A classification of combinatorial types of discriminantal arrangements

TL;DR

This paper classifies combinatorial types of discriminantal arrangements, providing explicit conditions for non very generic arrangements, advancing understanding of their geometric and combinatorial properties.

Contribution

It offers a classification of r-sets and a more explicit criterion for non very generic arrangements, improving on previous conditions.

Findings

Classification of r-sets for discriminantal arrangements

Explicit and tractable conditions for non very genericity

Enhanced understanding of the combinatorial structure of arrangements

Abstract

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