$G$-constellations and the maximal resolution of a quotient surface singularity

For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_\theta$ of $G$-constellations. It depends on the stability parameter $\theta$ and if $\theta$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_\theta$ for some generic $\theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.