On subspaces defining linear sets of maximum rank

Abstract: Let $V$ denote an $r$-dimensional $\mathbb{F}{q^n}$-vector space. Let $U$ and $W$ be $\mathbb{F}_q$-subspaces of $V$, $L_U$ and $L_W$ the projective points of $\mathrm{PG}\,(V,q^n)$ defined by $U$ and $W$ respectively. We address the problem when $L_W=L_U$ under the hypothesis that $U$ and $W$ have maximum dimension, i.e., $\dim{\mathbb{F}q} W=\dim{\mathbb{F}_q}U=$ $rn-n $, and we give a complete characterization for $r=2$.