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Title: | Tropical Mutation Schemes and Examples |
Authors: | Cook, Adrian |
Advisor: | Harada, Megumi |
Department: | Mathematics and Statistics |
Keywords: | Algebraic Geometry;Combinatorics |
Publication Date: | 2023 |
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. |
URI: | http://hdl.handle.net/11375/28464 |
Appears in Collections: | Open Access Dissertations and Theses |
Files in This Item:
File | Description | Size | Format | |
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Cook_Adrian_G_2023-04_MSc.pdf | 571.6 kB | Adobe PDF | View/Open |
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