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

For a finite subgroup $G$ of $\mathrm{GL}(2,\mathbb{C})$, 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 ${\mathbb{C}}^2\mathrm{/}G$. In this paper, we show that a resolution $Y$ of ${\mathbb{C}}^2\mathrm{/}G$ 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.