Projective embedding of projective spaces

In this paper, embeddings phi : M --> P from a linear space (M,M) in a projective space (P, L) are studied. We give examples for dim ill > dim P and show under which conditions equality holds. More precisely, we introduce propel-ties (G) (for a line L is an element of C and for a plane E subset of M it holds that \L boolean AND phi(M)\ not equal 1) and (E) (phi(E) = <(phi(E))over bar> boolean AND phi(M), whereby <(phi(E))over bar> denotes the by phi(E) generated subspace of P). If (G) and (E) are satisfied then dim M = dim P. Moreover we give examples of embeddings of m-dimensional projective spaces in n-dimensional projective spaces with m > n that map any n+l independent points onto n+1 independent points. This implies that for a proper subspace T of hi it holds phi(T) = <(phi(T))over bar> boolean AND phi(M) if and only if dimT less than or equal to n-1, in particular (E) holds for n greater than or equal to 3. (cf. 4.1).