Seshadri constants of singular loci of line arrangements with at most twelve lines
Abstract
Let ℒ be an arrangement of d lines in the complex projective plane and let Z=Sing(ℒ). A question of Pokora asks whether the multipoint Seshadri constant of 𝒪ℙ2(1) at Z is always computed by a line of the arrangement containing the largest number of points of Z. We prove that the answer is affirmative for every arrangement of at most twelve lines. The proof uses a Bézout-type covering criterion, elementary restrictions on the intersection lattice, and a detailed analysis of the two extremal combinatorial types that can occur for twelve lines. One type is the Hesse arrangement; it is handled using a net of quartics through its twelve double points. The other type would yield a uniquely determined incidence structure on thirteen lines. We show that this incidence structure is not realizable over any field.
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BibTeX
@misc{dumnicki2026seshadri,
title = {{Seshadri constants of singular loci of line arrangements with at most twelve lines}},
author = {Marcin Dumnicki and Łucja Farnik and Krishna Hanumanthu and Brian Harbourne and Piotr Pokora and Joaquim Roé and Halszka Tutaj-Gasińska},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03797},
eprint = {2611.03797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03797},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Seshadri constants of singular loci of line arrangements with at most twelve lines
AU - Marcin Dumnicki
AU - Łucja Farnik
AU - Krishna Hanumanthu
AU - Brian Harbourne
AU - Piotr Pokora
AU - Joaquim Roé
AU - Halszka Tutaj-Gasińska
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03797
UR - https://arxiv.org/abs/2611.03797
AB - Let ℒ be an arrangement of d lines in the complex projective plane and let Z=Sing(ℒ). A question of Pokora asks whether the multipoint Seshadri constant of 𝒪_ℙ^2(1) at Z is always computed by a line of the arrangement containing the largest number of points of Z. We prove that the answer is affirmative for every arrangement of at most twelve lines. The proof uses a Bézout-type covering criterion, elementary restrictions on the intersection lattice, and a detailed analysis of the two extremal combinatorial types that can occur for twelve lines. One type is the Hesse arrangement; it is handled using a net of quartics through its twelve double points. The other type would yield a uniquely determined incidence structure on thirteen lines. We show that this incidence structure is not realizable over any field.
ER -