On subspaces defining linear sets of maximum rank

Abstract

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