Families of $G$-constellations over Resolutions of Quotient Singularities

Let $G$ be a finite subgroup of $\operatorname{GL}_{n}(\mathbb{C})$. A study is made of the ways in which resolutions of the quotient space $\mathbb{C}^{n}/G$ can parametrise $G$-constellations, that is, $G$-regular finite length sheaves. These generalise $G$-clusters, which are used in the McKay correspondence to construct resolutions of orbifold singularities. A complete classification theorem is achieved, in which all the natural families of $G$-constellations are shown to correspond to certain finite sets of $G$-Weil divisors, which are a special sort of rational Weil divisor, introduced in this paper. Moreover, it is shown that the number of equivalence classes of such families is always finite. Explicit examples are computed throughout using toric geometry.