Compactifications of moduli of points and lines in the projective plane

TL;DR

This paper explores the relationship between two compactification methods of moduli spaces of points and lines in the projective plane, clarifying their connection and answering a question posed by Kapranov in 2003.

Contribution

It establishes the link between Kapranov's Chow quotient compactification and Gerritzen-Piweks' degenerations, providing new insights into the structure of these moduli spaces.

Findings

Connected different compactification approaches for moduli spaces.

Answered Kapranov's question from 2003.

Enhanced understanding of degenerations of the projective plane.

Abstract

Projective duality identifies the moduli spaces $\mathbf{B}_n$ and $\mathbf{X}(3,n)$ parametrizing linearly general configurations of $n$ points in $\mathbb{P}^2$ and $n$ lines in the dual $\mathbb{P}^2$, respectively. The space $\mathbf{X}(3,n)$ admits Kapranov's Chow quotient compactification $\overline{\mathbf{X}}(3,n)$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $\mathbb{P}^2$ with $n$ "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $\mathbb{P}^2$ with $n$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.