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 $$\phi :G\hookrightarrow G{L}_n(\mathbb{C}),\phi (1)=\mathrm{diag}(\underset{s}{\underbrace{\varpi ,\dots ,\varpi }},\underset{n-s}{\underbrace{{\varpi }^{-1},\dots ,{\varpi }^{-1}}}),$$ where $0<s<n$. Denote the corresponding GIT quotient by $${\mathcal{X}}_{s,n,r}=\mathrm{Spec}((\mathbb{C}[z_1,\dots ,z_n]{)}^G).$$ Then the varieties ${\mathcal{X}}_{s,n,r}$ is a cyclic quotient singularity of type $\frac{1}{r}(\underset{s}{\underbrace{1,\dots ,1}},\underset{n-s}{\underbrace{-1,\dots ,-1}})$. We show that the associated $G$-Hilbert schemes ${\mathcal{Y}}_{s,n,r}$ are smooth, connected, and irreducible. The natural morphism $${\rho }_{s,n,r}:{\mathcal{Y}}_{s,n,r}⟶{\mathcal{X}}_{s,n,r}$$ 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 $$\Psi :D^b({\mathrm{Coh}}_G({\mathbb{C}}^n))⟶D^b(\mathrm{Coh}({\mathcal{Y}}_{s,n,r})),$$ 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 ${\rho }_{s,n,r}^{-1}(0)$.