Tangent Spaces to Grassmannians

Abstract

We have seen that the Grassmannian 𝔾(k, n) is a smooth variety of dimension (k + 1) (n - k). This follows initially from our explicit description of the covering of 𝔾 (k, n) by open sets U _Λ ≅ 𝔸^(k+1)(n-k), though we could also deduce this from the fact that it is a homogeneous space for the algebraic group PGL _n+1 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 _Λ(𝔾;) in terms of the linear algebra of Λ ⊂ K ⁿ⁺¹. 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.