arxiv: 2605.07944 · v1 · submitted 2026-05-08 · 🧮 math.AG · math.CV· math.DS

Recognition: 2 theorem links

· Lean Theorem

Homogeneous pre-foliations of co-degree one and degree four on the projective plane

Carla Pracias, Maycol Falla Luza

Pith reviewed 2026-05-11 03:09 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DS

keywords homogeneous pre-foliationsprojective planeflat 4-websLegendre transformGauss map ramificationfoliations of degree 3normal formsalgebraic geometry

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The pith

All homogeneous pre-foliations of co-degree one and degree four on P² that induce flat 4-webs are classified up to automorphism.

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

The paper classifies, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on the complex projective plane P² whose Legendre transform defines a flat 4-web. It organizes the classification by the type of the underlying homogeneous foliation H of degree 3 and completes the cases where the tangent bundle of H has degree 3 or 4, after prior work handled degree 2. A sympathetic reader would care because the result supplies an explicit finite list of one-forms parametrized by the ramification data of the Gauss map of H, giving concrete normal forms in the study of webs and foliations on projective varieties. The proof rests on curvature-holomorphy criteria together with symbolic computation to exhaust the possibilities.

Core claim

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on P² whose Legendre transform defines a flat 4-web. The classification is organized according to the type of the underlying homogeneous foliation H of degree 3, distinguishing the cases deg(T_H)=2, 3, and 4. The case deg(T_H)=2 was treated by Bedrouni, while the cases deg(T_H)=3 and deg(T_H)=4 are completed here. The proof combines Bedrouni's curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrised by the ramification data of the Gauss map of H.

What carries the argument

Bedrouni's curvature-holomorphy criteria applied to the underlying homogeneous foliation H of degree 3, combined with explicit normal forms and symbolic computation of the pre-foliations.

Load-bearing premise

Bedrouni's curvature-holomorphy criteria remain valid and sufficient for tangent-bundle degrees 3 and 4, and the symbolic computation finds every possible family without omissions.

What would settle it

The appearance of a homogeneous pre-foliation of co-degree one and degree four on P², not projectively equivalent to any listed normal form, whose Legendre transform nonetheless defines a flat 4-web would falsify the classification.

read the original abstract

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on the complex projective plane $\Ptwo$ whose Legendre transform defines a flat $4$-web. The classification is organized according to the type of the underlying homogeneous foliation $\Hcal$ of degree~$3$, distinguishing the cases $\deg(\Tcal_{\Hcal})=2$, $3$, and~$4$. The case $\deg(\Tcal_{\Hcal})=2$ was treated by Bedrouni, while the cases $\deg(\Tcal_{\Hcal})=3$ and $\deg(\Tcal_{\Hcal})=4$ are completed here. The proof combines Bedrouni's curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrised by the ramification data of the Gauss map of~$\Hcal$.

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

This paper finishes the classification for the missing deg(T_H)=3 and 4 cases by listing explicit normal forms for the pre-foliations that yield flat 4-webs.

read the letter

This paper finishes the classification for the missing deg(T_H)=3 and 4 cases by listing explicit normal forms for the pre-foliations that yield flat 4-webs. It builds directly on Bedrouni's earlier work for the degree-2 case and uses the same curvature-holomorphy criteria plus normal forms and symbolic enumeration over the ramification data of the Gauss map of H. The output is a finite list of one-forms parametrized by that data, which the authors claim exhausts all possibilities up to projective automorphism. That is the concrete advance: the gap is closed with equations that specialists can now cite and compute with directly. The approach is standard for this subfield and the result is useful because it turns an incomplete program into a complete list for these degrees. The explicit forms are the part that will actually get used. The soft spot is the symbolic computation step. The paper states that the enumeration produces only the listed forms, but it does not include the CAS code, the full search tree, or the Gröbner or resultant steps. Without those, confirming that no additional families were missed requires re-implementing the search, which is doable in principle since the set of ramification configurations is finite but still leaves the claim harder to verify from the text alone. The extension of the curvature criteria from degree 2 to 3 and 4 is also taken as holding without extra obstructions being checked in detail. This is a narrow, technical classification aimed at people already working on holomorphic foliations and webs on P². A reader in that niche will find the normal forms and the completed list worth having for reference and further calculations. It is the sort of incremental but solid result that deserves a serious referee to check the computational exhaustiveness and confirm the normal forms are indeed complete and correctly derived.

Referee Report

2 major / 0 minor

Summary. The paper classifies, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on P² whose Legendre transform defines a flat 4-web. It organizes the classification by the type of the underlying homogeneous foliation H of degree 3, treating the case deg(T_H)=2 via Bedrouni's prior work and completing the cases deg(T_H)=3 and 4 by combining curvature-holomorphy criteria with explicit normal forms and symbolic computation over ramification data of the Gauss map, yielding a finite list of explicit one-forms.

