A new characterization of the desarguesian and the Figueroa plane

A result of Dempwolff [4] asserts that a projective plane Π of order q3 admitting G =PGL(3, q) as a collineation group contains a G-invariant subplane π_0 isomorphic to PG(2, q) on which G acts 2-transitively. Moreover, G splits the point set and the line set of Π into 3 orbits P_i(Π) and L_i(Π), i = 1, 2, 3, consisting of the points (respectively lines) of Π incident with q + 1, 1 or 0 lines (respectively points) of π_0. In this paper it is proven that Π is either the desarguesian or the Figueroa plane of order q^3 if, and only if, (P_3(Π), L_3(Π)) is isomorphic to (P_3(D), L_3(D)), where D = PG(2, q^3)