Derived categories of quadric bundles and moduli stacks of spinor sheaves

1. Problem Statement

The central problem addressed is the geometric identification of the Kuznetsov component ($Ku(Q)$) of the derived category of a flat family of even-dimensional quadrics ($\rho: Q \to S$).

• Context: For a quadric bundle, the derived category $D_{qc}(Q)$ admits a semiorthogonal decomposition where $Ku(Q)$ is the "interesting" residual component, orthogonal to the pullbacks of the base derived category twisted by line bundles.
• Algebraic Description: It is known (via Kuznetsov) that $Ku(Q)$ is equivalent to the derived category of the sheaf of even Clifford algebras of the quadric bundle.
• Geometric Goal: The authors seek to determine when $Ku(Q)$ is equivalent to the twisted derived category of an algebraic space (or scheme) $M$, i.e., $Ku(Q) \simeq D_{qc}(M, \beta)$ for some Brauer class $\beta$.
• Challenges:
  • Previous results were limited to specific cases (e.g., smooth quadrics, pencils, or specific dimensions) or required characteristic $\neq 2$.
  • In higher dimensions or with singular fibers (corank $\ge 2$), the moduli space of spinor sheaves often fails to be separated or well-behaved.
  • The behavior of spinor sheaves in characteristic 2 is subtle and requires a unified treatment.

2. Methodology

The authors employ a combination of derived category techniques, moduli theory, and geometric constructions:

• Hyperbolic Reduction: They utilize the process of hyperbolic reduction along a regular section (or isotropic subbundle). If $\rho: Q \to S$ admits a regular section, one can construct a lower-dimensional quadric bundle $\rho': Q' \to S$. They prove that $Ku(Q) \simeq Ku(Q')$ via a Fourier-Mukai transform. This allows inductive reduction of the dimension of the quadric fibers.
• Moduli Stacks of Spinor Sheaves: Instead of working directly with Clifford algebras, they construct and study the moduli stack of spinor sheaves ($\mathcal{M}$) on the quadric bundle.
  • They define spinor sheaves relative to isotropic subbundles.
  • They analyze the geometry of the stack $\mathcal{M}$, showing it is a $\mathbb{G}_m$-gerbe over an algebraic space (the coarse moduli space).
• Fourier-Mukai Transforms on Stacks: They develop technical tools to verify when a Fourier-Mukai functor induces an equivalence between the derived category of a stack (specifically a $\mathbb{G}_m$-gerbe) and a subcategory of the derived category of a scheme. This involves:
  • Descent theory for Fourier-Mukai functors (checking equivalence fppf locally).
  • Relating fully faithful functors to open immersions of moduli stacks of complexes (generalizing work by Lieblich, Orlov, and others).
• Handling Singularities: They carefully stratify the base $S$ by the corank of the fibers ($S_c = \{s \mid \text{corank}(Q_s) \ge c\}$). They address the specific geometric behavior of spinor sheaves over the corank 2 locus, where fibers are cones over smooth quadrics.

3. Key Contributions and Technical Results

A. General Descent and Equivalence Criteria

• Theorem 1.23: A descent result stating that a Fourier-Mukai functor from a weight component of a $\mathbb{G}_m$-gerbe to a scheme is an equivalence if it is fully faithful and essentially surjective after passing to an fppf cover.
• Theorem 2.8 & 3.12: A generalization of the "Fourier-Mukai open immersion" principle. If a Fourier-Mukai transform is fully faithful, it induces an open immersion between the moduli stacks of complexes. This links the geometry of the moduli stack directly to the categorical equivalence.

B. Hyperbolic Reduction and Spinor Preservation

• Theorem B (6.1): Establishes an $S$-linear equivalence $\Phi: Ku(Q') \to Ku(Q)$ where $Q'$ is the hyperbolic reduction of $Q$ along a regular section.
• Kernel Identification (6.2): Explicitly identifies the kernel of this equivalence as a twist of the ideal sheaf of the incidence correspondence (lines in $Q$ passing through the section).
• Spinor Compatibility (6.3): Proves that $\Phi$ maps dual spinor sheaves of $Q'$ to dual spinor sheaves of $Q$. This is crucial for inductively building the global equivalence.

