  We study the compactification of the locus parametrizing lines with a fixed
intersection to a given line, inside the moduli space of line arrangements in
the projective plane constructed for weight one by Hacking-Keel-Tevelev and
Alexeev for general weights. We show that this space is smooth, with normal
crossing boundary, and that it has a morphism to the moduli space of marked
rational curves which can be understood as a natural continuation of the blow
up construction of Kapranov. In addition, we prove that it is isomorphic to a
closed subvariety inside a non-reductive Chow quotient. The parametrized
objects are surfaces with broken lines, whose dual graphs are rooted trees with
possibly repeated markings.
