A linear condition for non-very generic discriminantal arrangements

TL;DR

This paper introduces a linear condition called weak linear independency that helps identify hyperplane configurations in discriminantal arrangements that are not in the generic Zariski open set, with three illustrative examples.

Contribution

It defines a new linear independence condition enabling the construction of non-generic hyperplane configurations in discriminantal arrangements.

Findings

The weak linear independency condition distinguishes non-generic configurations.

Three explicit examples of such configurations are provided.

The approach builds on recent results to identify non-generic arrangements.

Abstract

The discriminantal arrangement is the space of configurations of $n$ hyperplanes in generic position in a $k$ dimensional space (see \cite{MS}). Differently from the case $k=1$ in which it corresponds to the well known braid arrangement, the discriminantal arrangement in the case $k>1$ has a combinatorics which depends from the choice of the original $n$ hyperplanes. It is known that this combinatorics is constant in an open Zariski set $\mathcal{Z}$, but to assess wether or not $n$ fixed hyperplanes in generic position belongs to $\mathcal{Z}$ proved to be a nontrivial problem. Even to simply provide examples of configurations not in $\mathcal{Z}$ is still a difficult task. In this paper, moving from a recent result in \cite{SSc}, we define a $\textit{weak linear independency}$ condition among sets of vectors which, if imposed, allows to build configurations of hyperplanes not in $\mathcal{Z}$. We provide $3$ examples.