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

TL;DR

This paper proves that very general quintic surfaces in projective 3-space are algebraically hyperbolic by establishing lower bounds on the genus of curves of certain degrees, improving previous bounds by G. Xu.

Contribution

It establishes new lower bounds on the genus of curves on very general surfaces of degree at least 5, demonstrating algebraic hyperbolicity for the quintic case.

Findings

Lower bounds on genus of curves on degree d surfaces

Proof that very general quintic surfaces are algebraically hyperbolic

Improvement over previous bounds by G. Xu

Abstract

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ 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.