Séminaire Lotharingien de Combinatoire, 89B.44 (2023), 12 pp.

Daniel Corey and Dante Luber

Degenerations of the Grassmannian via Matroidal Subdivisions of the Hypersimplex

Abstract. We study initial degenerations of Gr₀(3,8)---the open part of the Grassmannian Gr(3,8) defined by the nonvanishing of all Plücker coordinates---using matroid subdivisions of the (3,8)-hypersimplex. We show that all initial degenerations are smooth and, outside of a single symmetry class, irreducible. As an application, we show 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 the projective plane in linear general position. This fully resolves a conjecture of Hacking, Keel, and Tevelev.

Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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