Non-very generic arrangements in low dimension

TL;DR

This paper investigates the structure and classification of non-very generic hyperplane arrangements in low-dimensional spaces, providing new examples and a comprehensive characterization of these arrangements.

Contribution

It characterizes, classifies, and supplies examples of non-very generic arrangements in low dimensions, expanding understanding beyond the well-studied very generic case.

Findings

Complete classification of non-very generic arrangements in low dimensions

Examples illustrating the diversity of non-very generic arrangements

New theoretical insights into the combinatorics of these arrangements

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 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.