A geometric approach to alternating k-linear forms

Denote by Gk (V) the Grassmannian of the k-subspaces of a vector space V over a field K. There is a natural correspondence between hyperplanes H of Gk (V) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of Gk (V), we define a subspace R up arrow(H) of G(k-1)(V) whose elements are the (k-1)subspaces A such that all k-spaces containing A belong to H. When n - k is even, R up arrow(H) might be empty; when n - k is odd, each element of G(k-2)(V) is contained in at least one element of R.(H). In the present paper, we investigate several properties of R up arrow(H), settle some open problems and propose a conjecture.