The Grassmannian of 3-planes in ℂ 8 is schön

Abstract

We prove that the open subvariety Gr 0 (3,8) of the Grassmannian Gr(3,8) determined by the nonvanishing of all Plücker coordinates is schön, 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 Gr(3,8) by the diagonal torus of PGL(8) is the log canonical compactification of the moduli space of 8 lines in ℙ 2 , resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes.