Algebraic hyperbolicity of the very general quintic surface in P 3

We prove that a curve of degree dk on a very general surface of degree d≥5 in P 3 has geometric genus at least [Formula presented]+1. This gives a substantial improvement on the celebrated genus bounds of Geng Xu. As a corollary, we deduce the algebraic hyperbolicity of a very general quintic surface in P 3 , resolving a long-standing conjecture of Demailly. This completely determines which very general hypersurfaces in P 3 are algebraically hyperbolic.