High Quality Content by WIKIPEDIA articles! Schubert varieties form one of the most important and best studied classes of singular algebraic varieties. A certain measure of singularity of Schubert varieties is provided by Kazhdan-Lusztig polynomials, which encode their local Goresky-MacPherson intersection cohomology. The algebras of regular functions on Schubert varieties have deep significance in algebraic combinatorics and are examples of algebras with a straightening law. (Co)homology of the Grassmanian, and more generally, of more general flag varieties, is spanned by the (co)homology classes of Schubert varieties, the Schubert cycles. The study of the intersection theory on the Grassmanian was initiated by Hermann Schubert and continued by Zeuthen in 19th century under the heading of enumerative geometry. This area was deemed by David Hilbert important enough to be included as the fifteenth of his celebrated 23 problems.

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