Moduli of \(G\)-constellations and crepant resolutions II: the Craw-Ishii conjecture

For any given finite subgroup \(G\subset SL_3(\mathbb{C})\), we show that every projective crepant resolution \(X\) of the quotient variety \(\mathbb{C}^3/G\) is isomorphic to the moduli space of \(\theta\)-stable \(G\)-constellations for a generic stability condition \(\theta\), as conjectured by Craw and Ishii. We also show that generators of the Cox ring of \(X\) can be obtained from semi-invariants for representations of the McKay quiver of \(G\).