Geometries arising from trilinear forms on low-dimensional vector spaces

Let ${\mathcal{G}}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal{G}}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J. Algebraic Combin. doi:10.1007/s10801-016-0730-6] a point-line subgeometry of $\mathrm{P}\mathrm{G}(V)$ called the {\it geometry of poles of $H$}. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for $k=3$ and $n\le 7$ and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi:10.1016/j.laa.2010.03.040] regarding the existence of line spreads of $\mathrm{P}\mathrm{G}(5,\mathbb{K})$ arising from hyperplanes of ${\mathcal{G}}_3(V)\mathrm{.}$