Symmetric spread sets

Abstract

Some new results on symplectic translation planes  are given using their representation by spread sets of symmetric matrices. We provide a general construction of symplectic planes of even order and then consider the special case of planes of order with kernel containing , stressing the role of Brown's theorem on ovoids containing a conic section. In particular we provide a criterion for a symplectic plane of even order  with  kernel containing to be desarguesian. As a consequence we prove  that a symplectic plane of even order with  kernel containing and  admitting an affine homology of order or a Baer involution fixing a totally isotropic -subspace  is desarguesian. Finally a short proof that symplectic semifield planes of even order with  kernel containing are desarguesian is given.