Compactifications of Moduli of Points and Lines in the Projective Plane

Projective duality identifies the moduli spaces Bn and X(3,n) parametrizing linearly general configurations of n points in P2 and n lines in the dual P2, respectively. The space X(3,n) admits Kapranov's Chow quotient compactification, 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 P2 with n "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of P2 with n smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.