{
  "abstract": "The Cassels--Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels--Swinnerton-Dyer conjecture for cubic surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15930",
  "author_total": 2,
  "authors": [
    {
      "name": "Valery Alexeev"
    },
    {
      "name": "Stefan Schreieder"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1zpz6hc9jxyxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15930",
  "fulltext_url": null,
  "id": "1zpz6hc9jxyxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15930",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213417630",
  "pdf_url": "https://arxiv.org/pdf/2609.15930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1zpz6hc9jxyxa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.15930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces",
  "title_text": "Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1zpz6hc9jxyxa",
  "venue": "arXiv",
  "versions": []
}