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arXiv · 1703.06821

Geometries arising from trilinear forms on low-dimensional vector spaces

Ilaria Cardinali·Luca Giuzzi
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Abstract

Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, 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 ${\mathrm{PG}}(V)$ 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. 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 ${\mathrm{PG}}(5,{\mathbb K})$ arising from hyperplanes of ${\mathcal G}_3(V).$

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Ilaria Cardinali, Luca Giuzzi. 2017-03-20. Geometries arising from trilinear forms on low-dimensional vector spaces. https://doi.org/10.1515/advgeom-2018-0027

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