Schubert variety

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 (reference) 2-dimensional subspace W₂:

${\displaystyle X\ =\ \{V\subset W\mid \dim(V)=2,\,\dim(V\cap W_{2})\geq 1\}.}$

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 ${\displaystyle \mathbb {P} (W)}$, we obtain an open subset X° ⊂ X. This is isomorphic to the set of all lines L (not necessarily through the origin) which meet the x-axis. Each such line L corresponds to a point of X°, and continuously moving L in space (while keeping contact with the x-axis) corresponds to a curve in X°. Since there are three degrees of freedom in moving L (moving the point on the x-axis, rotating, and tilting), 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 ${\displaystyle W_{1}\subset W_{2}\subset \cdots \subset W_{n}=W}$, where ${\displaystyle \dim W_{j}=j}$. (In the example above, this would mean requiring certain intersections of the line L with the x-axis and the xy-plane.)

In even greater generality, given a semisimple algebraic group G with a Borel subgroup B and a standard parabolic subgroup P, it is known that the homogeneous space X = G/P, which is an example of a flag variety, consists of finitely many B-orbits that may be parametrized by certain elements of the Weyl group W. The closure of the B-orbit associated to an element w of the Weyl group is denoted by X_w and is called a Schubert variety in G/P. The classical case corresponds to G = SL_n and P being the kth maximal parabolic subgroup of G.