A Generic Slice of the Moduli Space of Line Arrangements

TL;DR

This paper investigates the structure of a compactified moduli space of line arrangements with fixed intersections, demonstrating its smoothness, boundary properties, and relationships to other moduli spaces, thus advancing understanding of geometric configurations.

Contribution

It introduces a new compactification of the line arrangement moduli space, proving its smoothness, boundary normal crossings, and describing its morphism to the moduli space of rational curves.

Findings

The space is smooth with normal crossing boundary.

It admits a morphism to the moduli space of marked rational curves.

It is isomorphic to a closed subvariety inside a non-reductive Chow quotient.

Abstract

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.