On the non-very generic intersections in discriminantal arrangements

TL;DR

This paper investigates the combinatorial structure of discriminantal arrangements, specifically characterizing conditions under which the intersection lattice's complexity decreases in non-very generic cases.

Contribution

It extends previous work by providing sufficient conditions for the reduction of intersection cardinalities in discriminantal arrangements beyond rank 2.

Findings

Identifies conditions leading to fewer high-rank intersections

Builds on prior necessary and sufficient conditions for rank 2

Advances understanding of non-very generic arrangements

Abstract

In 1985 Crapo introduced in \cite{Crapo} a new mathematical object that he called $\textit{geometry of circuits}$. Four years later, in 1989, Manin and Schechtman defined in \cite{MS} the same object and called it $\textit{discriminantal arrangement}$, the name by which it is known now a days. Those discriminantal arrangements $\mathcal{B}(n,k,\mathcal{A}^0)$ are builded from an arrangement $\mathcal{A}^0$ of $n$ hyperplanes in general position in a $k$-dimensional space and their combinatorics depends on the arrangement $\mathcal{A}^0$. On this basis, in 1997 Bayer and Brandt (see \cite{BB}) distinguished two different type of arrangements $\mathcal{A}^0$ calling $\textit{very generic}$ the ones for which the intersection lattice of $\mathcal{B}(n,k,\mathcal{A}^0)$ has maximum cardinality and $\textit{non-very generic}$ the others. Results on the combinatorics of $\mathcal{B}(n,k,\mathcal{A}^0)$ in the very generic case already appear in Crapo \cite{Crapo} and in 1997 in Athanasiadis \cite{Atha} while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper \cite{LS} they provided a necessary and sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank 2 intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank r, $r \geq 2$, intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ decreases.