An improved bound for the degree of smooth surfaces in P not of general type

This is an addendum to the paper of Braun and Fløystad ([BF]) on the bound for the degree of smooth surfaces in P 4 not of general type. (In fact, any statements made here without comment will be found there.) Using their construction and the regularity of curves in P 3 , one may lower the bound a little more. We will prove the following: Theorem. Let S be a smooth surface of degree d in P 4 not of general type. Then either d ≤ 70 or S lies on a hypersurface in P 4 of degree 5 and d ≤ 80. Thus d ≤ 80. (We will also indicate how one may actually get the bound down to 76.) The main result of [BF] is to bound χO S from below using generic initial ideal theory and information coming from a generic hyperplane section of S.