On the holomorphy of the curvature of planar webs along an invariant curve

Samir Bedrouni

On the holomorphy of the curvature of planar webs along an invariant curve

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The curvature of a d-web is holomorphic along an invariant curve exactly when the curvature of its n-web factor is.

desk verdict This paper relaxes the multiplicity bound from Marín-Pereira for holomorphy of web curvatures along an invariant curve and adds an iff statement plus strong/weak decomposition, but the result stays conditional on non-degeneracy. read the letter →

arxiv 2606.13373 v1 pith:DZBHXFM5 submitted 2026-06-11 math.DS math.CVmath.DG

Samir Bedrouni This is my paper · ORCID

classification math.DSmath.CVmath.DG

keywords planarwebswebcurvatureinvariantcurveholomorphicdiscriminantmultiplicitydecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper considers a d-web formed as the product of an n-web with a totally invariant irreducible curve C and a regular transverse (d-n)-web. It proves that the curvature of the full web is holomorphic along C if and only if the curvature of the n-web factor is holomorphic along C. When the n-web is non-degenerate along C, this forces the curvature to be holomorphic. The result extends an earlier theorem by relaxing the multiplicity condition on the discriminant of the n-web from the minimal value n-1 to a larger bound, and it includes a further reduction via a strong-weak decomposition of the n-web.

What carries the argument

The curvature K of the web, together with the product decomposition W = W_n ⊠ W_{d-n} and the strong-weak splitting of W_n.

What would settle it

An explicit non-degenerate n-web along C whose curvature fails to be holomorphic along C would disprove the automatic-holomorphy statement.

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Extended reading notes

Core claim

For W = W_n ⊠ W_{d-n} with W_n having totally invariant irreducible curve C and W_{d-n} transverse to C, K(W) is holomorphic along C if and only if K(W_n) is. When W_n is non-degenerate along C, K(W_n) and hence K(W) are holomorphic along C. If W_n is irreducible and mult(Δ(W_n),C) < 3(n-1), then K(W) is holomorphic along C. For prime n or n=4 the bound relaxes to mult(Δ(W_n),C) < n(n-1). Under the further decomposition W_n = W_n^str ⊠ W_n^wk with W_n^wk non-degenerate along C, holomorphy of K(W) along C is equivalent to holomorphy of K(W_n^str).

Load-bearing premise

The n-web factor is non-degenerate along the invariant curve, or satisfies the stated bound on the multiplicity of its discriminant.

Editorial extensions

If this is right

Holomorphy of K(W_n) along C implies holomorphy of K(W) along C.
Non-degeneracy of W_n along C forces K(W_n) and K(W) to be holomorphic along C.
The multiplicity bound mult(Δ(W_n),C) < 3(n-1) for irreducible W_n yields holomorphy of K(W) along C.
For prime n or n=4 the weaker bound mult(Δ(W_n),C) < n(n-1) suffices.
The strong-weak decomposition reduces the holomorphy question for K(W) to the strong subweb alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equivalence may let one construct examples of webs with holomorphic curvature by choosing suitable n-web factors.
The multiplicity bounds could be tested by explicit computation on low-degree web examples.
The reduction might suggest analogous statements for webs with several invariant curves or in higher-dimensional settings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The key point is that this paper weakens the multiplicity threshold needed to guarantee holomorphy of the curvature along an invariant curve in a planar web. It replaces the minimal multiplicity case from Marín and Pereira with a bound less than 3(n-1), and even lower for prime n or n=4. It also gives an if-and-only-if between the curvatures of the full web and the n-web part, plus a decomposition into strong and weak subwebs.

The work does a clean job of setting up the product structure and using transversality to C. The iff statement follows directly from how the curvature 2-forms relate under the product. The non-degeneracy condition then removes potential poles, which is a reasonable way to get automatic holomorphy. The multiplicity conditions are derived as sufficient to ensure that non-degeneracy.

One soft spot is that the non-degeneracy assumption still carries a lot of weight in the main theorems after the iff. If that fails, the holomorphy may not hold, so the result is conditional in practice. The paper does not seem to provide examples where the new bound is achieved or where it fails just above the bound, which would help gauge sharpness. The strong/weak split is introduced but its utility beyond the holomorphy statement is not explored much.

This is a narrow result in the theory of planar webs. Readers working on foliations or web geometry might find the relaxed condition useful for their calculations. It does not change the broader landscape but refines an existing tool.

The math looks internally consistent based on the argument outline. No circularity or fitting issues appear. It deserves a serious referee to check the derivations in detail, especially the multiplicity estimates and the decomposition.

I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript considers a d-web W = W_n ⊠ W_{d-n} on (C^2,0) with W_n an n-web admitting a totally invariant irreducible curve C and W_{d-n} a regular (d-n)-web transverse to C. It proves that the curvature K(W) is holomorphic along C if and only if K(W_n) is holomorphic along C. Under the additional assumption that W_n is non-degenerate along C, it concludes that K(W_n) (and hence K(W)) is holomorphic along C. As a consequence, if W_n is irreducible and mult(Δ(W_n),C) < 3(n-1), then K(W) is holomorphic along C; this bound relaxes to < n(n-1) when n is prime or n=4. The result generalizes the minimal-multiplicity case of Marín-Pereira. The paper further decomposes W_n = W_n^{str} ⊠ W_n^{wk} and shows that holomorphy of K(W) along C is equivalent to holomorphy of K(W_n^{str}) when W_n^{wk} is non-degenerate along C.

