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 ${\mathbb{P}}^3$ at a point, augmenting our earlier work on ${\mathbb{P}}^3$. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurface in any variety admitting a group action with dense orbit.

Comments welcome!