A geometric approach to alternating k-linear forms

Denote by ({{mathcal {G}}}_k(V)) the Grassmannian of the k-subspaces of a vector space V over a field ({mathbb {K}}). There is a natural correspondence between hyperplanes H of ({mathcal {G}}_k(V)) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of ({{mathcal {G}}_k}(V)), we define a subspace (R^{uparrow }(H)) of ({{mathcal {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^{uparrow }(H)) might be empty; when (n-k) is odd, each element of ({mathcal {G}}_{k-2}(V)) is contained in at least one element of (R^{uparrow }(H)). In the present paper, we investigate several properties of (R^{uparrow }(H)), settle some open problems and propose a conjecture.