Initial degenerations of flag varieties

TL;DR

This paper studies the initial degenerations of flag varieties, showing they embed into limits of flag matroid strata and are smooth and irreducible for small n, with applications to Chow quotients and resolutions.

Contribution

It establishes a new embedding of initial degenerations into flag matroid strata and analyzes their geometric properties for small n, including smoothness and irreducibility.

Findings

Initial degenerations embed into inverse limits of flag matroid strata.

For n ≤ 4, initial degenerations are smooth and irreducible.

The Chow quotient by the diagonal torus is a log crepant resolution for n=4.

Abstract

We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\operatorname{F\ell}^{\circ}(n)$ -- the open subvariety of the complete flag variety $\operatorname{F\ell}(n)$ consisting of flags in general position -- are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\operatorname{F\ell}(n)$ by the diagonal torus of $\operatorname{PGL}(n)$, and show that, for $n=4$, this is a log crepant resolution of its log canonical model.