Initial degenerations of Grassmannians

TL;DR

This paper studies the initial degenerations of Grassmannians, constructing explicit embeddings into limits of thin Schubert cells, and applies these results to prove properties of specific Grassmannians and their quotients.

Contribution

It introduces a new construction of embeddings for initial degenerations of Grassmannians into limits of thin Schubert cells, advancing understanding of their geometric structure.

Findings

Constructed closed immersions for initial degenerations of Grassmannians.

Proved isomorphisms for specific cases: (2,n), (3,6), (3,7).

Established that (3,7) is schf6n and identified the Chow quotient as a log canonical compactification.

Abstract

We construct closed immersions from initial degenerations of $\operatorname*{Gr}_{0}(d,n)$---the open cell in the Grassmannian $\operatorname*{Gr}(d,n)$ given by the nonvanishing of all Pl\"ucker 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 $\operatorname*{Gr}_0(3,7)$ is sch\"on, and the Chow quotient of $\operatorname*{Gr}(3,7)$ by the maximal torus in $ \operatorname*{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.