ALGEBRAIC COMBINATORICS

no. 5
The Grassmannian of 3-planes in ℂ 8 is schön
Corey, Daniel1; Luber, Dante2
1 University of Nevada, Las Vegas 4505 S Maryland Pkwy Las Vegas, NV 89154
2 Technische Universität Berlin Institut für Mathematik, Sekr. MA 6-2 Strasse des 17 Juni 136 10623 Berlin
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1273-1299.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.302
Classification: 14T90, 05E14, 14C05, 52B40
Keywords: Chow quotient, Grassmannian, matroid, tight span
Corey, Daniel 1; Luber, Dante 2

1 University of Nevada, Las Vegas 4505 S Maryland Pkwy Las Vegas, NV 89154
2 Technische Universität Berlin Institut für Mathematik, Sekr. MA 6-2 Strasse des 17 Juni 136 10623 Berlin
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Corey, Daniel; Luber, Dante. The Grassmannian of $3$-planes in $\mathbb{C}^{8}$ is schön. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1273-1299. doi : 10.5802/alco.302. https://alco.centre-mersenne.org/articles/10.5802/alco.302/

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