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We prove that the open subvariety $\operatorname{Gr}_0(3,8)$ of theGrassmannian $\operatorname{Gr}(3,8)$ determined by the nonvanishing of allPl\"ucker coordinates is sch\"on, i.e., all of its initial degenerations aresmooth. Furthermore, we find an initial degeneration that has two connectedcomponents, and show that the remaining initial degenerations, up to symmetry,are irreducible. As an application, we prove that the Chow quotient of$\operatorname{Gr}(3,8)$ by the diagonal torus of $\operatorname{PGL}(8)$ isthe log canonical compactification of the moduli space of $8$ lines in$\mathbb{P}^2$, resolving a conjecture of Hacking, Keel, and Tevelev. Along theway we develop various techniques to study finite inverse limits of schemes.

The Grassmannian of 3-planes in $\mathbb{C}^{8}$ is sch\"on