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On the singular locus of a Schubert variety. (English) Zbl 0981.14021

In this survey article the authors collect many of the most important results on singular loci of Schubert varieties. The article can be used as a handbook by geometers and combinatorists. It covers the topics:
1. Generalities of \(G/B\) and \(G/Q\).
2. Schubert varieties in \(SL(n)/B\).
3. Tangent spaces and smoothness.
4. Rational smoothness.
5. Determination of the singular locus of \(X(w)\).
6. Descriptions of \(T(w, \tau)\).
7. Computationally efficient criteria for smoothness and rational smoothness.
8. Irreducible components of the singular locus of a Schubert variety.
9. Groups of rank 2.
10. Factoring the Poincaré polynomial of a Schubert variety.
11. Counting smooth Schubert varieties.
The treatment is short and technical. For a more complete and leisurely presentation the interested reader should consult the book “Singular loci of Schubert varieties” [Prog. Math. 182 (2000; Zbl 0959.14032)] by S. Billey and V. Lakshmibai.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14B05 Singularities in algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry

Citations:

Zbl 0959.14032
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