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

Source: arXiv:2606.13373 · Published 2026-06-11 · By Samir Bedrouni

TL;DR

This paper investigates the conditions under which the curvature of planar d-webs on (\mathbb{C}^2,0) remains holomorphic along an invariant irreducible curve C. Specifically, for a d-web decomposed as W = W_n \boxtimes W_{d-n}, where W_n is an n-web admitting C as a totally invariant curve and W_{d-n} a regular transverse web, the author establishes that K(W) is holomorphic on C if and only if K(W_n) is. When W_n is non-degenerate along C—a condition involving combinatorial restrictions on Puiseux indices and multiplicities—the curvature K(W_n), and thus K(W), is holomorphic along C. This generalizes prior results by Marín and Pereira that held under the more restrictive minimal multiplicity assumption for the discriminant of W_n. Moreover, the paper identifies a natural decomposition of W_n into strong and weak subwebs, showing that under non-degeneracy of the weak subweb along C, holomorphy of curvature reduces to that of the strong subweb.

Key results include weakened restrictions on multiplicities ensuring curvature holomorphy, equivalences between holomorphy conditions of composite webs and their components, and examples demonstrating the necessity of non-degeneracy conditions. The paper develops these using ramified Galois coverings, analytic expansions, and detailed local computations of fundamental forms and curvature 2-forms, refining and unifying previous partial understandings of curvature regularity in web geometry.

Key findings

• The curvature of W = W_n \boxtimes W_{d-n} is holomorphic along C if and only if the curvature of W_n is holomorphic along C (Theorem 1.6).
• If W_n is non-degenerate along C, then K(W_n) and hence K(W) are holomorphic along C (Proposition 1.7).
• If W_n is irreducible and the multiplicity of C in the discriminant satisfies mult(∆(W_n),C) < 3(n-1), then K(W) is holomorphic along C (Corollary 1.9), generalizing the known case for minimal multiplicity n-1.
• If n is prime or n=4, the bound on multiplicity can be relaxed to mult(∆(W_n),C) < n(n-1) for holomorphy of K(W) along C (Corollary 1.10).
• Decomposition W_n = W_n^{str} \boxtimes W_n^{wk} allows reduction of curvature holomorphy of W to that of W_n^{str} when W_n^{wk} is non-degenerate along C (Theorem 1.11).
• Counterexamples show failure of holomorphy when non-degeneracy conditions (on gcd(ν,κ), Puiseux exponent distinctness, etc.) are not met (Section 4).
• Key computation technique uses ramified Galois coverings to pull-back webs, allowing curvature analysis on completely decomposable coverings and descent arguments.
• Explicit local normal forms of foliations and webs with Puiseux indices κ and their relation to multiplicity bounds provide intrinsic geometric invariants governing holomorphy.

Methodology — deep read

1. Threat model and assumptions: The objects of study are germs of planar d-webs W on (\mathbb{C}^2,0), decomposable into W_n \boxtimes W_{d-n}, with W_n admitting a totally invariant irreducible curve C, and W_{d-n} a regular web transverse to C. The adversarial scenario is geometric: under what analytic/algebraic conditions on W and its components does the curvature 2-form K(W) remain holomorphic on C? There are no adversarial entities, but the threats are singularities or poles arising in curvature forms near discriminant divisors.

2. Data: The data are analytic local objects (webs, foliations) defined by symmetric 1-forms. The key analytic invariant is the discriminant divisor ∆(W) with multiplicities along C. The paper builds on known local normal forms, Puiseux expansions of slope parametrizations near C, and their interaction with the web's differential invariants.

3. Architecture/algorithm: The method heavily uses the decomposition of web W into irreducible subwebs, distinguishes between strong and weak invariance of curve C, and introduces Puiseux indices κ and exponents ρ to characterize local behavior. The author applies ramified cyclic Galois coverings π(x,y) = (x,y^ν) to completely decompose the web pull-backs into foliations, whose curvature and fundamental forms are easier to analyze. This approach leverages symmetry and factorization to study pole orders and residues of curvature forms on the covering space, then descends results back.

The core technical device is the fundamental 1-form η(W) satisfying dω_i = η(W) ∧ ω_i for foliations ω_i defining the web, whose exterior derivative K(W)=dη(W) gives curvature. Explicit local expansions of η(W) and K(W) are computed in terms of coordinates and Puiseux data, controlling singularities.

4. Training regime: Not applicable; this is analytic algebraic geometry.

5. Evaluation protocol: Holomorphy is tested by analyzing pole orders of curvature forms along C and their pull-backs. Key lemmas (e.g., Lemma 2.3) provide pole order bounds under assumptions (non-degeneracy conditions). The analysis extends inductively using decompositions into irreducible subwebs, applying previous curvature holomorphy results for 3-webs and completely decomposable webs as baselines.

