Initial degenerations of Grassmannians

Abstract

We construct closed immersions from initial degenerations of $\operatorname*{Gr}_{0}(d,n)$---the open cell in the Grassmannian $\operatorname*{Gr}(d,n)$ given by the nonvanishing of all Pl\"ucker coordinates---to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms when $(d,n)$ equals $(2,n)$, $(3,6)$ and $(3,7)$. As an application we prove $\operatorname*{Gr}_0(3,7)$ is sch\"on, and the Chow quotient of $\operatorname*{Gr}(3,7)$ by the maximal torus in $ \operatorname*{PGL}(7)$ is the log canonical compactification of the moduli space of 7 points in $\mathbb{P}^2$ in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev.