The generating and embedding rank of hyperplanes of Grassmannians

摘要

Let H be a hyperplane of the Grassmannian of the k-dimensional subspaces of the projective space PG(n, F), 0 <= k <= n - 1, and let (H) over tilde denote the subgeometry of the Grassmannian induced on the point set H. We prove that the embedding and generating ranks of (H) over tilde are always equal to ((n+1)(k+1)) - 1.