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Derived categories of quadric bundles and moduli stacks of spinor sheaves

Raymond Cheng, Noah Olander

Abstract

We prove that the Kuznetsov component of a flat family of even-dimensional quadrics of corank at most 2 is equivalent to the twisted derived category of an algebraic space whenever: (i) the open subset of the base over which the quadrics has corank at most 1 is scheme-theoretically dense; and (ii) a certain étale double cover of the closed complement admits a section. This provides the first general geometricity result for Kuznetsov components of higher dimensional quadrics, thereby generalizing works of Kapranov, Bondal, Orlov, Kuznetsov, Moschetti, Xie, and others. Our main tool is the moduli stack of spinor sheaves on a family of quadrics, which we define and study in detail. In the situation of our main result, we produce an open substack which is a $\mathbf{G}_m$-gerbe, and show that the associated twisted derived category is equivalent to the Kuznetsov component of the family of quadrics, thereby providing a geometric interpretation of the Brauer classes appearing in previous works.

Derived categories of quadric bundles and moduli stacks of spinor sheaves

Abstract

We prove that the Kuznetsov component of a flat family of even-dimensional quadrics of corank at most 2 is equivalent to the twisted derived category of an algebraic space whenever: (i) the open subset of the base over which the quadrics has corank at most 1 is scheme-theoretically dense; and (ii) a certain étale double cover of the closed complement admits a section. This provides the first general geometricity result for Kuznetsov components of higher dimensional quadrics, thereby generalizing works of Kapranov, Bondal, Orlov, Kuznetsov, Moschetti, Xie, and others. Our main tool is the moduli stack of spinor sheaves on a family of quadrics, which we define and study in detail. In the situation of our main result, we produce an open substack which is a -gerbe, and show that the associated twisted derived category is equivalent to the Kuznetsov component of the family of quadrics, thereby providing a geometric interpretation of the Brauer classes appearing in previous works.
Paper Structure (54 sections, 80 theorems, 276 equations)

This paper contains 54 sections, 80 theorems, 276 equations.

Table of Contents

  1. Derived categories of algebraic stacks
  2. Relative generators
  3. Relative generators of subcategories
  4. Fourier--Mukai transforms
  5. Gerbes
  6. Trivial gerbes
  7. Relatively perfect objects
  8. Relative orthogonal categories
  9. Moduli of complexes
  10. Glueability and simplicity
  11. Stack of complexes
  12. Autoequivalences
  13. Fourier--Mukai open immersion
  14. Residual category to an exceptional collection
  15. Relatively exceptional objects
  16. ...and 39 more sections

Key Result

Theorem A

Let $\rho \colon Q \to S$ be a quadric bundle of relative dimension $2\ell$. Assume that: Then there exists an algebraic space $M$, a proper morphism $M \to S$ which is finite of degree $2$ away from $S_2$, a Brauer class $\beta \in \operatorname{Br}(M)$, and an $S$-linear equivalence

Theorems & Definitions (162)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • proof
  • Lemma 1.3
  • proof
  • proof
  • proof
  • ...and 152 more