Geometries arising from trilinear forms on low-dimensional vector spaces

Ilaria Cardinali 1  and Luca Giuzzi 2
  • 1 Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy
  • 2 D.I.C.A.T.A.M., Section of Mathematics, University of Brescia, Via Branze 43, 25123, Brescia, Italy
Ilaria Cardinali
  • Corresponding author
  • Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy
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and Luca Giuzzi

Abstract

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).

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