{
  "abstract": "Let $\\mathcal{A}$ be an arrangement of $d$ lines in $\\mathbb{P}^2_\\mathbb{C}$ and let $\\mathcal{T}\\langle\\mathcal{A}\\rangle$ be its bundle of logarithmic vector fields. On a general line, $\\mathcal{T}\\langle\\mathcal{A}\\rangle$ splits as $\\mathcal{O}(-k)\\oplus\\mathcal{O}(k+1-d)$ with $k=\\min\\{\\mathrm{mdr}(\\mathcal{A}),\\lfloor\\frac{d-1}{2}\\rfloor\\}$, where $\\mathrm{mdr}$ is the minimal degree of a Jacobian syzygy. We prove that $k$ is determined by the intersection lattice whenever $d\\le16$, answering in this range a question of Cook, Harbourne, Migliore and Nagel. Since freeness can be read off from $k$ and $c_2(\\mathcal{T}\\langle\\mathcal{A}\\rangle)$, this strengthens Terao's freeness conjecture in this range; in particular we obtain a proof of Terao's conjecture for $14$ lines that uses it only for $13$ lines. For any number $d$ of lines, it follows that $\\mathrm{mdr}$ is determined by the lattice whenever $\\mathrm{mdr}\\le6$ and $2\\mathrm{mdr}\\le d-3$, and that Terao's conjecture holds for every free arrangement whose smaller exponent is at most $7$. The proof reduces the question to critical arrangements: $2s+3$ lines, each containing at most $s+1$ intersection points, with $\\mathrm{mdr}=s$. For these, the inflection polynomial of the minimal derivation factors as $Q_\\mathcal{A} R$ with $R\\in H^0(\\mathcal{I}_Z^2(s-3))$, where $Z$ is the zero scheme of the derivation, of length $ν=c_2(\\mathcal{T}\\langle\\mathcal{A}\\rangle)-s(s+2)$. For $s\\le5$ this forces $ν\\le1$, and Chern class arguments then reduce everything to Terao's conjecture for $13$ lines. For $s=6$ we show $ν\\le3$, using a Riccati equation along multiple components of $R$, Ziegler's multirestriction, and local computations verified by computer algebra; an exhaustive enumeration of the 26,265,318 candidate lattices with $ν\\in\\{2,3\\}$ that satisfy Bojanowski's inequality completes the proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31556",
  "author_total": 2,
  "authors": [
    {
      "name": "Cristian Anghel"
    },
    {
      "name": "Rares Stan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5n3d3dqf4tcxf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31556",
  "fulltext_url": null,
  "id": "5n3d3dqf4tcxf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31556",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223274334",
  "pdf_url": "https://arxiv.org/pdf/2610.31556",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5n3d3dqf4tcxf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31556",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31556"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is combinatorial",
  "title_text": "The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is combinatorial",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/5n3d3dqf4tcxf",
  "venue": "arXiv",
  "versions": []
}