On the Canonical Map of Smooth Surfaces in ℙ4

Abstract We prove that there exists an integer M such that if S ⊂ ℙ4 is a smooth surface with deg(S) > M, then the canonical map of S is birational. Then we consider surfaces, S, satisfying h i (ℐ S (3 − i)) = 0, 0 ≤ i ≤ 2 and show that they are regular and that their canonical system is base point free and very ample if deg(S) > M and S doesn't contain −2-curves.