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

TL;DR

This paper proves that a specific open subvariety of the Grassmannian is sch"on, meaning all its initial degenerations are smooth, and applies this to resolve a conjecture about moduli space compactification.

Contribution

It establishes the sch"on property for the open subvariety of the Grassmannian and connects it to the log canonical compactification of the moduli space of lines in projective space.

Findings

The subvariety (3,8) is schf6n.

An initial degeneration with two connected components exists.

Remaining initial degenerations are irreducible up to symmetry.

Abstract

We prove that the open subvariety $\operatorname{Gr}_0(3,8)$ of the Grassmannian $\operatorname{Gr}(3,8)$ determined by the nonvanishing of all Pl\"ucker coordinates is sch\"on, i.e., all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, 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 $\operatorname{PGL}(8)$ is the log canonical compactification of the moduli space of $8$ lines in $\mathbb{P}^2$, resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes.