A geometric approach to alternating k-linear forms

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