Linear spaces of rational integrable 1-forms

Gabriel Barbosa

arxiv: 2605.23749 · v1 · pith:2DAH6YRJnew · submitted 2026-05-22 · 🧮 math.CV · math.AG

Linear spaces of rational integrable 1-forms

Gabriel Barbosa This is my paper

Pith reviewed 2026-05-25 02:20 UTC · model grok-4.3

classification 🧮 math.CV math.AG

keywords rational one-formsintegrable locusprojective manifoldfinite-dimensional spacescomplex geometry

0 comments

The pith

Finite-dimensional spaces of rational one-forms on projective manifolds are studied via their integrable loci.

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

The paper investigates finite-dimensional vector spaces made up of rational one-forms defined on a projective manifold. It proposes examining these spaces by focusing on the subset of points where the forms satisfy the integrability condition. A sympathetic reader would expect this locus-based method to expose algebraic or geometric invariants that classify or constrain the possible spaces. The approach treats the integrable locus as the primary diagnostic tool rather than direct analysis of the forms themselves.

Core claim

Finite-dimensional spaces of rational one-forms on a projective manifold can be studied by means of their integrable locus.

What carries the argument

The integrable locus of a finite-dimensional space of rational one-forms, which serves as the object whose geometric or algebraic properties are used to understand the ambient space.

Load-bearing premise

That examining the integrable locus alone yields enough information to determine the structure or classification of the finite-dimensional spaces.

What would settle it

An explicit pair of distinct finite-dimensional spaces of rational one-forms whose integrable loci are identical as subvarieties.

read the original abstract

We study finite-dimensional spaces of rational one-forms on a projective manifold by means of their integrable locus.

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 abstract states only that the authors study finite-dimensional spaces of rational 1-forms via their integrable locus, with no theorems, examples, or results visible.

read the letter

The main thing here is that the abstract is one sentence long and announces a study without delivering any concrete theorem, classification, or example. Nothing new is shown or claimed in the provided text. The title suggests work on linear spaces of rational integrable 1-forms, but the body does not appear to contain derivations, dimension counts, or comparisons to existing results on rational forms in algebraic geometry. If the full manuscript has actual content, it is not reflected in what is available for review. The approach of examining the integrable locus might be a reasonable organizing idea for this narrow topic, but it remains untested here. The central assumption that this locus yields structural information about the spaces is stated but not supported by any calculation or counterexample. No equations, no proofs, and no engagement with prior literature are on display, so soundness cannot be checked and circularity is not an issue only because there is nothing to circle. This leaves the paper looking like a research plan rather than a completed piece of work. It is aimed at a tiny group of specialists already deep in rational 1-forms and integrability conditions on projective manifolds; almost no one else will find usable material. I would not send it to peer review in this form. The authors would need to add explicit results, examples, and comparisons before it merits referee time.

Referee Report

0 major / 1 minor

Summary. The manuscript proposes to study finite-dimensional spaces of rational one-forms on projective manifolds by means of their integrable locus.

Significance. The subject lies at the intersection of complex geometry and foliation theory. If concrete structural results or classifications were obtained, they could be of interest; however, the provided text contains no theorems, definitions, examples, or derivations, preventing any evaluation of potential impact.

minor comments (1)

The abstract is the only content supplied and states a general research direction without outlining methods, results, or examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The manuscript is currently a brief announcement of the research program; we address the evaluation concern below.

read point-by-point responses

Referee: the provided text contains no theorems, definitions, examples, or derivations, preventing any evaluation of potential impact.

Authors: We agree that the present version consists only of the title and a one-sentence abstract and therefore supplies none of the requested material. A revised manuscript will contain the necessary definitions of integrable loci for linear spaces of rational 1-forms, concrete examples on projective manifolds, and at least one structural theorem relating dimension of the space to properties of the integrable locus. revision: yes

Circularity Check

0 steps flagged

No circularity identifiable from provided text

full rationale

The abstract and context provide only a high-level description of studying finite-dimensional spaces of rational one-forms via their integrable locus, with no equations, derivations, theorems, self-citations, or load-bearing steps presented. Without any explicit chain of reasoning or definitions that could reduce to inputs by construction, no circular steps exist to flag. The derivation is not visible and thus cannot be evaluated as circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5511 in / 945 out tokens · 23868 ms · 2026-05-25T02:20:10.314471+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[1]

Pereira, Jorge Vit\'orio and Perrone, Carlo , TITLE =. Bull. Sci. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.bulsci.2009.09.005 , URL =

work page doi:10.1016/j.bulsci.2009.09.005 2010
[2]

Cerveau, Dominique , TITLE =. Publ. Mat. , FJOURNAL =. 2002 , NUMBER =. doi:10.5565/PUBLMAT\_46202\_06 , URL =

work page doi:10.5565/publmat 2002
[3]

Panasyuk, Andriy , title =. Rep. Math. Phys. , issn =. 2002 , language =. doi:10.1016/S0034-4877(02)80059-3 , keywords =

work page doi:10.1016/s0034-4877(02)80059-3 2002
[4]

