A characterisation of spreads ovally-derived from Desarguesian spreads

Abstract

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding “ovally-derived” planes are generalised André planes, of order 2^N, although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.