6. Reproducibility: All computations are analytic with explicit forms and expansions; code is not applicable. The paper references foundational web geometry texts and classical results ensuring standard definitions and tools are accessible.

One concrete example end-to-end: Consider W = W_6 from Example 4.1, irreducible 6-web with invariant curve C defined by w=0 and Puiseux index κ=3. The gcd(6,3)=3>2 violates non-degeneracy condition (a). Pulling back by π(x,y) = (x,y^6), the curvature K(π^*W) develops a polar term proportional to 1/y near y=0, hence is not holomorphic on the preimage curve ˜C. Descending back shows K(W) is non-holomorphic on C, illustrating the necessity of gcd conditions for holomorphy in Proposition 1.7. This example concretely demonstrates the method of using ramified coverings to detect curvature singularities.

Technical innovations

• Extension of curvature holomorphy results from minimal multiplicity invariant curves to neighboring multiplicities less than 3(n-1), weakening known assumptions from Marín and Pereira.
• Introduction and detailed use of Puiseux index and exponent invariants (κ, ρ) attached to irreducible subwebs and the invariant curve, providing intrinsic criteria for non-degeneracy along C.
• Identification of a natural decomposition of webs into strong and weak invariant subwebs along C, and proof that curvature holomorphy reduces to the strong factor assuming non-degeneracy of the weak factor (Theorem 1.11).
• New analytic techniques combining ramified Galois coverings with explicit computation of fundamental forms and curvature poles to analyze holomorphy, extending prior work focused on special cases and 3-webs.
• Provision of explicit counterexamples that showcase failures of curvature holomorphy when the gcd and exponent multiplicity conditions are violated, clarifying sharpness of assumptions.

Baselines vs proposed

• Marín and Pereira minimal multiplicity case (mult(∆(W_n),C) = n-1): curvature K(W) holomorphic on C (prior result) vs Bedrouni generalization: curvature K(W) holomorphic for mult(∆(W_n),C) < 3(n-1).
• For prime n or n=4, prior minimal multiplicity bound n-1 vs new relaxed bound mult(∆(W_n),C) < n(n-1) for curvature holomorphy.
• Example 4.1 baseline: gcd(6,3)=3 violating non-degeneracy gives K(W) non-holomorphic vs proposition 1.7 prediction that gcd ≤2 ensures holomorphy.

Limitations

• The results critically depend on non-degeneracy conditions (gcd bounds, distinctness of Puiseux exponents) which may exclude some interesting webs with singularities.
• Holomorphy conclusions pertain only along an irreducible totally invariant curve C; no results for curves that are only partially invariant or multiple components.
• The curvature is analyzed only locally near C; global properties of webs or behavior away from C are not considered.
• Quantitative estimates on the size of holomorphic neighborhoods or effective bounds on curvature are not provided.
• The approach requires differentiable ramified coverings which may complicate extension to more singular or higher dimensional webs.
• No experimental or numerical validation; all arguments are formal and analytic with explicit but complex symbolic computations.

Open questions / follow-ons

• Can the non-degeneracy conditions be further weakened or replaced by alternative geometric invariants guaranteeing curvature holomorphy along invariant curves?
• How do curvature holomorphy properties extend globally on compact complex surfaces or under more complicated web compositions?
• Is there a direct geometric interpretation or classification of webs violating non-degeneracy that produce curvature poles?
• Can these techniques extend to webs on higher-dimensional complex manifolds or to singularities with multiple invariant components?

Why it matters for bot defense

While this paper is firmly rooted in complex web geometry and analytic properties of curvature near invariant algebraic curves, the underlying theme of understanding singularities and invariants impacted by decomposition and local coverings resonates conceptually with bot-defense techniques leveraging structural decompositions of complex data. A bot-defense engineer might draw an analogy: just as this work decomposes webs into subwebs with well-understood local behavior to detect singularities in curvature, CAPTCHA systems seek invariants or signatures stable under certain adversarial transformations. The analytic tools using ramified covers to understand singular behavior might inspire novel strategies for capturing subtle local patterns or failures induced by automated scripts versus humans. However, direct application is abstract and would require significant adaptation from differential geometry to computational security domains.

Cite

@article{arxiv2606_13373,
  title={ On the holomorphy of the curvature of planar webs along an invariant curve },
  author={ Samir Bedrouni },
  journal={arXiv preprint arXiv:2606.13373},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.13373}
}

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• arXiv:2606.13373 (abstract)
• arXiv:2606.13373 (PDF)