1991 , ISSN =

Webs, Veronese curves, and bihamiltonian systems , JOURNAL =. 1991 , ISSN =. doi:https://doi.org/10.1016/0022-1236(91)90057-C , URL =

work page doi:10.1016/0022-1236(91)90057-c 1991
[5]

arXiv preprint arXiv:2505.14873 , year=

Unlikely intersections of codimension one foliations , author=. arXiv preprint arXiv:2505.14873 , year=

work page arXiv
[6]

Cerveau, Dominique and Lins-Neto, Alcides and Loray, Frank and Pereira, Jorge Vit\'orio and Touzet, Fr\'ed\'eric , TITLE =. Mosc. Math. J. , FJOURNAL =. 2007 , NUMBER =. doi:10.17323/1609-4514-2007-7-1-21-54 , URL =

work page doi:10.17323/1609-4514-2007-7-1-21-54 2007
[7]

and Dufour, Jean P

Bouetou, Thomas B. and Dufour, Jean P. , TITLE =. Int. J. Math. Math. Sci. , FJOURNAL =. 2006 , PAGES =. doi:10.1155/IJMMS/2006/93142 , URL =

work page doi:10.1155/ijmms/2006/93142 2006
[8]

Cousin, Ga\"el and Pereira, Jorge Vit\'orio , TITLE =. Math. Res. Lett. , FJOURNAL =. 2014 , NUMBER =. doi:10.4310/MRL.2014.v21.n5.a5 , URL =

work page doi:10.4310/mrl.2014.v21.n5.a5 2014
[9]

1993 , PAGES =

The. 1993 , PAGES =. doi:10.1007/978-1-4612-0345-2 , URL =

work page doi:10.1007/978-1-4612-0345-2 1993
[10]

Sc\'ardua, Bruno Azevedo , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1997 , NUMBER =. doi:10.1016/S0012-9593(97)89918-1 , URL =

work page doi:10.1016/s0012-9593(97)89918-1 1997
[11]

1979 , PAGES =

Jouanolou, Jean-Pierre , TITLE =. 1979 , PAGES =

1979

[1] [1]

Pereira, Jorge Vit\'orio and Perrone, Carlo , TITLE =. Bull. Sci. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.bulsci.2009.09.005 , URL =

work page doi:10.1016/j.bulsci.2009.09.005 2010

[2] [2]

Cerveau, Dominique , TITLE =. Publ. Mat. , FJOURNAL =. 2002 , NUMBER =. doi:10.5565/PUBLMAT\_46202\_06 , URL =

work page doi:10.5565/publmat 2002

[3] [3]

Panasyuk, Andriy , title =. Rep. Math. Phys. , issn =. 2002 , language =. doi:10.1016/S0034-4877(02)80059-3 , keywords =

work page doi:10.1016/s0034-4877(02)80059-3 2002

[4] [4]

1991 , ISSN =

Webs, Veronese curves, and bihamiltonian systems , JOURNAL =. 1991 , ISSN =. doi:https://doi.org/10.1016/0022-1236(91)90057-C , URL =

work page doi:10.1016/0022-1236(91)90057-c 1991

[5] [5]

arXiv preprint arXiv:2505.14873 , year=

Unlikely intersections of codimension one foliations , author=. arXiv preprint arXiv:2505.14873 , year=

work page arXiv

[6] [6]

Cerveau, Dominique and Lins-Neto, Alcides and Loray, Frank and Pereira, Jorge Vit\'orio and Touzet, Fr\'ed\'eric , TITLE =. Mosc. Math. J. , FJOURNAL =. 2007 , NUMBER =. doi:10.17323/1609-4514-2007-7-1-21-54 , URL =

work page doi:10.17323/1609-4514-2007-7-1-21-54 2007

[7] [7]

and Dufour, Jean P

Bouetou, Thomas B. and Dufour, Jean P. , TITLE =. Int. J. Math. Math. Sci. , FJOURNAL =. 2006 , PAGES =. doi:10.1155/IJMMS/2006/93142 , URL =

work page doi:10.1155/ijmms/2006/93142 2006

[8] [8]

Cousin, Ga\"el and Pereira, Jorge Vit\'orio , TITLE =. Math. Res. Lett. , FJOURNAL =. 2014 , NUMBER =. doi:10.4310/MRL.2014.v21.n5.a5 , URL =

work page doi:10.4310/mrl.2014.v21.n5.a5 2014

[9] [9]

1993 , PAGES =

The. 1993 , PAGES =. doi:10.1007/978-1-4612-0345-2 , URL =

work page doi:10.1007/978-1-4612-0345-2 1993

[10] [10]

Sc\'ardua, Bruno Azevedo , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1997 , NUMBER =. doi:10.1016/S0012-9593(97)89918-1 , URL =

work page doi:10.1016/s0012-9593(97)89918-1 1997

[11] [11]

1979 , PAGES =

Jouanolou, Jean-Pierre , TITLE =. 1979 , PAGES =

1979