Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

We study ordinary Gushel--Mukai varieties with one node in dimensions three, four, and five, under some generality hypotheses. The blowup of the node is related by a relative Atiyah flop to a quadric fibration over $\PP^2$. We identify the categorical resolution of the Kuznetsov component with the Clifford component of this fibration. Then, we study hyperbolic equivalence, the relation with Verra varieties, and birational Torelli for general nodal threefolds. For fourfolds, we prove that a countable union of divisorial families is dense in the complex topology, with rational members and untwisted degree-two $K3$ categories. Finally, we construct a spinor modification of the associated conic bundle and compute its trace-free endomorphism bundle, which is not a sum of line bundles.