A non-existence result on symplectic semifield spreads

TL;DR

This paper proves the non-existence of certain symplectic semifield spreads in projective geometry over large finite fields, by analyzing linear sets and secant varieties related to the Veronese variety.

Contribution

It establishes a non-existence result for non-Desarguesian symplectic semifield spreads in PG(5,q^2) for large even q, using geometric and algebraic methods.

Findings

No non-Desarguesian symplectic semifield spreads exist for q ≥ 2^14.

The only relevant linear set is a plane with three points outside the Veronese surface.

The result relies on properties of secant varieties and linear sets in projective space.

Abstract

We prove that there do not exist non-Desarguesian symplectic semifield spreads of PG$(5,q^2)$, $q\geq 2^{14}$ 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$(5,q^2)$.