Significance. If the result holds, the classification supplies an explicit, finite list of such pre-foliations parametrized by ramification data, extending the degree-2 case and providing concrete normal forms that may be useful for further study of flat 4-webs and homogeneous foliations. The methodological choice to reduce the problem to a finite symbolic search over ramification configurations and to output explicit one-forms is a clear strength when the computation can be independently verified.

major comments (2)

[Proof of the main theorem (cases deg(T_H)=3,4)] The central claim of a complete finite list for deg(T_H)=3 and 4 rests on the symbolic enumeration of ramification data being exhaustive. The manuscript asserts that the computation produces only the listed one-forms but does not exhibit the enumeration tree, the CAS code, the Gröbner-basis or resultant steps, or any independent verification that no additional solutions exist. This is load-bearing for the classification statement in the abstract and the proof combining Bedrouni's criteria with normal forms.
[Introduction and statement of the main result] The argument assumes Bedrouni's curvature-holomorphy criteria extend verbatim to deg(T_H)=3 and 4 without new obstructions. While the abstract states that the proof combines these criteria with explicit normal forms, no separate verification or reference is supplied showing that the criteria remain sufficient in the higher-degree setting; this extension is required for the reduction to the listed normal forms to be valid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We will revise the manuscript to strengthen the presentation of the computational verification and to clarify the applicability of the curvature-holomorphy criteria. We address the major comments point by point below.

read point-by-point responses

Referee: [Proof of the main theorem (cases deg(T_H)=3,4)] The central claim of a complete finite list for deg(T_H)=3 and 4 rests on the symbolic enumeration of ramification data being exhaustive. The manuscript asserts that the computation produces only the listed one-forms but does not exhibit the enumeration tree, the CAS code, the Gröbner-basis or resultant steps, or any independent verification that no additional solutions exist.

Authors: We agree that explicit details on the symbolic computation are needed to make the exhaustiveness verifiable. In the revised manuscript we will add an appendix describing the ramification configurations examined, the key Gröbner-basis and resultant computations performed in the CAS, and the arguments establishing that the search is complete. The full code will be supplied as supplementary material. revision: yes
Referee: [Introduction and statement of the main result] The argument assumes Bedrouni's curvature-holomorphy criteria extend verbatim to deg(T_H)=3 and 4 without new obstructions. While the abstract states that the proof combines these criteria with explicit normal forms, no separate verification or reference is supplied showing that the criteria remain sufficient in the higher-degree setting.

Authors: The curvature-holomorphy criteria are formulated in terms of general properties of the foliation and its Gauss map that hold independently of the tangent-sheaf degree. Nevertheless, to address the concern we will insert a short clarifying paragraph (or subsection) in the introduction that recalls the relevant statements from Bedrouni and explains why no additional obstructions arise for degrees 3 and 4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classification via external criteria and symbolic enumeration

full rationale

The derivation combines Bedrouni's curvature-holomorphy criteria (treated as external for deg(T_H)=2 and extended here) with explicit normal forms for homogeneous foliations of degree 3 and symbolic computation that enumerates ramification data of the Gauss map to produce the finite list of one-forms. No equation or step reduces the output classification to a definitional identity, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the result is generated from the enumeration process rather than presupposed by the input definitions. The paper is self-contained against the stated external benchmark and computational method.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard facts from algebraic geometry of the projective plane and on the curvature-holomorphy criteria established in prior literature. No new entities are postulated and no numerical parameters are fitted to data.

axioms (2)

domain assumption Homogeneous foliations of degree 3 on P² admit a well-defined tangent sheaf whose degree is an invariant
Invoked when the classification is organized by deg(T_H)
domain assumption Bedrouni's curvature-holomorphy criteria apply to the cases deg(T_H)=3 and 4
Used as the starting point for the proof

pith-pipeline@v0.9.0 · 5464 in / 1375 out tokens · 42738 ms · 2026-05-11T03:09:31.225790+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The proof combines Bedrouni’s curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrised by the ramification data of the Gauss map of H.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Table 1. Possible types TH for deg H = 3. ... deg(TH) ∈ {2,3,4}

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Bedrouni, Pre-foliations of co-degree one on ^2 with a flat Legendre transform, arXiv preprint arXiv:2309.xxxx (2023)

S. Bedrouni, Pre-foliations of co-degree one on ^2 with a flat Legendre transform, arXiv preprint arXiv:2309.xxxx (2023)

work page 2023
[2]

Bedrouni, Platitude des tissus duaux de certains pr\'e-feuilletages convexes du plan projectif complexe, arXiv preprint arXiv:2405.05464 (2024)

S. Bedrouni, Platitude des tissus duaux de certains pr\'e-feuilletages convexes du plan projectif complexe, arXiv preprint arXiv:2405.05464 (2024)

work page arXiv 2024
[3]

Beltr\'an, M

A. Beltr\'an, M. Falla Luza, and D. Mar\' n, Flat 3-webs of degree one on the projective plane, Ann.\ Fac.\ Sci.\ Toulouse Math.\ 23 (2014), no. 4, 779--796

work page 2014
[4]

Bedrouni and D

S. Bedrouni and D. Mar\' n, Tissus plats et feuilletages homog\`enes sur le plan projectif complexe, Bull.\ Soc.\ Math.\ France 146 (2018), no. 3, 479--516

work page 2018
[5]

Mar\' n and J

D. Mar\' n and J. V. Pereira, Rigid flat webs on the projective plane, Asian J.\ Math.\ 17 (2013), no. 1, 163--192

work page 2013
[6]

J. V. Pereira, Vector fields, invariant varieties and linear systems, Ann.\ Inst.\ Fourier 51 (2001), no. 5, 1385--1405

work page 2001
[7]

J. V. Pereira and L. Pirio, An invitation to web geometry, IMPA Monographs, vol. 2, Springer, Berlin, 2015

work page 2015
[8]

C. C. de Lima Pracias, Flat webs on the projective plane: constructions and classifications via convex and homogeneous foliations, Ph.D.\ Thesis, Universidade Federal Fluminense, Niter\'oi, 2025

work page 2025