly; without this check the counterexample does not necessarily address the conjecture about foliation tangent sheaves.

    Authors: We agree that the identification is central to transferring the counterexample to the setting of the 2018 conjecture. The general correspondence between the logarithmic tangent sheaf of the pair and the tangent sheaf of the associated codimension-one foliation follows from the standard construction in the literature on foliations in P^3, where the 1-form is defined directly from the pair of polynomials and integrability holds by the way the foliation is constructed. Nevertheless, to make the argument fully self-contained for this specific nearly-free example, we will add an explicit computation in the revised manuscript. This will verify that the defining 1-form satisfies the integrability condition dω ∧ ω = 0 and confirm that the stability and non-splitting properties transfer directly from the logarithmic sheaf to the foliation tangent sheaf. We believe this addition will address the concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructions and standard identifications support the claims.

full rationale

The paper introduces the invariants m and Bourbaki degree as new discrete quantities for pairs of homogeneous polynomials in four variables, explicitly defined and bounded via quadratic relations on degrees, drawing inspiration from but not reducing to the prior Jardim-Nejad-Simis framework for plane curves. Interplay results with syzygies of the Jacobian matrix are established directly from these definitions. The use of foliation theory rests on the stated fact that the logarithmic sheaf is (up to twist) the tangent sheaf of a codimension-one foliation in P^3, presented as an external tool rather than a self-derived or self-cited premise. The negative answer to the 2018 conjecture is obtained via explicit construction and computation for a specific nearly-free example of degree 3, without any fitted parameters renamed as predictions or load-bearing self-citation chains. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that the logarithmic sheaf is (up to twist) the tangent sheaf of a codimension-one foliation; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The logarithmic sheaf is, up to a twist, the tangent sheaf of a codimension-one foliation in projective three-space.
    Invoked when the authors adopt tools from foliation theory to study the sheaf and its syzygies.

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Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Plus-one Generated and Next to Free Arrangements of Hyperplanes

    Abe, Takuro (June 2019). “Plus-one Generated and Next to Free Arrangements of Hyperplanes”. In: Inter- national Mathematics Research Notices 2021.12, pp. 9233–9261. issn: 1073-7928. doi: 10.1093/imrn/ rnz099. Calvo-Andrade, Omegar, Maur´ ı cio Corrˆ ea, and Marcos Jardim (2018). “Codimension One Holomorphic Distributions on the Projective Three-space”. In...

    work page doi:10.1093/imrn/ 2019

  2. [2]

    Logarithmic sheaves of complete intersections

    doi: 10.1007/s00025-025-02371-z . Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry . Graduate texts in mathematics. Springer-Verlag. isbn: 9783540942696. Eisenbud, David and Joe Harris (2016). 3264 and all that: A second course in algebraic geometry . Cambridge University Press. Faenzi, Daniele, Marcos Jardim, Jean Vall` es, ...

    work page doi:10.1007/s00025-025-02371-z 1995