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Algebraic Combin. doi:10.1007/s10801-016-0730-6] a point-line subgeometry of ${\\mathrm{PG}}(V)$ 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 $n\\leq 7$ and propose some new constructions. 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