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A classification of combinatorial types of discriminantal arrangements

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Abstract

Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the discriminantal arrangements. Athanasiadis proved a conjecture by Bayer and Brandt providing a full description of the combinatorics of discriminantal arrangements in the case of very generic arrangements. Libgober and Settepanella described a sufficient geometric condition for given arrangements to be non-very generic in terms of the notion of dependency for a certain arrangement. Settepanella and the author generalized the notion of dependency introducing r-sets and \(K_{\mathbb {T}}\)-vector sets, and provided a sufficient condition for non-very genericity but still not convenient to verify by hand. In this paper, we give a classification of the r-sets, and a more explicit and tractable condition for non-very genericity.

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Acknowledgements

The author would like to thank Anatoly Libgober for useful discussions and for pointing out Corollaries 1 and 2, and Masahiko Yoshinaga for useful discussions. The author also would like to thank referees for many useful comments and suggestions. The author was supported by JSPS Research Fellowship for Young Scientists Grant Number 20J10012.

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  1. Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido, 060-0812, Japan

    So Yamagata

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Yamagata, S. A classification of combinatorial types of discriminantal arrangements. J Algebr Comb 57, 1007–1032 (2023). https://doi.org/10.1007/s10801-022-01207-1

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