Symmetric locally free resolutions and rationality problems

Abstract

We show that the birationality class of a quadric surface bundle over [Formula: see text] is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over [Formula: see text] with discriminant curves of fixed degree. In particular, we construct explicit models of these bundles for some discriminant data. Among others, we obtain various birational models of a nodal Gushel–Mukai fourfold, as well as of a cubic fourfold containing a plane. Finally, we prove stable irrationality of several types of quadric surface bundles.