Compact projective planes with homogeneous ovals

Abstract

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO2 or SO3, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.