Deformations of sandwiched surface singularities and the minimal model program

TL;DR

This paper explores the relationships between different deformation theories of rational surface singularities, providing explicit methods to translate between them and applying the minimal model program to prove Kollár's conjecture for certain cases.

Contribution

It introduces a method to connect three deformation theories of rational surface singularities and applies the minimal model program to prove Kollár's conjecture for sandwiched surface singularities.

Findings

Established explicit correspondence between deformation theories.

Provided a method to translate deformations across theories.

Proved Kollár's conjecture for various sandwiched surface singularities.

Abstract

We investigate the correspondence between three theories of deformations of rational surface singularities: de Jong and van Straten's picture deformations, Koll\'ar's P-resolutions, and Pinkham's smoothings of negative weights. We provide an explicit method for obtaining, from a given deformation in one theory, deformations in other theories that parameterize the same irreducible components of the deformation space of the singularity. We employ the semi-stable minimal model program significantly for this purpose. We prove Koll\'ar conjecture for various sandwiched surface singularities as an application.