Transitive Deficiency-One Baer Subgeometry Partitions

The translation planes of order 43 with spread in PG(5, 4) arising from Baer subgeometry partitions of PG(2, 16) found by Mathon and Hamilton are characterized within all Baer subgeometry partitions admitting groups fixing one Baer subgeometry and acting transitively on the remaining Baer subgeometries. When q is even, we show that a Baer subgeometry partition of PG(2, q2) with a group fixing one and transitive on the remaining PG(2, q)ʹs forces the partition to be classical or q=4 and the partition is one of the partitions of Mathon and Hamilton. When q is odd, we give a complete characterization of these Baer subgeometry partitions. Further, generalizations are given for similar Baer subgeometry partitions of PG(2m, q2).