Projective ovoids and generalized quadrangles

Abstract

<jats:title>Abstract</jats:title> <jats:p>We consider a simple condition defining a <jats:bold> <jats:italic>tetradic</jats:italic> </jats:bold> set of ovoids in a projective three-space over a finite field. By elementary counting and geometrical methods we establish the properties of a tetradic set and are able to give a purely synthetic construction of the class of generalized quadrangles of order (<jats:italic>s, s</jats:italic> <jats:sup>2</jats:sup>) satisfying Property (G) at a flag. This includes the class of dual flock generalized quadrangles due to Kantor and Payne in the 1980's. We also show that the dual flock generalized quadrangles are characterised by Property (G) at a line.</jats:p>