On finite Steiner surfaces

Unlike the real case, for each qq power of a prime it is possible to injectively project the quadric Veronesean of PG(5,q)PG(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 σσ of PG(2,q)PG(2,q) into PG(3,q)PG(3,q) mapping any line onto a non-singular conic. Its image PG(2,q)σPG(2,q)σ has a nucleus, say TσTσ, arising from three points of PG(2,q3)PG(2,q3) forming an orbit of the Frobenius collineation.

► Any finite quadric Veronesean can be injectively projected into a plane.
► Non-singular Steiner embeddings in finite three-dimensional projective spaces.
► A finite analogue of the Roman surface of J. Steiner.
► An embedding of PG(2,q)PG(2,q) into PG(3,q)PG(3,q) mapping any line into a non-singular conic.
► A permutation of PG(2,q)PG(2,q) mapping any line into a non-singular conic.