Codimension 2 and 3 pluricanonical embeddings in projective spaces

In this paper some non existence results are given for smooth varieties of dimension bigger then 1, embedded in projective spaces with low codimension, by the pluricanonical system |mK _X |, withm≥2. The only meaningful cases of codimension 2 are surfaces in P⁴ and threefolds in P⁵, whose existence is excluded. When the codimension is 3, for surfaces in P⁵ is proven thatm=2, while for three-folds in P⁶ and fourfolds in P⁷ only two numerical possibilities for the degree are given.