# Geometries arising from trilinear forms on low-dimensional vector spaces

## TL;DR

This paper classifies geometries of poles from hyperplanes in Grassmannians related to low-dimensional trilinear forms, revealing new structures and extending known results on line spreads in projective spaces.

## Contribution

It characterizes possible pole geometries for 3-forms in dimensions up to 7 and introduces new constructions, extending previous classifications and results.

## Key findings

- Classified geometries of poles for k=3, n≤7.
- Proposed new geometric constructions from hyperplanes.
- Extended results on line spreads in PG(5, K).

## 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 ${\mathcal G}_3(V).$