Notes from a family of smooth $G$-Hilbert schemes

Abstract

Let $ G = \mathbb{Z}/r\mathbb{Z}$ be the cyclic group of order $r$, and let $\varpi = e^{2\pi i / r}$ denote a primitive $r$ th root of unity. Consider the action of $G$ on $\mathbb{C}^n$ via the embedding $$ \varphi : G \hookrightarrow GL_n(\mathbb{C}), \qquad \varphi(1) = \mathrm{diag}\!\bigl( \underbrace{\varpi, \dots, \varpi}_{s},\, \underbrace{\varpi^{-1}, \dots, \varpi^{-1}}_{n - s} \bigr), $$ where $0 < s < n $. Denote the corresponding GIT quotient by $$ \mathcal{X}_{s,n,r} = \mathrm{Spec}\bigl((\mathbb{C}[z_1,\dots,z_n])^G\bigr). $$ Then the varieties $\mathcal{X}_{s,n,r}$ is a cyclic quotient singularity of type $\tfrac{1}{r}\bigl(\underbrace{1,\dots,1}_{s}, \underbrace{-1,\dots,-1}_{n-s}\bigr)$. We show that the associated $G$-Hilbert schemes $\mathcal{Y}_{s,n,r}$ are smooth, connected, and irreducible. The natural morphism $$ \rho_{s,n,r}:\mathcal{Y}_{s,n,r}\longrightarrow\mathcal{X}_{s,n,r} $$ is a projective resolution of $\mathcal{X}_{s,n,r}$, discrepant for $n \ge 3$. We establish that the irreducible components of the central fiber $\rho_{s,n,r}^{-1}(0)$ are in bijection with the nontrivial characters of \(G\), thereby realizing the classical McKay correspondence in this family of examples. Finally, we describe a canonical choice of this bijection via the Fourier--Mukai type functor $$ \Psi : D^b(\mathrm{Coh}_G(\mathbb{C}^n)) \longrightarrow D^b(\mathrm{Coh}(\mathcal{Y}_{s,n,r})), $$ by showing that, for each nontrivial irreducible representation of $G$, the corresponding skyscraper sheaf is mapped to a complex whose $0^{\text{th}}$ cohomology is supported on a unique irreducible component of the central fiber $\rho_{s,n,r}^{-1}(0)$.