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Bioriented flags and resolutions of Schubert varieties
Author(s) -
Cibotaru Daniel
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800440
Subject(s) - grassmannian , flag (linear algebra) , schubert variety , mathematics , generalized flag variety , variety (cybernetics) , schubert calculus , pure mathematics , type (biology) , resolution (logic) , flags register , algebra over a field , computer science , artificial intelligence , ecology , statistics , lie group , biology , operating system
Abstract We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W ‐flag varieties, algebro‐geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.