Tohoku Mathematical Journal
2025
September
SECOND SERIES VOL. 77, NO. 3

Tohoku Math. J.
77 (2025), 357-373

Title NON-VERY GENERIC ARRANGEMENTS IN LOW DIMENSION

Author Takuya Saito and Simona Settepanella

(Received March 13, 2023, revised September 21, 2023)
Abstract. The discriminantal arrangement $\mathcal{B}(n,k,\mathcal{A})$ has been introduced by Manin and Schectman in 1989 and it consists of all non-generic translates of a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. It is known that its combinatorics depends on the original arrangement $\mathcal{A}$ which, following Bayer and Brandt [3], is called very generic if the intersection lattice of the induced discriminantal arrangement has maximum cardinality, non-very generic otherwise. While a complete description of the combinatorics of $\mathcal{B}(n,k,\mathcal{A})$ when $\mathcal{A}$ is very generic is known (see [2]), very few is known in the non-very generic case. Even to provide examples of non very generic arrangements proved to be a non-trivial task (see [17]). In this paper, we characterize, classify and provide examples of non-very generic arrangements in low dimension.

Mathematics Subject Classification. Primary 52C35; Secondary 05B35, 14M15.
Key words and phrases. Hyperplane arrangements, intersection lattice, discriminantal arrangements, permutation groups.

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