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In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces V, specified using linear algebra, inside a fixed vector subspace W. Here W may be a vector space over an arbitrary field, though most commonly over the complex numbers.
A typical example is the set X whose points correspond to those 2-dimensional subspaces V of a 4-dimensional vector space W, such that V non-trivially intersects a fixed 2-dimensional subspace W2:
Over the real number field, this can be pictured in usual xyz-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of P {\displaystyle \mathbb {P} } , we obtain an open subset X° ⊂ X. This is isomorphic to the set of all lines L which meet the x-axis. Each such line L corresponds to a point of X°, and continuously moving L in space corresponds to a curve in X°. Since there are three degrees of freedom in moving L , X is a three-dimensional real algebraic variety. However, when L is equal to the x-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes L a singular point of X.
More generally, a Schubert variety is defined by specifying the minimal dimension of intersection between a k-dimensional V with each of the spaces in a fixed reference flag W 1 ⊂ W 2 ⊂ ⋯ ⊂ W n = W {\displaystyle W_{1}\subset W_{2}\subset \cdots \subset W_{n}=W} , where dim W j = j {\displaystyle \dim W_{j}=j}.