On finite Steiner surfaces

Unlike the real case, for each q power of a prime it is possible to injectively project the quadric Veronesean of P G ( 5 , q ) into a solid or even a plane. Here a finite analogue of the Roman surface of J. Steiner is described. Such an analogue arises from an embedding σ of P G ( 2 , q ) into P G ( 3 , q ) mapping any line onto a non-singular conic. Its image P G ( 2 , q ) σ has a nucleus, say T σ , arising from three points of P G ( 2 , q 3 ) forming an orbit of the Frobenius collineation.