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

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. We also give some partial results for Schubert varieties in Kac-Moody flag varieties. In particular, we show that rationally smooth Schubert varieties of affine type A are also iterated fibre bundles of rationally smooth Grassmannian Schubert varieties. As a consequence, we finish a conjecture of Billey-Crites that a Schubert variety in affine type A is smooth if and only if the corresponding affine permutation avoids the patterns 4231 and 3412