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- W2759031908 abstract "The main theme of this thesis is the interplay between algebraic and tropical intersectiontheory, especially in the context of enumerative geometry. We begin by exploitingwell-known results about tropicalizations of subvarieties of algebraic tori to give asimple proof of Nishinou and Siebert’s correspondence theorem for rational curvesthrough given points in toric varieties. Afterwards, we extend this correspondenceby additionally allowing intersections with psi-classes. We do this by constructinga tropicalization map for cycle classes on toroidal embeddings. It maps algebraiccycle classes to elements of the Chow group of the cone complex of the toroidalembedding, that is to weighted polyhedral complexes, which are balanced with respectto an appropriate map to a vector space, modulo a naturally defined equivalence relation.We then show that tropicalization respects basic intersection-theoretic operations likeintersections with boundary divisors and apply this to the appropriate moduli spacesto obtain our correspondence theorem.Trying to apply similar methods in higher genera inevitably confronts us with modulispaces which are not toroidal. This motivates the last part of this thesis, where weconstruct tropicalizations of cycles on fine logarithmic schemes. The logarithmic point ofview also motivates our interpretation of tropical intersection theory as the dualizationof the intersection theory of Kato fans. This duality gives a new perspective on thetropicalization map; namely, as the dualization of a pull-back via the characteristicmorphism of a logarithmic scheme." @default.
- W2759031908 created "2017-10-06" @default.
- W2759031908 creator A5031332984 @default.
- W2759031908 date "2016-11-10" @default.
- W2759031908 modified "2023-09-27" @default.
- W2759031908 title "An Intersection-Theoretic Approach to Correspondence Problems in Tropical Geometry" @default.
- W2759031908 hasPublicationYear "2016" @default.
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