Symplectic semifield spreads of \(\mathrm{PG}(5,q)\) and the Veronese surface (Q547273)

A spread of \(\mathrm{PG}(2n-1, q)\) is a partition \({\mathcal S}\) of the point set of \(\mathrm{PG}(2n-1, q)\) into \((n-1)\)-dimensional subspaces. With any spread \({\mathcal S}\) is associated a translation plane \(A({\mathcal S})\). The spread \({\mathcal S}\) is called Desarguesian if \(A({\mathcal S})\) is isomorphic to \(\mathrm{AG}(2,q^n)\). The spread \({\mathcal S}\) is a semifield spread with respect to an element \(X\in {\mathcal S}\) if there exists an elementary abelian subgroup \(G\) of \(\mathrm{PGL}(2n,q)\) of order \(q^n\) which fixes \(X\) pointwise and acts regularly on \({\mathcal S}\setminus \{X\}\). The spread \({\mathcal S}\) is a symplectic spread with respect to a nonsingular symplectic polarity \(\perp\) of \(\mathrm{PG}(2n-1, q)\) when all subspaces of \({\mathcal S}\) are totally isotropic with respect to \(\perp\). The authors show that starting from a symplectic semifield spread \({\mathcal S}\) of \(\mathrm{PG}(5, q)\), \(q\) odd, another symplectic semifield spread of \(\mathrm{PG}(5,q)\) can be obtained, called the symplectic dual of \({\mathcal S}\), and prove that the symplectic dual of a Desarguesian spread of \(\mathrm{PG}(5,q)\) is the symplectic semifield spread arising from a generalized twisted field. They construct a new symplectic semifield spread of \(\mathrm{PG}(5,q)\) (\(q = s^{2}\), \(s\) odd), describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, they determine the nuclei of the commutative pre-semifields constructed by \textit{Z. Zha, G. M. Kyureghyan} and \textit{X. Wang} [Finite Fields Appl 15, No. 2, 125--133 (2009; Zbl 1194.12003)].