On the non-very generic intersections in discriminantal arrangements

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 $\B(n,k,\A^0)$ are builded from an arrangement $\A^0$ of $n$ hyperplanes in general position in a $k$-dimensional space and their combinatorics depends on the arrangement $\A^0$. On this basis, in 1997 Bayer and Brandt (see \cite{BB}) distinguished two different type of arrangements $\A^0$ calling \textit{very generic} the ones for which the intersection lattice of $\B(n,k,\A^0)$ has maximum cardinality and \textit{non-very generic} the others. Results on the combinatorics of $\B(n,k,\A^0)$ 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 $\A^0$ for which the cardinality of rank 2 intersections in $\B(n,k,\A^0)$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $\A^0$ for which the cardinality of rank r, $r \geq 2$, intersections in $\B(n,k,\A^0)$ decreases.