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Hyperplane Sections of Kantor's Unitary Ovoids

Authors:	B N Cooperstein

Institution:	(1) Department of Mathematics, University of California, Santa Cruz

Abstract:	The projective plane $\mathbb{P}\mathbb{G}(2,q^2 )$ is embedded as a variety of projective points $\mathcal{V}$ in $\mathbb{P}\mathbb{G}(M)$ , where M is a nine dimensional $\mathbb{F}_q$ -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections $\mathbb{P}\mathbb{G}(D) \cap \mathcal{V}$ whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.

Keywords:	Projective space orthogonal space singular point non-singular point ovoid partial ovoid
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