A classification of combinatorial types of discriminantal arrangements

Abstract

Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the $\text{discriminantal arrangements}$. Athanasiadis proved a conjecture by Bayer and Brandt providing a full description of the combinatorics of discriminantal arrangements in the case of $\text{very generic}$ arrangements. Libgober and Settepanella described a sufficient geometric condition for given arrangements to be $\text{non very generic}$ 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.

18 pages, 6 figures

Cited by in corpus (2)

• Degeneration in discriminantal arrangements
• Arithmetic non-very generic arrangements