On the maximum field of linearity of linear sets

Abstract

Let $V$ denote an $r$-dimensional ${\mathbb{F}}_{q^n}$-vector space. For an $m$-dimensional ${\mathbb{F}}_q$-subspace $U$ of $V$ assume that ${\dim }_q\left(⟨\mathbf{v}{⟩}_{{\mathbb{F}}_{q^n}}\cap U\right)\ge 2$ for each non zero vector $\mathbf{v}\in U$. If $n\le q$ then we prove the existence of an integer $1<d\mid n$ such that the set of one-dimensional ${\mathbb{F}}_{q^n}$-subspaces generated by non-zero vectors of $U$ is the same as the set of one-dimensional ${\mathbb{F}}_{q^n}$-subspaces generated by non-zero vectors of $⟨U{⟩}_{{\mathbb{F}}_{q^d}}$. If we view $U$ as a point set of $\mathrm{AG}(r,q^n)$, it means that $U$ and $⟨U{⟩}_{{\mathbb{F}}_{q^d}}$ determine the same set of directions. We prove a stronger statement when $n\mid m$. In terms of linear sets it means that an ${\mathbb{F}}_q$-linear set of $\mathrm{PG}(r-1,q^n)$ has maximum field of linearity ${\mathbb{F}}_q$ only if it has a point of weight one. We also present some consequences regarding the size of a linear set.

We corrected the definition of the Segre embedding

References in corpus (1)

• The number of directions determined by less than $q$ points