On embedding a unitary block design as a polar unital and an intrinsic characterization of the classical unital

Abstract

A necessary and sufficient condition is given for embedding a unital into a projective plane as a polar unital. A strengthened version of the condition is introduced and is shown to be necessary for a classical unital. Using the strengthened condition and results of Wilbrink (1983) and Grundhöfer, Stroppel and Van Maldeghem (2013), a new intrinsic characterization of the classical unital is given without assuming the absence of OʼNan configurations. Finally, a unital of even order satisfying the first two intrinsic characterization conditions of Wilbrink is shown to satisfy the strengthened condition by an elementary (combinatorial-geometric) proof and without invoking deep results from group theory.