C. Geometry of the Moduli Stack of Spinors

• Smoothness (7.2): The morphism from the relative Fano scheme of $\ell$-planes to the stack of spinor sheaves is smooth.
• Coarse Moduli Space (7.14): The coarse moduli space $M$ of spinor sheaves behaves as follows:
  • Over $S \setminus S_1$: A double cover (two points).
  • Over $S_1 \setminus S_2$: A single point (non-reduced structure).
  • Over $S_2$ (corank 2): A disjoint union of two $\mathbb{P}^1$s.
• Separation Issue: The coarse moduli space is generally not separated over $S$ if $S_2 \neq \emptyset$, because points over $S \setminus S_2$ can specialize to either component of the $\mathbb{P}^1$ over $S_2$.

D. The Main Geometricity Theorem

Theorem A (7.25): Let $\rho: Q \to S$ be a quadric bundle of relative dimension $2\ell$. Assume:

1. $S_3 = \emptyset$ (no fibers of corank $\ge 3$).
2. $S_2$ contains no weakly associated points of $S$ (ensures hyperbolic reduction is well-behaved).
3. The étale double cover $\tilde{S}_2 \to S_2$ (parameterizing the two families of spinor sheaves over corank 2 fibers) splits (admits a section).

Conclusion: There exists an algebraic space $M_\sigma$ (an open subspace of the coarse moduli space), a Brauer class $\beta \in Br(M_\sigma)$, and an equivalence:
$$\Phi_\sigma: D_{qc}(M_\sigma, \beta) \xrightarrow{\sim} Ku(Q)$$
The space $M_\sigma$ is generically a double cover of $S$, finite over $S \setminus S_1$, and a $\mathbb{P}^1$-bundle over $S_2$.

4. Applications

A. Intersections of Quadrics

The authors apply their results to complete intersections of $m$ quadrics in $\mathbb{P}^{2\ell+1}$.

• Theorem D (8.7): For $m \le 4$, the Kuznetsov component of the intersection is equivalent to the twisted derived category of a curve ($m=2$), surface ($m=3$), or algebraic space ($m=4$).
• Significance: This generalizes the classical Bondal-Orlov result (intersection of two quadrics $\to$ hyperelliptic curve) to arbitrary base fields (including characteristic 2) and higher dimensions.

B. Cubic Fourfolds

• Theorem E (8.10): For a smooth cubic fourfold $X$ containing a plane $P$ over an algebraically closed field $k$ (any characteristic), the residual category $A_X$ is equivalent to $D_{coh}(S, \alpha)$, where $S$ is a K3 surface of degree 2.
• Characteristic 2 Case: The paper provides a new proof for characteristic 2. Specifically, for the Fermat cubic fourfold in characteristic 2, the associated K3 surface $S$ is the supersingular K3 surface of Artin invariant 1. This connects the derived category to the most special K3 surface in that characteristic.

5. Significance

• Generalization: This is the first general geometricity result for Kuznetsov components of higher-dimensional quadric bundles over arbitrary base schemes (including non-reduced and characteristic 2).
• Moduli-Theoretic Approach: It shifts the perspective from algebraic Clifford algebras to the geometry of spinor sheaves, providing a concrete geometric interpretation of the Brauer classes that appear in previous algebraic descriptions.
• Resolution of Singularities: It rigorously handles the "bad" behavior of spinor moduli spaces at corank 2 fibers by introducing the condition of splitting the étale double cover, which is necessary to select a consistent family of spinors and obtain a separated moduli space.
• Unification: It unifies the theory of quadric bundles across all characteristics, resolving open questions regarding the rationality and derived categories of cubic fourfolds in characteristic 2.

In summary, the paper establishes that the "interesting" part of the derived category of a quadric bundle is generated by its spinor sheaves, provided one can consistently choose a family of these sheaves over the singular locus. This choice is encoded by the splitting of a specific étale double cover, leading to a geometric realization of the Kuznetsov component as a twisted derived category.