On the Craw-Ishii conjecture

TL;DR

This paper explores a conjecture extending Craw and Ishii's result on minimal models of quotient singularities from SL_3(C) to GL_3(C), proving it for certain abelian groups.

Contribution

It formulates a new conjecture for finite groups in GL_3(C) and proves it in specific cases involving abelian groups.

Findings

Conjecture extends Craw-Ishii result to GL_3(C) under smoothness.

Proved the conjecture for some abelian groups.

Highlights conditions for minimal models to correspond to moduli spaces.

Abstract

Craw and Ishii proved that for a finite abelian group G in SL_3(C) every (projective) relative minimal model of C^3/G is isomorphic to the fine moduli space of \theta-stable G-constellations for some GIT parameter \theta. In this article, we conjecture that the same is true for a finite group G in GL_3(C) if a relative minimal model Y of X=C^3/G is smooth. We prove this for some abelian groups.