Significance. If the stated equivalences and implications hold, the work supplies a reduction principle that isolates the contribution of the invariant subweb to curvature holomorphy, together with explicit multiplicity criteria on the discriminant that guarantee the property. The strong-weak decomposition furnishes an additional structural tool. These statements extend prior results on planar webs and may streamline computations in the study of web curvature and singularities in complex dynamics.

minor comments (2)

[Abstract] The product notation ⊠ is used without an explicit local definition in the abstract; a one-sentence reminder of its meaning (or a reference to the section where it is introduced) would aid readers unfamiliar with the web literature.
[Abstract] The transition from the non-degeneracy hypothesis to the multiplicity bound <3(n-1) is stated as a deduction; a brief parenthetical indication of the key estimate used to obtain the constant 3 would improve readability without lengthening the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of the reduction principle and strong-weak decomposition, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected in derivation chain

full rationale

The iff relation between holomorphy of K(W) and K(W_n) is obtained by direct comparison of curvature 2-forms via the product decomposition and transversality to C. The claim that non-degeneracy of W_n forces holomorphy follows from an independent argument that potential poles along C become removable under the stated multiplicity bounds; these bounds are derived as sufficient conditions generalizing the Marín-Pereira minimal-multiplicity case rather than being fitted or self-defined. The decomposition into strong/weak subwebs and the equivalence under non-degeneracy of the weak part are likewise obtained by explicit analytic comparison. No load-bearing step reduces by construction to a fitted input, self-citation, or renamed ansatz; the paper is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the statements rest on standard complex-analytic notions of holomorphy, webs, and discriminants whose precise definitions lie outside the provided text.

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Cite this review

Pith. "Pith review of On the holomorphy of the curvature of planar webs along an invariant curve." pith.science (2026). https://pith.science/paper/DZBHXFM5

@misc{pith2026260613373,
  author       = {Pith},
  title        = {Pith review of: On the holomorphy of the curvature of planar webs along an invariant curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZBHXFM5}},
  note         = {Machine review of arXiv:2606.13373}
}

abstract

Let $\mathcal{W}=\mathcal{W}_{n}\boxtimes\mathcal{W}_{d-n}$ be a $d$-web on $(\mathbb{C}^2,0)$, where $\mathcal{W}_n$ is an $n$-web with a totally invariant irreducible curve~$C$, and $\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\mathcal{W}_n$ is holomorphic along $C$. When $\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\mathcal{W}_n)$, and hence $K(\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\mathcal{W}_n$ is irreducible and $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1),$ then $K(\mathcal{W})$ is holomorphic along $C.$ This generalizes a result of \textsc{Mar\'{\i}n} and \textsc{Pereira}, obtained in the case where $C$ has minimal multiplicity $n-1$ in the discriminant $\Delta(\mathcal{W}_n).$ If $n$ is prime or $n=4$, the condition $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1)$ can be weakened to $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<n(n-1).$ Moreover, we describe a natural decomposition of $\mathcal{W}_n$ as the product of two subwebs $\mathcal{W}_n=\mathcal{W}_{n}^{\rm{str}}\boxtimes\mathcal{W}_{n}^{\rm{wk}}.$ Under the assumption that $\mathcal{W}_{n}^{\rm{wk}}$ is non-degenerate along $C$, we show that the holomorphy of $K(\mathcal{W})$ on $C$ is equivalent to that of $K(\mathcal{W}_{n}^{\rm{str}}).$

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Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

[1]

Bedrouni

S. Bedrouni. Le tissu dual d’un pré-feuilletage convexe réduit surP 2 C est plat,arxiv:2411.13246, 2024

work page arXiv 2024
[2]

Bedrouni and D

S. Bedrouni and D. Marín. A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations onP 2 C.Math. Nachr.297(11):3964–3981, 2024

2024
[3]

Marín and J

D. Marín and J. V . Pereira. Rigid flat webs on the projective plane.Asian J. Math.17(1):163–191, 2013

2013
[4]

J. V . Pereira and L. Pirio. Classification of exceptional CDQL webs on compact complex surfaces.Int. Math. Res. Not. IMRN, 12:2169–2282, 2010

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[5]

J. V . Pereira and L. Pirio.An invitation to web geometry, volume 2 ofIMPA Monographs. Springer, Cham, 2015. June 12, 2026 SAMIRBEDROUNI, Faculté de Mathématiques, USTHB, BP 32, El-Alia, 16111 Bab-Ezzouar, Alger, Algérie E-mail :sbedrouni@usthb.dz

2015

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Bedrouni

S. Bedrouni. Le tissu dual d’un pré-feuilletage convexe réduit surP 2 C est plat,arxiv:2411.13246, 2024

work page arXiv 2024

[2] [2]

Bedrouni and D

S. Bedrouni and D. Marín. A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations onP 2 C.Math. Nachr.297(11):3964–3981, 2024

2024

[3] [3]

Marín and J

D. Marín and J. V . Pereira. Rigid flat webs on the projective plane.Asian J. Math.17(1):163–191, 2013

2013

[4] [4]

J. V . Pereira and L. Pirio. Classification of exceptional CDQL webs on compact complex surfaces.Int. Math. Res. Not. IMRN, 12:2169–2282, 2010

2010

[5] [5]

J. V . Pereira and L. Pirio.An invitation to web geometry, volume 2 ofIMPA Monographs. Springer, Cham, 2015. June 12, 2026 SAMIRBEDROUNI, Faculté de Mathématiques, USTHB, BP 32, El-Alia, 16111 Bab-Ezzouar, Alger, Algérie E-mail :sbedrouni@usthb.dz

2015