2022/04/11 by A. I. Štern, Stern, Arié
Mathematics · #14 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2204.04976
openalex publication_date 2022/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a surface cyclic quotient singularity Q∈ 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 Urzúa studied certain irreducible components of the versal deformation space of Y, and within these components, they found one parameter smoothings Y → \mathbbA1 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.