Dongsoo Shin, Deformations of Sandwiched Surface Singularities and the Semistable Minimal Model Program

A sandwiched surface singularity is a rational surface singularity that admits a birational map to the complex projective plane. de Jong and van Straten [Duke Math J 1998] prove that deformations of sandwiched surface singularities are induced from special deformations of germs of plane curve singularities (called picture deformations). On the other hand, Kollár and Shepherd-Barron [Invent Math 1988] conjecture that deformations of any rational surface singularities are described by special partial modifications (called P-modifications). So there are two different descriptions of deformations of sandwiched surface singularities. We provide a way to find a correspondence between picture deformations and P-resolutions (roughly speaking, normal P-modifications with mild singularities) using the semistable minimal model program for complex 3-folds. As applications, 1. we provide correspondence between various deformation theories of cyclic quotient surface singularities 2. We give proofs of Kollár conjecture for certain weighted homogeneous surface singularities.