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Author ORCID Identifier

https://orcid.org/0000-0002-1361-0153

AccessType

Open Access Dissertation

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2022

Month Degree Awarded

February

First Advisor

Jenia Tevelev

Second Advisor

Paul Hacking

Third Advisor

Eyal Markman

Fourth Advisor

David Barrington

Subject Categories

Algebraic Geometry

Abstract

In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm \cite{M02}. We study one parameter smoothings in these components that admit an anticanonical model with canonical but non-terminal singularities with the goal of classifying them completely. We identify certain class of ``diagonal" smoothings where the total space is a toric threefold and we construct the anticanonical model explicitly using the toric MMP.

DOI

https://doi.org/10.7275/27487488.0

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