PreprintCC BY 4.0arXiv
On Terao's conjecture for at most nine planes
Abstract
In this note we show that Terao's freeness conjecture holds for arrangements of at most nine planes in the complex projective $3$-space. Moreover, we show that every essential and irreducible free arrangement of at most nine planes in ℙ3ℂ is divisionally free, and we list all intersection lattices of such arrangements. Our approach is based on Ziegler's restriction theorem and a simple count involving the numbers of multiple lines. As a by-product, we show that Cynk--Szemberg octic arrangements are never free. Finally, motivated by a recent work of Dimca and Pokora, we present examples of weak Ziegler pairs of plane arrangements, including pairs of octic arrangements leading to rigid double octic Calabi--Yau threefolds.
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BibTeX
@misc{jaroslawski2026teraos,
title = {{On Terao's conjecture for at most nine planes}},
author = {Bartosz Jarosławski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40028},
eprint = {2610.40028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40028},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Terao's conjecture for at most nine planes
AU - Bartosz Jarosławski
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40028
UR - https://arxiv.org/abs/2610.40028
AB - In this note we show that Terao's freeness conjecture holds for arrangements of at most nine planes in the complex projective $3$-space. Moreover, we show that every essential and irreducible free arrangement of at most nine planes in ℙ^3_ℂ is divisionally free, and we list all intersection lattices of such arrangements. Our approach is based on Ziegler's restriction theorem and a simple count involving the numbers of multiple lines. As a by-product, we show that Cynk--Szemberg octic arrangements are never free. Finally, motivated by a recent work of Dimca and Pokora, we present examples of weak Ziegler pairs of plane arrangements, including pairs of octic arrangements leading to rigid double octic Calabi--Yau threefolds.
ER -