Symmetric locally free resolutions and rationality problems

We show that the birationality class of a quadric surface bundle over $\mathbb{P}^2$ is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality very general bundles with discriminant curves fixed degree. In particular, construct explicit models these for some data. Among others, obtain various birational nodal Gushel-Mukai fourfold, as well cubic fourfold containing plane. Finally, prove stable irrationality several types bundles.