  The moduli space of hyperplanes in projective space has a family of geometric
and modular compactifications that parametrize stable hyperplane arrangements
with respect to a weight vector. Among these, there is a toric compactification
that generalizes the Losev-Manin moduli space of points on the line. We study
the first natural wall crossing that modifies this compactification into a
non-toric one by varying the weights. As an application of our work, we show
that any $\mathbb{Q}$-factorialization of the blow up at the identity of the
torus of the generalized Losev-Manin space is not a Mori dream space for a
sufficiently high number of hyperplanes. Additionally, for lines in the plane,
we provide a precise description of the wall crossing.
