Tropical Mutation Schemes and Examples

Abstract

This thesis provides an introduction to the theory of tropical mutation schemes, and computes explicit examples. Tropical mutation schemes generalize toric geometry. The study of toric varieties is a popular area of algebraic geometry, due to toric varieties' strong combinatorial interpretations. In particular, the characters and one-parameter subgroups of the rank $r$ algebraic torus form a pair of dual lattices of rank $r$, isomorphic to $\mathbb{Z}^r$. We can then construct toric varieties from fans in these lattices, and compactifications of the algebraic torus are parametrized by full dimensional convex polytopes.

A tropical mutation scheme is a finite collections of lattices, equipped with bijective piecewise-linear functions between each pair of lattices, where these functions satisfy certain compatibility conditions. They generalize lattices in the sense that a lattice can be viewed as the trivial tropical mutation scheme. We also introduce the space of points of a tropical mutation scheme, which is the set of functions from a tropical mutation scheme to $\mathbb{Z}$ which satisfy a minimum condition. A priori, the structure of the space of points of a tropical mutation scheme is unknown, but in certain cases can be identified by the elements of another tropical mutation scheme, inducing a dual pairing between the two tropical mutation schemes. When we have a strict dual pairing of tropical mutation schemes, we can sometimes construct an algebra to be a detropicalization of the pairing. In the trivial case, the coordinate ring of the algebraic torus is a detropicalization of a single lattice and its dual. Thus, when we can construct a detropicalization for a non-trivial strict dual pairing, we recover much of the useful combinatorics from the toric case.

This thesis shows that all rank 2 tropical mutation schemes on two lattice charts are autodual, meaning there is a dual pairing between the tropical mutation scheme and its own space of points. Furthermore, we construct a detropicalization for these tropical mutation schemes. We end the thesis by reviewing open questions and future directions for the theory of tropical mutation schemes.