Let 𝓖k(V) be the k-Grassmannian of a vector space V with dim V = n. Given a hyperplane H of 𝓖k(V), we define in  a point-line subgeometry of PG(V) called the geometry of poles ofH. 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  regarding the existence of line spreads of PG(5, 𝕂) arising from hyperplanes of 𝓖3(V).
B. De Bruyn, Hyperplanes of embeddable Grassmannians arise from projective embeddings: a short proof. Linear Algebra Appl. 430 (2009), 418–422. MR2460527 Zbl 1161.5100210.1016/j.laa.2008.08.003)| false