Symplectic semifield spreads of PG(5, qsupt), q even - Pepe, Valentina (Ars mathematica contemporanea, 2019)
Let ?$q > 2 · 3^{4t}$? be even. We prove that the only symplectic semifield spread of PG(5, ?$q^t$?), whose associate semifield has center containing ?$\mathbb{F}_q$?, is the Desarguesian spread. Equivalently, a commutative semifield of order ?$q^{3t}$?, with middle nucleus containing ?$\mathbb{F}_{q^t}$? and center containing ?$\mathbb F_q$?, is a field. We do that by proving that the only possible ?$\mathbb F_q$?-linear set of rank ?$3t$? in PG(5, ?$q^t$?) disjoint from the secant variety of the Veronese surface is a plane of PG(5, ?$q^t$)?.
Keywords: linear sets; linearne množice; polobsegi; razdelitve; semifields; simplektična polarnost; spreads; symplectic polarity; Veronese variety; Veronesejeva raznoterost