Algebraic Geometry A First Course (4).pdf

LECTURE 16
Tangent Spaces to Grassmannians
Example . Tangent Spaces to Grassmannians
We have seen that the Grassmannian G(k, n) is a smooth variety of dimension
(k + l)(n - k). This follows initially from our explicit description of the covering of
G(k, n) by open sets U,, E A (k+-k), though we could also deduce this from the
fact that it is a homogeneous space for the algebraic group PGL,+l K. The Zariski
tangent spaces to G are thus all vector spaces of this dimension. For many reasons,
however, it is important to have a more intrinsic description of the space T’,(G) in
terms of the linear algebra of A c K”+l. We will derive such an expression here and
then use it to describe the tangent spaces of the various varieties constructed in Part
I with the use of the Grassmannians. c
C
To begin with, let us reexamine the basic open sets covering the Grassman-
nian. Recall from Lecture 6 that for any (n - k)-plane I- c K”+l the open set
U, c G is defined to be the subset of planes A c K”+plementary to r, .,
such that A n I- = (0). These open sets were seen to be isomorphic to affine spaces
A(k+l)(n-k) as follows. Fixing any subspace A E Ur, a subspace A’ E Ur is the graph
of a homomorphism cp: A --+ I-, so that
1
U, = Hom(A, r). ti
In particular, this isomorphism of spaces induces an isomorphism of tangent spaces
T,(G) = Hom(A, I-).
A
Now suppose we start with a subspace A and do not specify the subspace r.
To what extent is this identification independent of the choice of r? The answer is H
straightforward: any subspace plementary to A may naturally be identified
K “‘
with the quotient vector space l/A, and if we view the isomorphism of tangent n
to
16. Tangent Spaces Grassmannians 201
spaces in this light, it is independent of I. We thus have a natural identification
T,(G) = Horn@, K*+l/A).
In order to see this better, and because it will be extremely useful in applications,
we will a