  Discriminantal arrangements are hyperplane arrangements, which are
generalized braid ones. They are constructed from given hyperplane
arrangements, but their combinatorics are not invariant under combinatorial
equivalence. However, it is known that the combinatorics of the discriminantal
arrangement are constant on a Zariski open set of the space of hyperplane
arrangements. In the present paper, we introduce non-very generic varieties in
the space of hyperplane arrangements to classify discriminantal arrangements
and show that the Zariski open set is the complement of non-very generic
varieties. We study their basic properties and construction and provide
examples, including infinite families of non-very generic varieties. In
particular, the construction we call degeneration is a powerful tool for
constructing non-very generic varieties. As an application, we provide lists of
non-very generic varieties for spaces of small line arrangements.
