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PreprintCC BY 4.0arXiv

A generalized Terao Conjecture for line arrangements

  • Alexandru Dimca
  • Piotr Pokora
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Abstract

For a line arrangement π’œ : f=0 in β„™2, let Ξ½(π’œ) be the maximal dimension of a graded piece of the Jacobian module of f. We study the conjecture that Ξ½(π’œ) depends only on the intersection lattice of π’œ. It is known that π’œ is free if and only if Ξ½(π’œ)=0, and hence this conjecture is a strengthening of Terao's conjecture for line arrangements. We show that the conjecture holds for arrangements of at most $13$ lines, except possibly for arrangements of exactly $13$ lines whose intersection points have maximal multiplicity $5$.

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Cite

BibTeXRIS
BibTeX
@misc{dimca2026generalized,
  title = {{A generalized Terao Conjecture for line arrangements}},
  author = {Alexandru Dimca and Piotr Pokora},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.33584},
  eprint = {2609.33584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33584},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A generalized Terao Conjecture for line arrangements
AU  - Alexandru Dimca
AU  - Piotr Pokora
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.33584
UR  - https://arxiv.org/abs/2609.33584
AB  - For a line arrangement π’œ : f=0 in β„™^2, let Ξ½(π’œ) be the maximal dimension of a graded piece of the Jacobian module of f. We study the conjecture that Ξ½(π’œ) depends only on the intersection lattice of π’œ. It is known that π’œ is free if and only if Ξ½(π’œ)=0, and hence this conjecture is a strengthening of Terao's conjecture for line arrangements. We show that the conjecture holds for arrangements of at most $13$ lines, except possibly for arrangements of exactly $13$ lines whose intersection points have maximal multiplicity $5$.
ER  -