Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

Abstract

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. We also describe the relation with Verra threefolds and fourfolds at the birational and categorical level. In particular, in the three-dimensional case, we investigate alternative birational models by hyperbolic equivalence and by Kuznetsov's spinor modifications. We show that the categorical resolution of the Kuznetsov component of a one-nodal Gushel-Mukai threefold determines its birational class, and we explicitly construct another birational model, which is again a conic fibration over $\mathbb{P}^2$ branched over a sextic, with the same Kuznetsov component up to autoequivalences.