  Manin and Schechtman introduced a family of arrangements of hyperplanes
generalizing classical braid arrangements, which they called the
$\textit{discriminantal arrangements}$. Athanasiadis proved a conjecture by
Bayer and Brandt providing a full description of the combinatorics of
discriminantal arrangements in the case of $\textit{very generic}$
arrangements. Libgober and Settepanella described a sufficient geometric
condition for given arrangements to be $\textit{non very generic}$ in terms of
the notion of dependency for a certain arrangement. Settepanella and the author
generalized the notion of dependency introducing $r$-sets and
$K_\mathbb{T}$-vector sets, and provided a sufficient condition for non very
genericity but still not convenient to verify by hand. In this paper we give a
classification of the $r$-sets, and a more explicit and tractable condition for
non very genericity.
