Abstract
We prove that a curve of degree dk on a very general surface of degree d≥5 in P3 has geometric genus at least dk(d−5)+k2+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 P3, resolving a long-standing conjecture of Demailly. This completely determines which very general hypersurfaces in P3 are algebraically hyperbolic.
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