On smooth surfaces in $mathbb{P}^4$ containing a plane curve

摘要

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 $deg(S) leq 40$.