{
  "abstract": "Exact calculations for smooth toric Fano varieties in dimensions at most seven suggest a sharp dimension-dependent lower bound for the positive difference between the rational and integral Gongyo indices. We compute both indices for the smooth toric Fano models obtained from blow-ups of projective space at torus-invariant points by the anticanonical minimal model program. For pseudo-symmetric smooth toric Fano varieties, we prove the proposed lower bound and determine all equality cases. We also apply a dimension-raising construction to obtain an explicit infinite family of smooth toric Fano varieties with positive gap from odd-dimensional weak Fano varieties with zero gap. For the standard models considered, we determine the smallest dimension increase needed to obtain a Fano variety by iterating this construction and compute both indices of the minimal lifts. Finally, every positive rational number occurs as a Gongyo-index gap, even within the pseudo-symmetric class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22997",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroshi Sato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4nxteyqe5j5q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22997",
  "fulltext_url": null,
  "id": "d4nxteyqe5j5q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22997",
  "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": "W7214028354",
  "pdf_url": "https://arxiv.org/pdf/2609.22997",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d4nxteyqe5j5q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22997",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22997"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the gap between the integral and rational Gongyo indices of toric Fano varieties",
  "title_text": "On the gap between the integral and rational Gongyo indices of toric Fano varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/d4nxteyqe5j5q",
  "venue": "arXiv",
  "versions": []
}