Schubert variety

2020 Mathematics Subject Classification: Primary: 14M15 [MSN][ZBL]

A Schubert variety is the set of all $m$-dimensional subspaces $W$ of an $n$-dimensional vector space $V$ over a field $k$ satisfying the Schubert conditions: $\dim(W\cap V_j) \ge j$, $j=1,\dots,m$, where $V_1\subset\cdots\subset V_m$ is a fixed flag of subspaces of $V$. 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 $G_{n,m}$. Schubert varieties define a basis of the Chow ring $A(G_{n,m})$, and for $k=\C$ — a basis for the homology group $H_*(G_{n,m},\Z)$.

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 [Kl]).

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

References

[Bo]	A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201
[De]	M. Demazure, "Désingularisation des variétés de Schubert généralisées" Ann. Sci. Ecole Norm. Sup., 7 (1974) pp. 53–87 MR0354697 Zbl 0312.14009
[GrHa]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry", 1, Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[HoPe]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry", 2, Cambridge Univ. Press (1954) MR0061846 Zbl 0055.38705
[Kl]	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 MR0429938
[LaSe]	V. Lakshmibai, C. Seshadri, "Geometry of $G/P$ V." J. of Algebra, 100 (1986) pp. 462–557 MR840589 Zbl 0618.14026
[Sc]	H. Schubert, "Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension" Mitt. Math. Gesellschaft Hamburg, 1 (1889) pp. 134–155 Zbl 18.0631.01