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 [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 $\mathcal{Y}\to {\mathbb{A}}^1$ 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 [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.