On smooth surfaces in $\bold P\sp 4$ containing a plane curve

Abstract

Let $\Sigma \subset \mathbb{P}^4$ be an integral hypersurface of degree $s$ with a $(s-2)$-uple plane. We show that the degrees of smooth surfaces $S \subset \Sigma$ with $q(S)=0$ are bounded by a function of $s$. We also show that if $S \subset \mathbb{P}^4$ is a smooth surface with $q(S)=0$ and if $S$ lies on a quartic hypersurface $\Sigma$ such that $\dim(\Sing(\Sigma))=2$, then $\deg(S) \leq 40$.