# 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.

## Key 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.