An Intersection-Theoretic Approach to Correspondence Problems in Tropical Geometry

The main theme of this thesis is the interplay between algebraic and tropical intersection theory, especially in the context of enumerative geometry. We begin by exploiting well-known results about tropicalizations of subvarieties of algebraic tori to give a simple proof of Nishinou and Siebert’s correspondence theorem for rational curves through given points in toric varieties. Afterwards, we extend this correspondence by additionally allowing intersections with psi-classes. We do this by constructing a tropicalization map for cycle classes on toroidal embeddings. It maps algebraic cycle classes to elements of the Chow group of the cone complex of the toroidal embedding, that is to weighted polyhedral complexes, which are balanced with respect to an appropriate map to a vector space, modulo a naturally defined equivalence relation. We then show that tropicalization respects basic intersection-theoretic operations like intersections with boundary divisors and apply this to the appropriate moduli spaces to obtain our correspondence theorem. Trying to apply similar methods in higher genera inevitably confronts us with moduli spaces which are not toroidal. This motivates the last part of this thesis, where we construct tropicalizations of cycles on fine logarithmic schemes. The logarithmic point of view also motivates our interpretation of tropical intersection theory as the dualization of the intersection theory of Kato fans. This duality gives a new perspective on the tropicalization map; namely, as the dualization of a pull-back via the characteristic morphism of a logarithmic scheme