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Free paths of arrangements of hyperplanes

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arxiv 2306.11310 v1 pith:OYZ7ZYSQ submitted 2023-06-20 math.CO math.AC

Takuro Abe , Toru Yamaguchi This is my paper
classification math.COmath.AC
keywords freearrangementsmathcalarrangementhyperplanesthemthenconnect
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abstract

We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A} \setminus \{H,L\}$ are free, then at least one of two among them is free. When $\mathcal{A}$ is in the three dimensional arrangement, we show a stronger statement.

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  1. Extendability of the $B_2$-arrangement

    math.CO 2025-06 conditional novelty 7.0 of 10

    For the B2 Coxeter arrangement with multiplicities (2,k,1,k), k>=4, no free extension exists, and this obstruction transfers to Bn and to explicit B3 multiplicities.

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