A non-existence result on symplectic semifield spreads

Abstract

We prove that there do not exist non-Desarguesian symplectic semifield spreads of PG$(5,q^2)$, $q\ge 2^{14}$ even, whose associated semifield has center containing ${\mathbb{F}}_q$, by proving that the only ${\mathbb{F}}_q$-linear set of rank 6 disjoint from the secant variety of the quadric Veronese variety of PG$(5,q^2)$ is a plane with three points of the Veronese surface of PG$(5,q^6)\setminus$PG$(5,q^2)$.