Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

An ordinary Gushel-Mukai variety with a single isolated node is the intersection of the Grassmannian $G(2,5)$ with a nodal quadric and a linear space. We consider such intersections in dimension three, four and five. We describe a flop between the blowup of such a variety and a quadric fibration over ${\mathbb{P}}^2$: at the level of derived categories, this flop establishes an equivalence between the categorical resolution of the Kuznetsov component of the Gushel–Mukai variety and the derived category of modules on the even part of the Clifford algebra of the quadric fibration. As a first application, we extend a result of Kuznetsov and Perry to the nodal case, and we describe a subfamily of rational, nodal Gushel–Mukai fourfolds whose Kuznetsov components admit a categorical resolution of singularities by an actual $K3$ surface of degree two without a Brauer twist. This produces evidence for a version of Kuznetsov’s rationality conjecture. Then, we describe the relation with Verra threefolds and fourfolds at the categorical level. As a further application, we show that the categorical resolution of the Kuznetsov component of a 1-nodal Gushel-Mukai threefold determines its birational class.