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Special vs Essential

Yukari Ito, Kohei Sato, Yusuke Sato

TL;DR

This paper addresses the dictionary between the geometry of $G$-Hilb$(\mathbb{C}^3)$ for $G=\frac{1}{r}(1,a,r-a)$ and the representation theory of $G\subset GL(3,\mathbb{C})$, extending Reid's recipe to non-Gorenstein threefolds. It constructs an explicit small resolution $\widetilde{G\mathrm{-}Hilb}(\mathbb{C}^3)$ using Kedzierski's toric combinatorics and introduces essential and special characters that encode algebraic and toric data, respectively. The main result establishes a perfect correspondence with SC$(D)=EC(D)\otimes\chi_1$ for every compact divisor $D$ and SC$(G)=EC(1)$, thereby extending the Special McKay correspondence and Reid's recipe to threefold terminal singularities, with a conjectural generalization to arbitrary cyclic $G\subset GL(3,\mathbb{C})$. By bridging $G$-Hilb geometry with nontrivial irreducible representations, the work offers a concrete, computable dictionary for terminal quotient singularities.

Abstract

We show a correspondence between the compact exceptional curves and divisors on $G-{\rm Hilb}(\mathbf{C}^3)$ and some non-trivial irreducible representations of $G \subset GL(n,C)$ which are special (or essential). Moreover, we provide an explicit construction of the small resolution of $G-{\rm Hilb}(\mathbf{C}^3)$ and, using this resolution, we construct a correspondence between special and essential representations. These results are an extension of ``Special McKay correspondence'' and ``Reid's recipe''.

Special vs Essential

TL;DR

This paper addresses the dictionary between the geometry of -Hilb for and the representation theory of , extending Reid's recipe to non-Gorenstein threefolds. It constructs an explicit small resolution using Kedzierski's toric combinatorics and introduces essential and special characters that encode algebraic and toric data, respectively. The main result establishes a perfect correspondence with SC for every compact divisor and SC, thereby extending the Special McKay correspondence and Reid's recipe to threefold terminal singularities, with a conjectural generalization to arbitrary cyclic . By bridging -Hilb geometry with nontrivial irreducible representations, the work offers a concrete, computable dictionary for terminal quotient singularities.

Abstract

We show a correspondence between the compact exceptional curves and divisors on and some non-trivial irreducible representations of which are special (or essential). Moreover, we provide an explicit construction of the small resolution of and, using this resolution, we construct a correspondence between special and essential representations. These results are an extension of ``Special McKay correspondence'' and ``Reid's recipe''.
Paper Structure (5 sections, 8 theorems, 32 equations, 14 figures, 3 tables)

This paper contains 5 sections, 8 theorems, 32 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Small resolutions over $G\mathrm{\text{-}Hilb}(\mathbb{C}^{3})$
  3. A generalization of Special McKay correspondence
  4. Special and Essential representations
  5. Special character for toric terminal $3$-fold

Key Result

Theorem 1.1

Let $G=\frac{1}{r}(1,a,r-a)$ be cyclic with $\gcd(r,a)=1$. Then

Figures (14)

  • Figure 1: Two valleys and socle
  • Figure 2: $G$-igsaw transformations $T_{UL}$ and $T_{UR}$
  • Figure 3: Construction of $G\mathrm{\text{-}Hilb}(\mathbb{C}^{3})$
  • Figure 7: The small resolution $\widetilde{G\mathrm{\text{-}Hilb}(\mathbb{C}^{3})}$ over $G\mathrm{\text{-}Hilb}(\mathbb{C}^{3})$
  • Figure 8: The degree of the vertex
  • ...and 9 more figures

Theorems & Definitions (30)

  • Theorem 1.1: Theorem \ref{['thm:es-sc-terminal']}
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Theorem 2.11 in Nakamura
  • Definition 2.4
  • Lemma 2.5: Lemma 3.13 in Kedzierski
  • Remark 2.6
  • Definition 2.7: Definition 5.8 in Kedzierski
  • Example 2.8
  • Lemma 2.9
  • ...and 20 more