## 14T05 Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

- An Intersection-Theoretic Approach to Correspondence Problems in Tropical Geometry (2016)
- 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.

- Moduli spaces of rational tropical stable maps into smooth tropical varieties (2013)
- This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.

- Intersection Theory of the Tropical Moduli Spaces of Curves (2007)
- Tropical geometry is a very new mathematical domain. The appearance of tropical geometry was motivated by its deep relations to other mathematical branches. These include algebraic geometry, symplectic geometry, complex analysis, combinatorics and mathematical biology. In this work we see some more relations between algebraic geometry and tropical geometry. Our aim is to prove a one-to-one correspondence between the divisor classes on the moduli space of n-pointed rational stable curves and the divisors of the moduli space of n-pointed abstract tropical curves. Thus we state some results of the algebraic case first. In algebraic geometry these moduli spaces are well understood. In particular, the group of divisor classes is calculated by S. Keel. We recall the needed results in chapter one. For the proof of the correspondence we use some results of toric geometry. Further we want to show an equality of the Chow groups of a special toric variety and the algebraic moduli space. Thus we state some results of the toric geometry as well. This thesis tries to discover some connection between algebraic and tropical geometry. Thus we also need the corresponding tropical objects to the algebraic objects. Therefore we give some necessary definitions such as fan, tropical fan, morphisms between tropical fans, divisors or the topical moduli space of all n-marked tropical curves. Since we need it, we show that the tropical moduli space can be embedded as a tropical fan. After this preparatory work we prove that the group of divisor classes in v classical algebraic geometry has it equivalence in tropical geometry. For this it is useful to give a map from the group of divisor classes of the algebraic moduli space to the group of divisors of the tropical moduli space. Our aim is to prove the bijectivity of this map in chapter three. On the way we discover a deep connection between the algebraic moduli space and the toric variety given by the tropical fan of the tropical moduli space.