Reid's recipe and derived categories

TL;DR

This paper proves two conjectures linking the geometric McKay correspondence with Reid's recipe for certain singularities, using derived categories and toric combinatorics to deepen understanding of the correspondence.

Contribution

It provides a categorification of Reid's recipe by connecting derived category techniques with toric combinatorics in the context of the McKay correspondence.

Findings

Proved two conjectures relating McKay correspondence and Reid's recipe

Established a link between Fourier-Mukai transforms and toric combinatorics

Enhanced understanding of singularities in SL3(C) quotients

Abstract

We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of computing a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reid's recipe', effectively providing a categorification of the latter.