2014/08/31 by Edward Richmond, William Slofstra · 18 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Fiber bundle #Grassmannian #Iterated function #Schubert calculus #Schubert polynomial #Schubert variety #Variety (cybernetics) #math.AG #math.CO
paper · pdf · doi:10.1007/s00208-015-1299-4
published in Mathematische Annalen 366(1-2), 31-55 (Springer Nature) · 22 pages. Substantial changes for publication; in particular, results for affine type A now appear in arXiv:1702.02236. This version does contain some schematic diagrams which are not in the published version, and which may be helpful
openalex publication_date 2015/09/29 · openalex created_date 2016/06/24 · arxiv created 2017/02/09 · arxiv updated 2017/02/10 · openalex updated_date 2026/08/05
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.