On Very Generic Discriminantal Arrangements

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.