Generalized Veronesean embeddings of projective spaces

We classify all embeddings theta: PG(n, q) -> PG(d, q), with d >= n(n+3)/2 such that theta maps the set of points of each line to a set of coplanar points and such that the image of theta generates PG(d, q). It turns out that d = 1/2n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.We classify all embeddings theta: PG(n, q) -> PG(d, q), with d >= n(n+3)/2 such that theta maps the set of points of each line to a set of coplanar points and such that the image of theta generates PG(d, q). It turns out that d = 1/2n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.A