The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is…

Abstract

Let 𝒜 be an arrangement of d lines in ℙ²C and let 𝒯⟨𝒜⟩ be its bundle of logarithmic vector fields. On a general line, 𝒯⟨𝒜⟩ splits as 𝒪(−k)⊕𝒪(k+1−d) with k=min{mdr(𝒜),⌊(d−1)/2⌋}, where mdr is the minimal degree of a Jacobian syzygy. We prove that k is determined by the intersection lattice whenever d≤16, answering in this range a question of Cook, Harbourne, Migliore and Nagel. Since freeness can be read off from k and c₂(𝒯⟨𝒜⟩), 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 mdr is determined by the lattice whenever mdr≤6 and 2mdr≤ 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 mdr=s. For these, the inflection polynomial of the minimal derivation factors as QA R with R∈ H⁰(ℐ_Z²(s−3)), where Z is the zero scheme of the derivation, of length ν=c₂(𝒯⟨𝒜⟩)−s(s+2). For s≤5 this forces ν≤1, and Chern class arguments then reduce everything to Terao's conjecture for $13$ lines. For s=6 we show ν≤3, 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∈{2,3} that satisfy Bojanowski's inequality completes the proof.

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BibTeX RIS

BibTeX

@misc{anghel2026generic,
  title = {{The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is combinatorial}},
  author = {Cristian Anghel and Rares Stan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31556},
  eprint = {2610.31556},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31556},
}

RIS

TY  - UNPB
TI  - The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is combinatorial
AU  - Cristian Anghel
AU  - Rares Stan
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31556
UR  - https://arxiv.org/abs/2610.31556
AB  - Let 𝒜 be an arrangement of d lines in ℙ^2_C and let 𝒯⟨𝒜⟩ be its bundle of logarithmic vector fields. On a general line, 𝒯⟨𝒜⟩ splits as 𝒪(−k)⊕𝒪(k+1−d) with k=min{mdr(𝒜),⌊(d−1)/2⌋}, where mdr is the minimal degree of a Jacobian syzygy. We prove that k is determined by the intersection lattice whenever d≤16, answering in this range a question of Cook, Harbourne, Migliore and Nagel. Since freeness can be read off from k and c_2(𝒯⟨𝒜⟩), 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 mdr is determined by the lattice whenever mdr≤6 and 2mdr≤ 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 mdr=s. For these, the inflection polynomial of the minimal derivation factors as Q_A R with R∈ H^0(ℐ_Z^2(s−3)), where Z is the zero scheme of the derivation, of length ν=c_2(𝒯⟨𝒜⟩)−s(s+2). For s≤5 this forces ν≤1, and Chern class arguments then reduce everything to Terao's conjecture for $13$ lines. For s=6 we show ν≤3, 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∈{2,3} that satisfy Bojanowski's inequality completes the proof.
ER  -

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