{
  "abstract": "For a smooth cubic surface $S/\\mathbb{Q}$, the absolute Galois group acts on the $27$ lines of $S_{\\overline{\\mathbb{Q}}}$, giving a subgroup \\[ G_S\\subseteq W(E_6). \\] We study the distribution of $G_S$ when the twenty coefficients of a defining cubic form are chosen independently from a large interval $[-H,H]$. We prove that, for every $\\varepsilon>0$, the number of smooth integral cubic forms of height at most $H$ for which \\[ G_S\\neq W(E_6) \\] is \\[ O_\\varepsilon\\!\\left(H^{19+1/21+\\varepsilon}\\right). \\] The principal obstruction is the unique subgroup $W(E_6)^+$ of index $2$. By a theorem of Saito, membership in this subgroup is equivalent to the signed discriminant of the cubic surface being a square. A recent square-value estimate of Bonolis, Pierce, and Woo then gives the exponent $19+1/21$. For each of the other five conjugacy classes of maximal subgroups of $W(E_6)$, recent results on rational points in projective thin sets give the sharper bound $O(H^{19})$. We also prove the lower bound \\[ N_{\\mathrm{exc}}(H)\\gg H^{16} \\] by counting cubic surfaces containing a fixed rational line, and exhibit an $H^{14}$-sized exceptional family with no rational line, arising from rational Eckardt points. These examples lead us to conjecture that \\[ N_{\\mathrm{exc}}(H)=H^{16+o(1)}. \\] Finally, we investigate whether Bhargava's proof of van der Waerden's conjecture for $S_n$ can be adapted to the index-$2$ $E_6$-problem. A double-discriminant argument gives the desired $H^{19+\\varepsilon}$ bound whenever the product of the strongly square-dividing primes is sufficiently large. The remaining difficulty comes from weak square divisibility.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37945",
  "author_total": 1,
  "authors": [
    {
      "name": "Evan O'Dorney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pq2c3stfvefg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37945",
  "fulltext_url": null,
  "id": "6pq2c3stfvefg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37945",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37945",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pq2c3stfvefg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37945",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37945"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Van der Waerden's Problem for Cubic Surfaces",
  "title_text": "Van der Waerden's Problem for Cubic Surfaces",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6pq2c3stfvefg",
  "venue": "arXiv",
  "versions": []
}