Symmetric spread sets


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 q^2 with kernel containing \GF(q), 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 q^2  with  kernel containing \GF(q) to be desarguesian. As a consequence we prove  that a symplectic plane of even order q^2 with  kernel containing \GF(q) and  admitting an affine homology of order q-1 or a Baer involution fixing a totally isotropic 2-subspace  is desarguesian. Finally a short proof that symplectic semifield planes of even order q^2 with  kernel containing \GF(q) are desarguesian is given.

DOI Code: 10.1285/i15900932v29n1supplp153

translation plane; symplectic spread; line-oval; affine homology; Baer involution

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