Symmetric locally free resolutions and rationality problems

We show that the birationality class of a quadric surface bundle over ℙ2ℙ2 is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over ℙ2ℙ2 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 bundle