Algebraic hyperbolicity of very general surfaces

Abstract

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $${\mathbb{P}^1} \times {\mathbb{P}^1} \times {\mathbb{P}^1},\,\,{\mathbb{P}^2} \times {\mathbb{P}^1},\,\,{\mathbb{F}_e} \times {\mathbb{P}^1}$$ and the blowup of ℙ3 at a point, augmenting our earlier work on ℙ3. Most importantly, we treat the boundary cases which are the hardest cases from the point of view of positivity. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurfaces in any threefold admitting a group action with dense orbit.