  This paper aims to undertake an exploration of the behavior of the moduli
space of line arrangements while establishing its combinatorial interplay with
the incidence structure of the arrangement. In the first part, we investigate
combinatorial classes of arrangements whose moduli space is connected. We unify
the classes of simple and inductively connected arrangements appearing in the
literature. Then, we introduce the notion of arrangements with a rigid pencil
form. It ensures the connectedness of the moduli space and is less restrictive
that the class of $C_3$ arrangements of simple type. In the last part, we
obtain a combinatorial upper bound on the number of connected components of the
moduli space. Then, we exhibit examples with an arbitrarily large number of
connected components for which this upper bound is sharp.
