A linear condition for non-very generic discriminantal arrangements

The discriminantal arrangement is the space of configurations of n hyperplanes in generic position in a k-dimensional space. Unlike the case k = 1, where it coincides with the wellknown braid arrangement, the discriminantal arrangement for k > 1 has a combinatorial structure that depends on the choice of the original n hyperplanes. It is known that this combinatorics remains constant on a Zariski-open set Z, but determining whether a given configuration of n generic hyperplanes belongs to Z has proved to be a nontrivial problem. Even providing explicit examples of configurations not contained in Z remains a challenging task. In this paper, building on a recent result by the present authors, we introduce the notion of weak linear independence among sets of vectors, which, when imposed, allows us to construct configurations of hyperplanes not lying in Z. We also present three explicit examples illustrating this construction. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).