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The McKay Correspondence for Dihedral Groups: The Moduli Space and the Tautological Bundles

John Ashley Navarro Capellan

TL;DR

This work extends the McKay correspondence to dihedral groups by realizing the maximal resolution Y_{max} of C^2/D_{2n} as a moduli space M_θ of D_{2n}-constellations and establishing a derived-equivalence framework on the associated stack. It shows that every resolution dominated by the maximal one arises as M_θ for some generic stability parameter θ, and develops two parallel lenses—tautological bundles on the stack and tops/socles—to realize the correspondence. The analysis deploys Z_n-Hilb/C^2, the 2nd root stack, and Fourier–Mukai transforms to compare stack versus coarse moduli descriptions, yielding explicit odd/even n distinctions in the tautological structures and top/socle data. The results provide a concrete, stack-centered McKay correspondence for complex reflection dihedral groups, with explicit geometric and categorical constitutions via moduli spaces and derived equivalences.

Abstract

A conjecture in [Ish20] states that for a finite subgroup $G$ of $GL(2; \mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_θ$ of $G$-constellations for some generic stability parameter $θ$ if and only if $Y$ is dominated by the maximal resolution. This paper affirms the conjecture in the case of dihedral groups as a class of complex reflection groups, and offers an extension of McKay correspondence (via [IN1], [IN2], and [Ish02]). To appear in Hiroshima Mathematical Journal.

The McKay Correspondence for Dihedral Groups: The Moduli Space and the Tautological Bundles

TL;DR

This work extends the McKay correspondence to dihedral groups by realizing the maximal resolution Y_{max} of C^2/D_{2n} as a moduli space M_θ of D_{2n}-constellations and establishing a derived-equivalence framework on the associated stack. It shows that every resolution dominated by the maximal one arises as M_θ for some generic stability parameter θ, and develops two parallel lenses—tautological bundles on the stack and tops/socles—to realize the correspondence. The analysis deploys Z_n-Hilb/C^2, the 2nd root stack, and Fourier–Mukai transforms to compare stack versus coarse moduli descriptions, yielding explicit odd/even n distinctions in the tautological structures and top/socle data. The results provide a concrete, stack-centered McKay correspondence for complex reflection dihedral groups, with explicit geometric and categorical constitutions via moduli spaces and derived equivalences.

Abstract

A conjecture in [Ish20] states that for a finite subgroup of , a resolution of is isomorphic to a moduli space of -constellations for some generic stability parameter if and only if is dominated by the maximal resolution. This paper affirms the conjecture in the case of dihedral groups as a class of complex reflection groups, and offers an extension of McKay correspondence (via [IN1], [IN2], and [Ish02]). To appear in Hiroshima Mathematical Journal.
Paper Structure (8 sections, 27 theorems, 80 equations, 4 figures, 9 tables)

This paper contains 8 sections, 27 theorems, 80 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-constellations on $\mathbb{C}^n$
  4. Definitions
  5. The Maximal Resolution
  6. Realizing Blow-ups as Moduli Spaces
  7. The Tautological Bundles and the McKay Correspondence
  8. The McKay Correspondence via the Top and the Socles

Key Result

Theorem 1.2

The maximal resolution $Y_{max}$ of $(\mathbb{C}^2/G, \hat{B})$, defined as the smooth variety which has unique maximal coefficients satisfying the inequality in Definition maxdef, is isomorphic to the quotient variety $\mathbb{Z}_n\operatorname{-Hilb}(\mathbb{C}^2)/\mathbb{Z}_2 := \langle \sigma \r

Figures (4)

  • Figure 3.1: Configuration of Exceptional Divisors and Boundary Divisors on $X_1$ and $Y_1$ for Odd $n$ Case
  • Figure 3.2: Configuration of Exceptional Divisors and Boundary Divisors on $X_1$ and $Y_1$ for Even $n$ Case
  • Figure 3.3: Dual graph of the exceptional divisors and boundary divisors of $f_2$
  • Figure 3.4: The process of flop in relation to the surface containing the exceptional curve

Theorems & Definitions (70)

  • Conjecture 1.1
  • Theorem 1.2: Theorem \ref{['maxemb']}
  • Theorem 1.3: Theorem \ref{['maxmain']}
  • Remark 1
  • Theorem 1.4: Theorem \ref{['mainthm2']}
  • Theorem 1.5: Theorem \ref{['mainthm3']}
  • Remark 2
  • Definition 2.1
  • Example 1
  • Definition 2.2
  • ...and 60 more