Compactifications of Moduli of Points and Lines in the Projective Plane

Abstract

Abstract Projective duality identifies the moduli spaces $\textbf{B}_n$ and $\textbf{X}(3,n)$ parametrizing linearly general configurations of $n$ points in $\mathbb{P}^2$ and $n$ lines in the dual $\mathbb{P}^2$, respectively. The space $\textbf{X}(3,n)$ admits Kapranov’s Chow quotient compactification $\overline{\textbf{X}}(3,n)$, 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.