{
  "abstract": "In this paper we study irreducible simplicial arrangements of projective planes in $\\mathbb{P}^{3}(\\mathbb{R})$ from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Grünbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types $A_4$ and $B_4$ admit a special vertex, whereas those of types $D_4$ and $F_4$ do not. A simplicial deletion chain inside $B_4$ supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Grünbaum-Shephard defects for these arrangements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28254",
  "author_total": 2,
  "authors": [
    {
      "name": "Marek Janasz"
    },
    {
      "name": "Piotr Pokora"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7f1jwhwe2s1m7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28254",
  "fulltext_url": null,
  "id": "7f1jwhwe2s1m7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28254",
  "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",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204768918",
  "pdf_url": "https://arxiv.org/pdf/2608.28254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7f1jwhwe2s1m7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Simplicial arrangements in real projective three-space revisited",
  "title_text": "Simplicial arrangements in real projective three-space revisited",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/7f1jwhwe2s1m7",
  "venue": "arXiv",
  "versions": []
}