Point-balanced arrangements in the real projective plane and the Hirzebruch property
Abstract
A collection of m ≥ 2 lines through the origin in ℝ2 is balanced if the sum of the orthogonal projections onto these lines equals m/2 times the identity. A line arrangement in ℝℙ2, 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 𝒜 in ℝℙ2 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 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 ℝ2 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.
Related
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BibTeX
@misc{borbon2026pointbalanced,
title = {{Point-balanced arrangements in the real projective plane and the Hirzebruch property}},
author = {Martin de Borbon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09826},
eprint = {2610.09826},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09826},
}
RIS
TY - UNPB
TI - Point-balanced arrangements in the real projective plane and the Hirzebruch property
AU - Martin de Borbon
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09826
UR - https://arxiv.org/abs/2610.09826
AB - A collection of m ≥ 2 lines through the origin in ℝ^2 is balanced if the sum of the orthogonal projections onto these lines equals m/2 times the identity. A line arrangement in 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 𝒜 in 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 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 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.
ER -