Derived categories of quadric bundles and moduli stacks of spinor sheaves

TL;DR

This paper establishes a geometric equivalence between Kuznetsov components of certain quadric families and twisted derived categories, using moduli stacks of spinor sheaves, thus generalizing previous results in higher dimensions.

Contribution

It introduces a new geometricity result for Kuznetsov components of higher-dimensional quadrics via moduli stacks of spinor sheaves, extending prior work in the field.

Findings

Proves equivalence of Kuznetsov components and twisted derived categories under specific conditions.

Defines and analyzes the moduli stack of spinor sheaves on quadrics.

Identifies an open substack as a $ extbf{G}_m$-gerbe related to the Kuznetsov component.

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 \'etale 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.