A categorical flop in dimension one

1. Problem Statement

The paper addresses the non-uniqueness of crepant resolutions in algebraic geometry through the lens of derived categories. While the Bondal–Orlov and Kawamata conjectures suggest that different crepant resolutions of a singularity are derived equivalent, the behavior of categorical resolutions for irrational singularities (specifically nodal curves) is less understood.

The central problem is to understand the categorical structure arising from the "flop" between different non-commutative resolutions of a nodal curve $X = \{xy=0\} \subset \mathbb{A}^2$. Specifically, the author investigates:

• How the categorical structure of a 3-fold flop (like the Atiyah flop) manifests in dimension one.
• The precise nature of the "flop-flop" autoequivalence in this context.
• The distinction between weakly crepant and strongly crepant categorical resolutions, and whether a minimal resolution exists that satisfies a stronger condition than weak crepancy but weaker than strong crepancy.

2. Methodology

The author employs a multi-faceted approach combining non-commutative algebra, derived category theory, and mirror symmetry (Landau–Ginzburg models):

• Non-Commutative Resolutions: The study focuses on the Auslander order $A = H^0 \text{End}_X(\mathcal{O}_X \oplus \pi_* \mathcal{O}_{\tilde{X}})$, where $\pi: \tilde{X} \to X$ is the normalization of the nodal curve. The derived category $D^b(A)$ serves as a categorical resolution.
• Partial Resolutions: The author analyzes "smaller" resolutions $D^b(A_x)$ and $D^b(A_y)$ obtained by contracting specific exceptional objects within $D^b(A)$. These correspond to partial versions of the Auslander order.
• Spherical Twists: The relationship between these categories is analyzed using spherical functors and spherical twists. The author identifies the "flop" as a derived equivalence induced by a spherical twist around a specific object.
• Landau–Ginzburg (LG) Models: The paper utilizes the "exoflop" construction (following Aspinwall) to interpret these algebraic categories as categories of B-branes on partial compactifications of LG models (specifically matrix factorizations of $xyz$ on $\mathbb{A}^3$).
• Homological Mirror Symmetry (A-side): The results are cross-verified using the wrapped Fukaya category of a pair-of-pants surface with "stops" (boundary conditions), providing a symplectic geometric interpretation of the categorical contractions.

3. Key Contributions

A. The Categorical Flop in Dimension One

The paper demonstrates that the diagram of categories for the nodal curve mimics the structure of a 3-fold flop:
$$D^b(A) \xrightarrow{\Phi} D^b(A_x) \quad \text{and} \quad D^b(A) \xrightarrow{\Phi'} D^b(A_y)$$

• $D^b(A)$ acts as the "roof" (analogous to the blow-up of the node in 3-folds).
• $D^b(A_x)$ and $D^b(A_y)$ are the two "flopped" resolutions.
• The composition of the flop and its inverse ($\Phi \circ \Phi^{-1}$) on $D^b(A_x)$ is an autoequivalence given by a spherical twist around a 3-spherical object.

B. Introduction of "Fair Crepancy"

The author identifies a gap in the existing definitions of crepancy:

• Weak Crepancy: The relative Serre functor preserves the kernel of the resolution. (Satisfied by all resolutions discussed).
• Strong Crepancy: The relative Serre functor is the identity. (Failed by these resolutions; they are not Calabi–Yau).
• Fair Crepancy (New Definition): A resolution is fairly crepant if the kernel subcategory $\ker(\pi^*)$ is Calabi–Yau with respect to the relative Serre functor. Specifically, the Serre functor restricted to the kernel is a homological shift $[n]$.
  • The "smaller" resolutions ($D^b(A_x)$) are fairly crepant (index 2).
  • The "roof" resolution ($D^b(A)$) is weakly crepant but not fairly crepant.

C. Geometric Interpretation via LG Models

The paper provides a geometric realization of these algebraic structures:

• The categorical resolutions correspond to matrix factorizations of $xyz$ on partial compactifications of $\mathbb{A}^3$.
• The "roof" corresponds to a compactification with an orbifold point ($\mathbb{P}^1_{1:2}$), while the "smaller" resolutions correspond to removing specific orbifold points or exceptional objects.
• The failure of strong crepancy is explained by the non-triviality of the canonical bundle at the orbifold point or the global toric compactification.

4. Key Results

1. Equivalence of Constructions: The paper proves that three distinct constructions—the Auslander order approach ([BD]), the categorical resolution approach ([KL]), and the LG-model exoflop approach ([Asp])—all yield the same derived category $D^b(A)$.
2. Spherical Objects:
   • In $D^b(A)$, the kernel is generated by two exceptional objects $E_x, E_y$.
   • In the smaller resolution $D^b(A_x)$, the kernel is generated by a single object $F_x = \text{cone}(E_x[-1] \to E_y[1])$.
   • Crucially, $F_x$ is a 3-spherical object (despite the base curve being 1-dimensional), which prevents the resolution from being strongly crepant.
3. Index of Crepancy: The author defines an "index of crepancy" $n$ such that $S|_{\ker} = [n]$.
   • For the nodal curve ($\dim X = 1$), the index is $2$.
   • Table 1 in the paper generalizes this: for a node in dimension $d$, the index is $2-d$ (or related parity shifts). The index is 0 only when geometric crepant resolutions exist (e.g., $d=2, 3$).
4. A-side Interpretation: The categorical flop corresponds to "swapping stops" in the wrapped Fukaya category of a pair-of-pants surface. The flop-flop autoequivalence corresponds to a "wrap-once" autoequivalence (a Dehn twist) of the stopping divisor.

5. Significance

• Unification of Dimensions: The paper successfully extends the theory of categorical flops, previously understood primarily in dimension 3 (Atiyah flop), to dimension 1. It shows that the "flop" phenomenon is intrinsic to the categorical structure of singularities, not just a high-dimensional geometric feature.
• Refinement of Minimality: By introducing fair crepancy, the author proposes a criterion that likely identifies minimal categorical resolutions. The result suggests that while strong crepancy (Calabi–Yau condition) is too strict for irrational singularities, "fair crepancy" (Calabi–Yau kernel) captures the essence of minimality.
• Bridge Between Fields: The work creates a robust link between non-commutative algebra (Auslander algebras), derived categories, Landau–Ginzburg models, and symplectic topology (Fukaya categories), demonstrating that these different mathematical languages describe the same underlying "flop" structure.
• Conjectural Framework: The paper lays the groundwork for a conjecture that fairly crepant resolutions are the unique minimal categorical resolutions for isolated singularities, offering a new direction for the classification of resolutions of irrational singularities.

In summary, Crossley provides a comprehensive analysis of how a nodal curve admits a "categorical flop" structure, defines a new intermediate condition (fair crepancy) to characterize minimal resolutions, and validates these findings through geometric and symplectic dualities.