Sphericality and Smoothness of Schubert Varieties

TL;DR

This paper investigates the geometric properties of Schubert varieties, demonstrating conditions for smoothness and sphericity, and classifying smooth Schubert varieties in rank two cases.

Contribution

It establishes that smooth Schubert varieties are either homogeneous spaces or have smooth divisors, and classifies smooth Schubert varieties in rank two algebraic groups.

Findings

Smooth Schubert varieties are homogeneous or have smooth divisors

All smooth Schubert varieties in rank two groups are spherical

Provides criteria for smoothness and sphericity in Schubert varieties

Abstract

We consider the action of the Levi subgroup of a parabolic subgroup that stabilizes a Schubert variety. We show that a smooth Schubert variety is a homogeneous space for a parabolic subgroup, or it has a smooth Schubert divisor. Further, we show that all smooth Schubert varieties in a (partial) flag variety of a rank two simple algebraic group are spherical.