Compactifications of moduli of points and lines in the projective plane

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 $\bar{\mathbf{X}}(3,n)$, 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.