Algebraic hyperbolicity of very general surfaces

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.