Projective embedding of projective spaces

In this paper, embeddings φ : M → P from a linear space (M,M) in a projective space (P,L) are studied. We give examples for dimM > dimP and show under which conditions equality holds. More precisely, we introduce properties (G) (for a line L ∈ L and for a plane E ⊂ M it holds that |L ∩ φ(M)| 6 = 1) and (E) (φ(E) = φ(E) ∩ φ(M), whereby φ(E) denotes the by φ(E) generated subspace of P ). If (G) and (E) are satisfied then dimM = dimP . Moreover we give examples of embeddings of m-dimensional projective spaces in n-dimensional projective spaces with m > n that map any n+ 1 independent points onto n+ 1 independent points. This implies that for a proper subspace T of M it holds φ(T ) = φ(T )∩ φ(M) if and only if dimT ≤ n− 1, in particular (E) holds for n ≥ 3. (cf. 4.1)