Initial degenerations of flag varieties
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 $\opera...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2207.08094 |