{
  "abstract": "In this note we show that Terao's freeness conjecture holds for arrangements of at most nine planes in the complex projective $3$-space. Moreover, we show that every essential and irreducible free arrangement of at most nine planes in $\\mathbb{P}^{3}_{\\mathbb{C}}$ is divisionally free, and we list all intersection lattices of such arrangements. Our approach is based on Ziegler's restriction theorem and a simple count involving the numbers of multiple lines. As a by-product, we show that Cynk--Szemberg octic arrangements are never free. Finally, motivated by a recent work of Dimca and Pokora, we present examples of weak Ziegler pairs of plane arrangements, including pairs of octic arrangements leading to rigid double octic Calabi--Yau threefolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40028",
  "author_total": 1,
  "authors": [
    {
      "name": "Bartosz Jarosławski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/be8pyzrwgp4zd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40028",
  "fulltext_url": null,
  "id": "be8pyzrwgp4zd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40028",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40028",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/be8pyzrwgp4zd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40028",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40028"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Terao's conjecture for at most nine planes",
  "title_text": "On Terao's conjecture for at most nine planes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/be8pyzrwgp4zd",
  "venue": "arXiv",
  "versions": []
}