For Hirzebruch surfaces, the algebraic exceptional set of a three-component normal-crossing curve with big K+B and no fiber or negative self-intersection component is finite and equals the hyper-bitangent set.
Algebraic Hyperbolicity of Complements of Generic Hypersurfaces in Projective Spaces
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abstract
We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines.
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Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces
For Hirzebruch surfaces, the algebraic exceptional set of a three-component normal-crossing curve with big K+B and no fiber or negative self-intersection component is finite and equals the hyper-bitangent set.