Algebraic hyperbolicity for surfaces in smooth complete toric threefolds with Picard rank 2 and 3

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. In this thesis, we are interested in generalizing the study of surfaces in projective space to surfaces in toric threefolds with Picard rank 2 or 3. Towards this goal, we explored the combinatorial description of toric threefolds with Picard rank 2 and 3 by following the works of Kleinschmidt and Batyrev. Then we used the method of finding algebraically hyperbolic surfaces in toric threefolds by Haase and Ilten. As a result, we were able to determine several algebraically hyperbolic surfaces in each of these varieties.