{
  "abstract": "Let $\\F$ be a codimension-one holomorphic foliation of degree $d$ on $\\PP^n$, $n\\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\\cap\\Sing(\\F))_{\\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, \\[ Ω=Q\\,dt-\\frac{t}{d+1}\\,dQ , \\] where $Q$ is homogeneous of degree $d+1$. Thus $Q/t^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\\subset\\PP^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14807",
  "author_total": 2,
  "authors": [
    {
      "name": "Víctor León"
    },
    {
      "name": "Bruno Scárdua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dt9ds89qa1xn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14807",
  "fulltext_url": null,
  "id": "5dt9ds89qa1xn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14807",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213291013",
  "pdf_url": "https://arxiv.org/pdf/2609.14807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dt9ds89qa1xn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An arithmetic integrability result for codimension-one foliations on complex projective spaces",
  "title_text": "An arithmetic integrability result for codimension-one foliations on complex projective spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/5dt9ds89qa1xn",
  "venue": "arXiv",
  "versions": []
}