Value Distributions

We fix n, and for each k, 0 ≤ k ≤ n, consider ∧k+1Cn+1. Set $$n(k) = (\begin{array}{*{20}{c}} {n + 1} \\ {k + 1} \end{array}) - 1$$ so that ∧k+1Cn+1 ≃ Cn(k)+1 Let G (n, k) be the Grassmannian of k-planes in P _n C, i.e., the Grassmannian of (k + 1)-dimensional subspaces in Cn+1 Then dim G (n, k) = (n − k)(k + 1). To a (k + 1)-dimensional subspace spanned by a₀, …, a _k ∈ Cn+1, we assign a decomposable (k + 1)-vector A = a₀ ∧ … ∧ a _k ∈ ∧k+1Cn+1 which is determined, up to a constant factor, by the subspace. Conversely, each decomposable (k + 1)-vector A determines a k-plane in P _n C, i.e., a (k + 1)-dimensional vector subspace of Cn+1 both of which will be denoted by the same symbol [A]. This correspondence defines the Plücker imbedding8.1.1 $$G(n,k) \subset {P_{n(k)}}C.$$