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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 $\mathcal{B}(n,k,\mathcal{A}^0)$ 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.