{
  "abstract": "For an almost simple complex algebraic group $G$ of semisimple rank $m$, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most $m+2$ members does. We determine the resulting Helly number $(h(G))$ for almost simple groups of types $A$, $B$, $C$, $G_2$, $F_4$, $E_7$, and $E_8$, and obtain bounds differing by one in types $D$ and $E_6$. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal $G$-bundle splits equivariantly. In particular, for $d \\ge 2$, this splitting property holds on $\\mathbb{P}^d$ if and only if $d \\ge h(G)$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11340",
  "author_total": 1,
  "authors": [
    {
      "name": "Boris Tsvelikhovskiy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jazgb1fdzp2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11340",
  "fulltext_url": null,
  "id": "7jazgb1fdzp2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11340",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221102763",
  "pdf_url": "https://arxiv.org/pdf/2610.11340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jazgb1fdzp2p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles",
  "title_text": "Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/7jazgb1fdzp2p",
  "venue": "arXiv",
  "versions": []
}