Ishii's conjecture and Bridgeland stability conditions for dihedral reflection groups

TL;DR

This paper proves Ishii's conjecture for dihedral reflection groups using Bridgeland stability conditions, employing derived McKay correspondence and geometric constructions on root stacks.

Contribution

It offers a new proof of Ishii's conjecture for dihedral groups through Bridgeland stability conditions and geometric methods, expanding understanding of stability conditions in this context.

Findings

Proof of Ishii's conjecture for dihedral groups

Reduction to geometric construction on root stacks

Application of derived McKay correspondence

Abstract

We provide a new proof of Ishii's conjecture for any dihedral reflection group $G\subset GL_2(\mathbb{C})$ from the viewpoint of Bridgeland stability conditions. Our strategy is to reduce the problem, via the derived McKay correspondence, to a geometric construction of Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor.