Algebraic hyperbolicity of the very general quintic surface in \(\mathbb{P}^3\). (English) Zbl 1499.32047
Summary: 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 \(\frac{d k(d - 5) + k}{2} + 1\). 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 \(\mathbb{P}^3\), resolving a long-standing conjecture of Demailly. This completely determines which very general hypersurfaces in \(\mathbb{P}^3\) are algebraically hyperbolic.
MSC:
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 14H50 | Plane and space curves |
| 14J70 | Hypersurfaces and algebraic geometry |
| 14J29 | Surfaces of general type |
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