Compact moduli of points in the plane via tropical geometry and Mustafin varieties

Kapranov's compactification of the moduli space of $n$ lines in the projective plane gives an example of a moduli space of stable surfaces: it carries a family of certain reducible degenerations of the plane with $n$ "broken lines". For $n=6$, Luxton proved that this compactification is tropical and that it is associated to the tropical Grassmannian of Speyer and Sturmfels. In this talk we consider a dual perspective and construct a compact moduli space parametrizing $n$-pointed degenerations of the plane arising from Mustafin varieties, which was originally proposed by Gerritzen and Piwek. Moreover, for $n=6$ we show that this compactification is tropical and associated to a specific refinement of the tropical Grassmannian.

This is joint work with Jenia Tevelev.