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ANTICANONICAL MODELS OF SMOOTHINGS OF CYCLIC QUOTIENT SINGULARITIES

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.
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