Initial degenerations of flag varieties

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 ${\mathrm{F\ell }}^{\circ }(n)$ -- the open subvariety of the complete flag variety $\mathrm{F\ell }(n)$ consisting of flags in general position -- are smooth and irreducible when $n\le 4$. We also study the Chow quotient of $\mathrm{F\ell }(n)$ by the diagonal torus of $\mathrm{PGL}(n)$, and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

arXiv comment26 pages, 13 figures

References in corpus (6)

• Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Crystal bases and Newton-Okounkov bodies
• Monodromy Filtrations and the Topology of Tropical Varieties
• Toric degenerations of cluster varieties and cluster duality
• Generalized permutahedra and positive flag Dressians