Geometries arising from trilinear forms on low-dimensional vector spaces

Let Gk(V) be the k-Grassmannian of a vector space V with dimV=n. Given a hyperplane H of Gk(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 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 [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 PG(5,K) arising from hyperplanes of G3(V).