Kapranov's compactification of the moduli space of 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 "broken lines". For , 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 -pointed degenerations of the plane arising from Mustafin varieties, which was originally proposed by Gerritzen and Piwek. Moreover, for we show that this compactification is tropical and associated to a specific refinement of the tropical Grassmannian.
This is joint work with Jenia Tevelev.