  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.
Subsequently in 1997 Bayer and Brandt ( see \cite{BB} ) distinguished two
different type of those arrangements calling \textit{very generic} the ones
having intersection lattice of maximum cardinality and \textit{non very
generic} the others. Results on the combinatorics of very generic arrangements
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 studied the
combinatorics of non very generic case in rank $2$ . In this paper we further
develop their result providing a sufficient condition for the discriminantal
arrangement to be non very generic which holds in rank $r \geq 2$.
