An improved bound for the degree of smooth surfaces in P4 not of general type.

Abstract

This is an addendum to the paper of Braun and Fløystad ([BF]) on the bound for the degree of a smooth surface in $\pfour$ not of general type. Using their construction and the regularity of curves in $\pthree$, one may lower the bound a little more. We will prove the following: Theorem. Let $S$ be a smooth surface of degree $d$ in $\pfour$ not of general type. Then either $d\le 70$ or $S$ lies on a hypersurface in $\pfour$ of degree 5 and $d\le 80$. Thus $d\le 80$. We will also indicate how to lower the bound to 76.

5 pages, Latex version 2.09, no figures.