  Algebraic hyperbolicity serves as a bridge between differential geometry and
algebraic geometry. Generally, it is difficult to show that a given projective
variety is algebraically hyperbolic. However, it was established recently that
a very general surface of degree at least five in projective space is
algebraically hyperbolic. We are interested in generalizing the study of
surfaces in projective space to surfaces in smooth projective toric threefolds
with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the
combinatorial description of smooth projective toric threefolds with Picard
rank 2 and 3. We then use Haase and Ilten's method of finding algebraically
hyperbolic surfaces in toric threefolds. As a result, we determine many
algebraically hyperbolic surfaces in each of these varieties.

Algebraic Geometry