Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties

TL;DR

This paper generalizes a theorem relating smoothness of Schubert varieties to their fiber bundle structure across all finite types, using combinatorial methods to characterize smoothness and rational smoothness.

Contribution

It extends Ryan and Wolper's theorem to arbitrary finite types, linking rational smoothness of Schubert varieties to their fiber bundle structure, and classifies smooth and rationally smooth cases.

Findings

Characterization of smooth and rationally smooth Grassmannian Schubert varieties

Extension of Ryan and Wolper's theorem to all finite types

New proof of Peterson's theorem on simply-laced rationally smooth Schubert varieties

Abstract

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly complete geometric description of smooth and rationally smooth Schubert varieties using primarily combinatorial methods.