{
  "abstract": "String Theory provides powerful tools for constructing and studying five-dimensional Quantum Field Theories, notably through geometric engineering and five-brane webs. Understanding how these two descriptions are related in the presence of 7-branes requires incorporating the s-rule into the geometry. To address this question, we focus on basic three-legged five-brane webs ending on 7-branes. Building on the local M-theory description of five-branes ending on 7-branes, we consider triangular toric diagrams with a single non-primitive edge. We show that the corresponding local complex-structure deformation is well defined on the overlaps with the neighboring toric patches, while the corresponding gluing map remains well behaved as a function of the holomorphic coordinates throughout the neighboring patches precisely when the s-rule is satisfied. At saturation, the geometry is a T-cone and the 7-brane deformation becomes its Q-Gorenstein smoothing. Our results thus identify the s-rule with the obstruction to globalizing the local 7-brane deformation and provide a direct geometric link between T-cones and five-branes ending on 7-branes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07154",
  "author_total": 2,
  "authors": [
    {
      "name": "Sebastian Franco"
    },
    {
      "name": "Diego Rodriguez-Gomez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8mt07yej8f488.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07154",
  "fulltext_url": null,
  "id": "8mt07yej8f488",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07154",
  "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",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7220907060",
  "pdf_url": "https://arxiv.org/pdf/2610.07154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8mt07yej8f488.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.07154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07154"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "The s-Rule from Geometry",
  "title_text": "The s-Rule from Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:22Z",
  "url": "https://research.draconitic.com/paper/8mt07yej8f488",
  "venue": "arXiv",
  "versions": []
}