Initial degenerations of Grassmannians

Abstract

We construct closed immersions from initial degenerations of ${\mathrm{Gr}}_0(d,n)$---the open cell in the Grassmannian $\mathrm{Gr}(d,n)$ given by the nonvanishing of all Plücker coordinates---to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms when $(d,n)$ equals $(2,n)$, $(3,6)$ and $(3,7)$. As an application we prove ${\mathrm{Gr}}_0(3,7)$ is schön, and the Chow quotient of $\mathrm{Gr}(3,7)$ by the maximal torus in $\mathrm{PGL}(7)$ is the log canonical compactification of the moduli space of 7 points in ${\mathbb{P}}^2$ in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev.

arXiv comment43 pages, 3 TikZ figures