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

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.

v3, 26 pages. Section 2.4 on Ext and base-change is expanded. The proof of Lemma 2.9 is rewritten. Added remarks (6.5, 6.8 and 6.9), in particular relating to some added references concerning the Homological MMP. Some further additions and changes. The article is published (Adv. Math. 340 (2018), 1108--1140) with open access