In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} of k {\displaystyle k} -dimensional subspaces of a vector space V {\displaystyle V} , usually with singular points. Like the Grassmannian, it is a kind of moduli space, whose elements satisfy conditions giving lower bounds to the dimensions of the intersections of its elements w ⊂ V {\displaystyle w\subset V} , with the elements of a specified complete flag. Here V {\displaystyle V} may be a vector space over an arbitrary field, but most commonly this taken to be either the real or the complex numbers. A typical example is the set X {\displaystyle X} of 2 {\displaystyle 2} -dimensional subspaces w ⊂ V {\displaystyle w\subset V} of a 4-dimensional space V {\displaystyle V} that intersect a fixed (reference) 2-dimensional subspace V 2 {\displaystyle V_{2}} nontrivially.
X = { w ⊂ V ∣ dim ( w ) = 2 , dim ( w ∩ V 2 ) ≥ 1 } . {\displaystyle X\ =\ \{w\subset V\mid \dim(w)=2,\,\dim(w\cap V_{2})\geq 1\}.}
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