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

TL;DR

The paper analyzes a family of cyclic quotient singularities arising from a specific $G=\mathbb{Z}/r\mathbb{Z}$-action on $\mathbb{C}^n$, showing that the associated $G$-Hilbert schemes $\mathcal{Y}_{s,n,r}$ are smooth, connected, and irreducible, with a projective resolution $\rho_{s,n,r}$ of the quotient $\mathcal{X}_{s,n,r}$ that is generally discrepant for $n\ge 3$. It establishes a toric framework for both the quotient and its Hilbert scheme, computes the central-fiber components in bijection with nontrivial $G$-characters, and realizes a McKay-type correspondence in this setting. The work also furnishes a canonical labeling of exceptional divisors via a Fourier–Mukai functor and realizes $\mathcal{Y}_{s,n,r}$ as a quiver variety, linking geometry, derived-category perspectives, and representation theory. Overall, the results extend the McKay correspondence to a new family of higher-dimensional cyclic quotients and provide a concrete quiver-theoretic and categorical picture of the resolution data.

Abstract

Let $ G = \mathbb{Z}/r\mathbb{Z}$ be the cyclic group of order $r$, and let $\varpi = e^{2πi / r}$ 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 $$ ρ_{s,n,r}:\mathcal{Y}_{s,n,r}\longrightarrow\mathcal{X}_{s,n,r} $$ 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 $ρ_{s,n,r}^{-1}(0)$ 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 $$ Ψ: D^b(\mathrm{Coh}_G(\mathbb{C}^n)) \longrightarrow D^b(\mathrm{Coh}(\mathcal{Y}_{s,n,r})), $$ 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 $ρ_{s,n,r}^{-1}(0)$.

Figures (4)

• Figure 1: The product $\mathcal{Y}_{s,n,r} \times \mathbb{C}^n$ together with its two natural projections.
• Figure 2: McKay quiver $Q=(G,\mathbb{C}^n)$
• Figure 3: Divisor $E_2$ and intersection of two divisors $E_2\cap E_3$
• Figure 4: Fixed point $I^{ij}_{r-2}$

Theorems & Definitions (29)

• Remark 2.1
• Theorem 2.2
• proof
• Definition 2.3
• Proposition 3.1
• Example 3.2
• proof : Proof of Proposition \ref{['Smooth']}
• Theorem 3.3
• proof
• Definition 3.4