## Doctoral Dissertations

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#### Author ORCID Identifier

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

#### AccessType

Open Access Dissertation

dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

2022

February

Jenia Tevelev

Paul Hacking

Eyal Markman

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

COinS