{
  "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 admitting an irreducible totally invariant~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. For each irreducible subweb $\\mathcal{W}_{\\nu}$ of $\\mathcal{W}_{n}$, we distinguish two cases depending on whether $C$ is strongly or weakly invariant by $\\mathcal{W}_{\\nu}$; in the latter case, we introduce the \\textsc{Puiseux} index and exponent $\\mathfrak{i}(\\mathcal{W}_{\\nu},C)$ and $\\rho(\\mathcal{W}_{\\nu},C).$ These invariants allow us to define the notion of non-degeneracy of $\\mathcal{W}_{n}$ 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 $C,$ generalizing a result of \\textsc{Mar\\'{\\i}n} and~\\textsc{Pereira}. Moreover, we obtain 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 $K(\\mathcal{W}_{n}^{\\rm{str}}).$ As an application, we establish a flatness result for products of dual webs of reduced convex foliations on $\\mathbb{P}^{2}_{\\mathbb{C}}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.13373",
  "author_total": 1,
  "authors": [
    {
      "name": "Samir Bedrouni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/escgnc81n5stj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.13373",
  "fulltext_url": null,
  "id": "escgnc81n5stj",
  "issns": [],
  "landing_url": "https://hal.science/hal-05621001v3",
  "language": "fr",
  "licence": "CC-BY-NC-4.0",
  "licence_label": "CC BY-NC 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7164502277",
  "pdf_url": "https://hal.science/hal-05621001/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/escgnc81n5stj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:34Z",
      "native_id": "hal-05621001",
      "source": "hal",
      "url": "https://hal.science/hal-05621001v3"
    },
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2606.13373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.13373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on $\\mathbb{P}^{2}_{\\mathbb{C}}$",
  "title_text": "Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on ℙ^2_ℂ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/escgnc81n5stj",
  "venue": "HAL",
  "versions": []
}