UC San Diego

Mirror symmetry is a curious duality, first noticed by physicists and then excitedly embraced by mathematicians, between certain manifolds (the A-side) and their "mirror" spaces (the B-side). On the A-side, certain counts of objects on a manifold are carried out (the purview of enumerative geometry), while on the B-side, special functions are integrated. Interestingly, the same numbers come out on both sides. This dissertation will consider mirror symmetry on toric surfaces, which are two dimensional varieties with certain convenient combinatorial properties and include many well-known surfaces such as the affine and projective planes and P1xP1. These surfaces are especially suited to being exploited by tropical geometry, which is a form of algebraic geometry over the "tropical semi-ring." Dramatically, key information about both sides of mirror symmetry on toric surfaces can be gleaned from the tropics. The mirror symmetry of a certain subclass of toric varieties, those that are Fano, has seen much progress. In the case of toric surfaces, being Fano means that tropical geometry can "see" all the curves of the surface; this makes enumerative problems more straight- forward. This dissertation attempts a generalization to non-Fano toric surfaces, where some curves are "hidden" from the viewpoint of tropical geometry. The generalization is accomplished following the Gross-Seibert Program wherein singularities are added to the tropical picture in order to "pull" these curves into view. We verify these methods by carrying out explicit calculations on the second Hirzebruch surface, which is a non-Fano toric variety