## Abstract

Let đť“–_{k}(*V*) be the *k*-Grassmannian of a vector space *V* with dim *V* = *n*. Given a hyperplane *H* of đť“–_{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* â‰¤ 7 and propose some new constructions. We also extend a result of [6] regarding the existence of line spreads of PG(5, đť•‚) arising from hyperplanes of đť“–_{3}(*V*).

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