Algebraic hyperbolicity of the very general quintic surface in ${\mathbb{P}}^3$

Abstract

We prove that a curve of degree $dk$ on a very general surface of degree $d\ge 5$ in ${\mathbb{P}}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2}+1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very general quintic surface in ${\mathbb{P}}^3$ is algebraically hyperbolic.