A geometric approach to alternating k-linear forms

Denote by ${\mathcal cG}_k(V)$ the Grassmannian of the $k$-subspaces of a vector space $V$ over a field $\KK$. There is a natural correspondence between hyperplanes $H$ of $\cG_k(V)$ and alternating $k$-linear forms on $V$ defined up to a scalar multiple. Given a hyperplane $H$ of ${\cG_k}(V)$, 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 $\cG_{k-2}(V)$ 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.