{
  "abstract": "We study the geometric exceptional set in Manin's conjecture for smooth Fano threefolds. We prove that every adjoint rigid surface with $a$-invariant one on a non-product smooth Fano threefold is normal. This yields a description of the geometric exceptional set in terms of the higher-$a$ locus, anticanonical conics, and del Pezzo fibrations. Zariski density arise only from the latter two contributions. For del Pezzo fibrations, we construct a finite collection of parameter covers which controlling the associated breaking maps. We classify their low-genus components for general smooth Fano threefolds yielding examples with potentially Zariski dense geometric exceptional sets in $22$ of the $105$ deformation families. Extending Batyrev-Tschinkel's example, we construct Fano fourfolds with quartic del Pezzo fibrations and an explicit Fano threefold with two degree-six del Pezzo fibrations, all with Zariski dense geometric exceptional sets. For this threefold, the original form of Manin's conjecture fails on every non-empty open subset for every anticanonical height.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23024",
  "author_total": 1,
  "authors": [
    {
      "name": "Runxuan Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40ca0kv4pdnek.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23024",
  "fulltext_url": null,
  "id": "40ca0kv4pdnek",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23024",
  "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": "W7222977822",
  "pdf_url": "https://arxiv.org/pdf/2610.23024",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40ca0kv4pdnek.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23024",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23024"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The geometric exceptional set for Fano threefolds",
  "title_text": "The geometric exceptional set for Fano threefolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/40ca0kv4pdnek",
  "venue": "arXiv",
  "versions": []
}