Deformations of rational surface singularities and reflexive modules with an application to flops

TL;DR

This paper explores how deformations of rational surface singularities paired with reflexive modules relate to their resolutions and flops, providing insights into conjectures on small resolutions and birational transformations.

Contribution

It establishes a connection between deformations of singularities with reflexive modules and their partial resolutions, advancing understanding of flops and small resolutions.

Findings

Deformations of (singularity, module) correspond to deformations of partial resolutions.

Results support recent conjectures on small resolutions and flops.

Provides a framework linking singularity deformations to resolution transformations.

Abstract

Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.