Natural $G$-constellation families

Abstract

Let $G$ be a finite subgroup of ${\mathrm{GL}}_n(\mathbb{C}).G$-constellations are a scheme-theoretic generalization of orbits of $G$ in ${\mathbb{C}}^n$. We study flat families of $G$-constellations parametrised by an arbitrary resolution of the quotient space ${\mathbb{C}}^n/G$. We develop a geometrical naturality criterion for such families, and show that, for an abelian $G$, the number of equivalence classes of these natural families is finite. The main intended application is the derived McKay correspondence.