Schubert variety

The set of all -dimensional subspaces of an -dimensional vector space over a field satisfying the Schubert conditions: , , where is a fixed flag of subspaces of . In Grassmann coordinates these conditions are given by linear equations; a Schubert variety is an irreducible (generally speaking, singular) algebraic subvariety of the Grassmann manifold . Schubert varieties define a basis of the Chow ring , and for — a basis for the homology group .

The Schubert conditions were considered by H. Schubert in connection with enumeration problems for geometric objects with given incidence properties. Hilbert's 15th problem concerns a foundation for the enumeration theory developed by Schubert (see [2]).

References

[1]	H. Schubert, "Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension" Mitt. Math. Gesellschaft Hamburg , 1 (1889) pp. 134–155
[2]	S.L. Kleiman, "Problem 15. Rigorous foundation of Schubert's enumerative calculus" F.E. Browder (ed.) , Mathematical developments arising from Hilbert problems , Proc. Symp. Pure Math. , 28 , Amer. Math. Soc. (1976) pp. 445–482
[3]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978)
[4]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1954)

Comments

The notion of a Schubert variety has been generalized to any complete homogeneous space of a semi-simple linear algebraic group . It is the Zariski closure of any Bruhat cell ([a1]). The geometry of Schubert varieties was studied, e.g., in [a2], [a3].

References

[a1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff
[a2]	M. Demazure, "Désingularisation des variétés de Schubert généralisés" Ann. Sci. Ecole Norm. Sup. , 7 (1974) pp. 53–87
[a3]	V. Laksnibai, C. Seshadri, "Geometry of - V" J. of Algebra , 100 (1986) pp. 462–557