Residual categories of quadric surface bundles

Abstract

We show the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme in two situations: (1) the quadric surface bundle has a smooth section; (2) the total space and the base of the quadric surface bundle are smooth and the base is a surface. We provide two proofs in situation (1) describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In situation (2) the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. As an application we apply the results to certain complete intersections of quadrics.