A generic slice of the moduli space of line arrangements

Abstract

We study the compactification of the locus parametrizing lines having a fixed intersection with a given line, inside the moduli space of line arrangements in the projective plane constructed for weight one by Hacking, Keel and Tevelev and for general weights by Alexeev. 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 our space is isomorphic to a closed subvariety inside a nonreductive Chow quotient.