The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq3 of index 3 and rank 4 (and so in PG(6,q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the ‘second kind’) is a dual ‘generalised Desarguesian translation plane’, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,q) with no fixed points, there exists such a projective plane of order q3 , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q3 is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.