Reid's Recipe for Crepant Resolutions

• Reid's Recipe is a combinatorial procedure that assigns group-theoretic marks to exceptional divisors and curves in crepant resolutions of toric Gorenstein singularities.
• It leverages the McKay correspondence, dimer models, and derived-categorical techniques to translate representation theory into geometric insights.
• The method underpins the construction of moduli spaces and derived equivalences, guiding modern approaches in noncommutative resolutions and wall-crossing phenomena.

Reid's recipe is a combinatorial procedure originally formulated to relate the geometry of crepant resolutions of quotient singularities—particularly those arising from finite subgroups of $\SL(3, \CC)$—to the representation theory of the group via the McKay correspondence. Its generalizations and derived-categorical refinements underpin much of modern progress in the study of toric Gorenstein singularities, dimer models, and noncommutative resolutions. The recipe enables the explicit marking of exceptional divisors and curves by group-theoretic data and governs the construction of moduli spaces and derived equivalences between geometric and algebraic categories.

1. Core Definitions and Setup

Let $G\subset \SL(3,\CC)$ be a finite (typically abelian or explicitly presented non-abelian) subgroup. The singularity $X = \CC^3/G$ is an affine Gorenstein 3-fold toric singularity. Reid's recipe forms a bridge between the geometry of crepant resolutions $Y\to X$—classically realized by $G$-Hilbert schemes—and the representation theory of $G$, particularly via the McKay quiver and its moduli of representations (Cautis et al., 2012, Wormleighton, 2019).

• Dimer models: More generally, consistent dimer models on $T^2$ encode toric singularities via an oriented quiver $Q$ whose Jacobian algebra $A = kQ/(\partial_a W)$ for a superpotential $W$ is a noncommutative crepant resolution of its center. The vertex set $Q_0$ indexes both combinatorial data and line bundles on the distinguished crepant resolution $Y$ (Bocklandt et al., 2013).
• Quiver moduli: For abelian $G$, the moduli space $Y = G\text{-Hilb}(\CC^3)$ parametrizes $\theta$-stable quiver representations, where $\theta$ is a stability parameter in King’s chamber $\Theta$ satisfying $\sum_{i \in Q_0}\theta_i = 0$ (Wormleighton, 2019).

2. The Marking Process: Exceptional Curves and Divisors

Reid’s recipe prescribes a marking of irreducible components of the exceptional locus $E \subset Y$ by irreducible representations of $G$. The marking process is combinatorial but reflects subtle geometric data (Cautis et al., 2012, Wormleighton, 2019, Celis, 2021):

• Curves: Each exceptional curve $C$ in $Y$—typically a rational curve or a $\mathbb{P}^1$, corresponding to an edge $\sigma$ in the triangulation of the junior simplex—receives a mark from the character $\chi$ of $G$ acting on a binomial $M_1/M_2$, determined by the primitive normal vector of $\sigma$. The collection of all such marked edges forms $\chi$-chains.
• Divisors: Every interior vertex of the triangulation gives an exceptional divisor $D$, marked by combinatorial rules: trivalent ($\mathbb{P}^2$) vertices are labeled by $\chi^2$; higher valency cases depend on tensor products or combinations of characters, as dictated by the configuration of incident curves and divisors (Wormleighton, 2019).

For non-abelian cases, the marking generalizes via the action on G-cluster socles and G-igsaw pieces, determined by explicit basis choices for tautological bundles on affine covers of $Y$ (Celis, 2021). The age of a conjugacy class (sum of shifts in eigenvalues of group elements) precisely enumerates one-dimensional (age 1) and two-dimensional (age 2) components of $E$.

3. Derived McKay Correspondence and Fourier–Mukai Equivalences

The modern formulation of Reid’s recipe is intrinsically derived-categorical:

• There exists a Fourier–Mukai equivalence between the bounded derived category $D^G(\CC^3)$ and $D^b(\Coh(Y))$, realized via kernels built from universal G-clusters or tautological bundles $\mathcal{T}$ (Cautis et al., 2012). Explicitly,

$\Phi = \mathbb{R}\pi_{\CC^3 *}\bigl( \mathcal{M} \otimes^{L} \pi_Y^*(-) \bigr)$

and its quasi-inverse involves the dual family $\widetilde{\mathcal{M}}$.

