Grzegorz Kapustka Title: Symmetric locally free resolutions of symmetric sheaves Abstract: We shall discuss symmetric locally free resolutions of a given two-torsion or theta characteristic sheaf on a curve. We shall see that different such symmetric resolutions give rise to the same Brauer classes so to birational quadric bundles. As an application we will prove that a general one-nodal Gushel-Mukai fourfold is birational to a Verra fourfold and find more unexpected pairs of birational varieties. This is a joint work in progress with G. Bini, M. Kapustka, R. Laterveer.