Geometries arising from trilinear forms on low-dimensional vector spaces

Let G(k)(V) be the k-Grassmannian of a vector space V with dim V = n. Given a hyperplane H of G(k)(V), we define in [3] a point-line subgeometry of PG(V) called the 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 <= 7 and propose some new constructions. We also extend a result of [6] regarding the existence of line spreads of PG(5, K) arising from hyperplanes of G(3)(V).