• Vertex simples and pure sheaves: Under the equivalence, vertex simple modules (for $A$ in the dimer setting) or skyscrapers $\mathcal{O}_0 \otimes \chi$ map to pure sheaves supported on marked exceptional loci, up to cohomological shift:
  • Marking a divisor $E$: $$\Psi(\mathcal{O}_0 \otimes \chi) \cong \mathcal{L}_\chi^{-1} \otimes \mathcal{O}_E$$ in degree $0$.
  • Marking a chain or tree: $[\cdots][1]$-shifted sheaf with support on unions of divisors or vector bundle over chains.
  • For the trivial character, the image is the dualizing complex of the full exceptional fibre (Cautis et al., 2012, Bocklandt et al., 2013).

4. Wall Structures and Stability Chambers

The moduli-theoretic aspect relies on the structure of the GIT stability space:

• Types of walls: The chamber of stability parameters $\Theta$ admits three types of bounding walls (Bocklandt et al., 2013, Wormleighton, 2019):
  • Type 0: Flop or bundle-contraction; wall locus is a union of compact torus-invariant divisors.
  • Type I: Small contraction of $(–1,–1)$ curve $\ell$; locus is $\ell$.
  • Type III: Contraction of a Hirzebruch surface $F_n$.
• Inequalities: Each kind of wall corresponds to explicit linear inequalities:
  • Curve-based: $\sum_{\rho \in \mathcal{G}(C)}\theta_\rho > 0$ for chains of marked characters.
  • Divisor-based: $\theta_\psi > 0$ for divisors marked by $\psi$.
  • The structure of genuine walls is determined by combinatorial analysis of the triangulation and the marking process (Wormleighton, 2019).
• Recipe implementation: For each nonzero vertex (or nontrivial character) $i \neq 0$, one tests if its associated hyperplane is a wall by checking the existence of a module with socle containing $S_i$. If so, the marking prescribes pure sheaf support and cohomology vanishing outside degree $0$.

5. Explicit Examples

Conifold Dimer (Bocklandt et al., 2013)

For the conifold case,

• Quiver: two vertices, four arrows, potential $W$.
• Algebra: $A = \mathrm{Jac}(Q, W)$, $X = \{xy = zw\} \subset \CC^4$.
• Resolution: $Y$ is the total space of $$\mathcal{O}_{\mathbb{P}^1}(-1) \oplus \mathcal{O}_{\mathbb{P}^1}(-1)$$, equipped with tautological bundles $\mathcal{L}_0, \mathcal{L}_1$.
• Recipe: Only one wall $H_1$ occurs; $\Psi(S_1) \cong \mathcal{O}_\ell$ ($\ell \cong \mathbb{P}^1$ exceptional fibre), $\Psi(S_0) \cong \omega_E[3]$.

Non-Abelian Examples (Celis, 2021)

• Dihedral group $D_{5,2} \subset \SL(3, \CC)$: 5 compact divisors, 11 rational curves, explicit marking via affine covers and tautological sections; Picard and cohomology structure enumerated via Chern classes.
• Trihedral group of order 39: 2 compact divisors, 6 rational curves; marking via monomial basis, tautological bundles, and socle computations.

6. Generalizations and Impact

Reid's recipe generalizes the classical McKay correspondence, which in two dimensions matches irreducible representations with $(-1)$-curves and their minus-one line bundles under derived equivalence (Cautis et al., 2012). In three dimensions, it orchestrates a more nuanced categorification involving chains, trees, and higher cohomological shifts. Key impacts:

• Provides an explicit algorithmic marking of exceptional subvarieties for both abelian and non-abelian $G$.
• Specifies basis generators for Picard group and cohomological invariants $H^2(Y)$, $H^4(Y)$, yielding an integral realization of the McKay correspondence.
• Underpins the study of derived equivalences and wall-crossing phenomena in birational geometry and moduli theory.

A plausible implication is that the combinatorial and derived-categorical methodology of Reid's recipe will continue to inform the structure of crepant resolutions and noncommutative geometry, extending to cluster categories and further generalizations in higher dimensions (Bocklandt et al., 2013).

7. Summary and Directions for Further Research

Reid's recipe establishes a detailed correspondence between representation-theoretic and geometric data for crepant resolutions of toric Gorenstein singularities. Its dimer-model generalization extends the paradigm to all consistent dimer models, subsuming the classical and derived McKay correspondences. The marking of walls in stability space, the derived functor formulae, and the explicit combinatorial rules position the recipe as a foundational tool in algebraic and noncommutative geometry.

Further directions include:

• Clarification of moduli-theoretic interpretations in non-abelian and higher-rank cases.
• Extension of combinatorial techniques such as CT-subdivisions, sink–source graphs, and “unlocking” procedures in the study of wall-crossing and birational models.
• Exploration of connections to cluster categories, virtual bundle invariants, and categorical birational geometry (Bocklandt et al., 2013, Cautis et al., 2012, Wormleighton, 2019, Celis, 2021).