Schubert variety

In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, ${\displaystyle \mathbf {Gr} _{k}(V)}$ of ${\displaystyle k}$-dimensional subspaces of a vector space ${\displaystyle V}$, 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 ${\displaystyle w\subset V}$, with the elements of a specified complete flag. Here ${\displaystyle V}$ 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 ${\displaystyle X}$ of ${\displaystyle 2}$-dimensional subspaces ${\displaystyle w\subset V}$ of a 4-dimensional space ${\displaystyle V}$ that intersect a fixed (reference) 2-dimensional subspace ${\displaystyle V_{2}}$ nontrivially.

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

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 ${\displaystyle \mathbb {P} (V)}$, 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 in ${\displaystyle \mathbf {Gr} _{k}(V)}$ is defined by specifying the minimal dimension of intersection of a ${\displaystyle k}$-dimensional subspace ${\displaystyle w\subset V}$ with each of the spaces in a fixed reference complete flag ${\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}$, where ${\displaystyle \dim V_{j}=j}$. (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 ${\displaystyle G}$ with a Borel subgroup ${\displaystyle B}$ and a standard parabolic subgroup ${\displaystyle P}$, it is known that the homogeneous space ${\displaystyle G/P}$, which is an example of a flag variety, consists of finitely many ${\displaystyle B}$-orbits, which may be parametrized by certain elements ${\displaystyle w\in W}$ of the Weyl group ${\displaystyle W}$. The closure of the ${\displaystyle B}$-orbit associated to an element ${\displaystyle w\in W}$ is denoted ${\displaystyle X_{w}}$ and is called a Schubert variety in ${\displaystyle G/P}$. The classical case corresponds to ${\displaystyle G=SL_{n}}$, with ${\displaystyle P=P_{k}}$, the ${\displaystyle k}$th maximal parabolic subgroup of ${\displaystyle SL_{n}}$, so that ${\displaystyle G/P=\mathbf {Gr} _{k}(\mathbf {C} ^{n})}$ is the Grassmannian of ${\displaystyle k}$-planes in ${\displaystyle \mathbf {C} ^{n}}$.