Generalized Veronesean embeddings of projective spaces

AbstractWe classify all embeddings θ: PG(n, q) → PG(d, q), with $$d \geqslant \tfrac{{n(n + 3)}} {2}$$ , such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ½n(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.