{
  "abstract": "A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, and successive work of Bokowski, Grünbaum, Schewe, Pilaud and Cuntz settled every case but $n=23$. We settle this last case by bounding the symmetry of such configurations and classifying the most symmetric ones exactly. We prove that for every prime $n\\equiv3\\pmod4$ the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, explaining why symmetric constructions failed at $n=23$, and the same argument bounds the symmetry of $(n_5)$ configurations for primes $n\\not\\equiv1\\pmod5$. Reducing the Klein four-group case to 4021 bilinear polynomial systems and solving all of them exactly with Gröbner bases, we show that exactly two $(23_4)$ configurations reach this bound up to projective equivalence and that there are no others even over $\\mathbb{C}$. Both are self-polar; the first has integer coordinates and the second is defined over $\\mathbb{Q}(\\sqrt{17})$ but not over $\\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\\ge18$ and $n\\ne19$. The same classification recovers Cuntz's $(22_4)$ and $(26_4)$ configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$. Among self-polar $(23_4)$ configurations with a single involution we find $50$ more, defined over number fields of degree at most $7$. Four of them deform in one-parameter families, and computing three of these exactly, one birational to the modular curve $X_1(11)$, we prove that there are infinitely many geometric $(23_4)$ configurations, among them infinitely many with rational coordinates that are self-polar and have no projective symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14099",
  "author_total": 1,
  "authors": [
    {
      "name": "Wenhao Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vjc63j65xgpv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14099",
  "fulltext_url": null,
  "id": "3vjc63j65xgpv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14099",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221751302",
  "pdf_url": "https://arxiv.org/pdf/2610.14099",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vjc63j65xgpv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.14099",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14099"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Symmetric, rational and infinitely many $(23_4)$ configurations, and the end of the existence problem for geometric $(n_4)$ configurations",
  "title_text": "Symmetric, rational and infinitely many (23_4) configurations, and the end of the existence problem for geometric (n_4) configurations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3vjc63j65xgpv",
  "venue": "arXiv",
  "versions": []
}