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A theorem of Ryan and Wolper states that a type A Schubert variety is smoothif and only if it is an iterated fibre bundle of Grassmannians. We extend thistheorem to arbitrary finite type, showing that a Schubert variety in ageneralized flag variety is rationally smooth if and only if it is an iteratedfibre bundle of rationally smooth Grassmannian Schubert varieties. The proofdepends on deep combinatorial results of Billey-Postnikov on Weyl groups. Wedetermine all smooth and rationally smooth Grassmannian Schubert varieties, andgive a new proof of Peterson's theorem that all simply-laced rationally smoothSchubert varieties are smooth. Taken together, our results give a fairlycomplete geometric description of smooth and rationally smooth Schubertvarieties using primarily combinatorial methods.

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