{
  "abstract": "A collection of $m \\geq 2$ lines through the origin in $\\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\\mathcal{A}$ in $\\mathbb{RP}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf--Ness convexity argument, we show that an irreducible arrangement $\\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in $\\mathbb{R}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov's classification of real Hirzebruch arrangements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09826",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin de Borbon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1t1cfa1d15x2s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09826",
  "fulltext_url": null,
  "id": "1t1cfa1d15x2s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09826",
  "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.CO"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220966537",
  "pdf_url": "https://arxiv.org/pdf/2610.09826",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1t1cfa1d15x2s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09826",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09826"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Point-balanced arrangements in the real projective plane and the Hirzebruch property",
  "title_text": "Point-balanced arrangements in the real projective plane and the Hirzebruch property",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/1t1cfa1d15x2s",
  "venue": "arXiv",
  "versions": []
}