Papers › The Grassmannian of 3-planes in ℂ⁸ is schön
The Grassmannian of 3-planes in ℂ⁸ is schön
Daniel Corey, Dante Luber
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We prove that the open subvariety Gr₀(3,8) of the Grassmannian 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 Gr(3,8) by the diagonal torus of PGL(8) is the log canonical compactification of the moduli space of 8 lines in ℙ², resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes.
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