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Author ORCID Identifier

https://orcid.org/0000-0001-6861-4242

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2020

Month Degree Awarded

February

First Advisor

Paul Hacking

Subject Categories

Algebraic Geometry

Abstract

In this thesis we identify certain cluster varieties with the complement of a union of closures of hypertori in a toric variety. We prove the existence of a compactification $Z$ of the Fock--Goncharov $\mathcal{X}$-cluster variety for a root system $\Phi$ satisfying some conditions, and study the geometric properties of $Z$. We give a relation of the cluster variety to the toric variety for the fan of Weyl chambers and use a modular interpretation of $X(A_n)$ to give another compactification of the $\mathcal{X}$-cluster variety for the root system $A_n$.

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