On Very Generic Discriminantal Arrangements

TL;DR

This paper investigates the properties of very generic discriminantal arrangements, proving limitations on certain six-line configurations and exploring the relationship between hyperplane arrangements and their associated convex cones.

Contribution

It provides new insights into the structure of discriminantal arrangements, including proofs of non-existence results and a detailed combinatorial and geometric analysis of their intersection lattices.

Findings

Six-line arrangements with three perpendicular pairs do not produce very generic discriminantal arrangements.

The number of simplex cells in a hyperplane arrangement may not match the number of boundary hyperplanes of the associated convex cone.

The intersection lattice of very generic discriminantal arrangements can be described combinatorially as sets of hyperplane concurrencies.

Abstract

In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement in the sense of C. A. Athanasiadis \cite{MR1720104}. We give two proofs of the first result. The second result is as follows. The codimension-one boundary faces of (a region) a convex cone of a very generic discriminantal arrangement has not been characterized and is not known even though the intersection lattice of a very generic discriminantal arrangement is known. So secondly, we show that the number of simplex cells of the very generic hyperplane arrangement $\mathcal{H}^m_n=\{H_i:\underset{j=1}{\overset{m}{\sum}}a_{ij}x_j=c_i,1\leq i\leq n\}$ may not be not precisely equal to the number of codimension-one boundary hyperplanes of $\mathbb{R}^n$ of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. That is, for $1\leq i_1<i_2<\ldots<i_m<i_{m+1}\leq n$, if $\Delta^m H_{i_1}H_{i_2}\ldots H_{i_m}H_{i_{m+1}}$ is a simplex cell of the hyperplane arrangement $\mathcal{H}^m_n$ then it need not give rise to a codimension-one boundary hyperplane of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. We finally mention an interesting open-ended remark before the appendix section. In the appendix section we give a self contained exposition and describe combinatorially the intersection lattice of a (Zariski open and dense) class of "very generic or sufficiently general" discriminantal arrangements. As a consequence, we give a geometric description of the lattice elements as sets of concurrencies of the hyperplane arrangements which give the same "very generic or sufficiently general" discriminantal arrangement.