The smooth torus orbit closures in the Grassmannians

TL;DR

This paper characterizes smooth torus orbit closures in Grassmannians, showing they are products of projective spaces and are uniquely determined by simple matroid polytopes, linking combinatorics and geometry.

Contribution

It proves that simple matroid polytopes are products of simplices and that smooth orbit closures correspond to these products, providing a complete classification.

Findings

Smooth torus orbit closures are products of complex projective spaces.

Simple matroid polytopes are exactly products of simplices.

Smooth orbit closures are uniquely determined by simple matroid polytopes.

Abstract

It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We prove that simple matroid polytopes are products of simplices and smooth torus orbit closures in the Grassmannians are products of complex projective spaces. Moreover, it turns out that the smooth torus orbit closures are uniquely determined by the corresponding simple matroid polytopes.