Algebraic hyperbolicity of very general surfaces

TL;DR

This paper completes the classification of algebraically hyperbolic surfaces in certain toric threefolds, introduces techniques for proving hyperbolicity, and extends results to hypersurfaces with group actions.

Contribution

It finalizes the classification of hyperbolic surfaces in specific toric threefolds and develops methods applicable to hypersurfaces with dense group actions.

Findings

Complete classification for several toric threefolds.

Development of techniques for proving algebraic hyperbolicity.

Extension of results to hypersurfaces with dense group actions.

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.