Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

TL;DR

This paper explores the birational geometry and categorical resolutions of nodal Gushel-Mukai varieties, establishing derived equivalences and extending rationality results for these special algebraic varieties.

Contribution

It introduces a new flop relating blowups of nodal Gushel-Mukai varieties to quadric fibrations, and extends Kuznetsov-Perry's rationality results to the nodal case.

Findings

Derived equivalence between categorical resolutions and Clifford algebra modules.

Identification of a subfamily of rational nodal GM fourfolds with K3 surface resolutions.

Categorical resolution determines the birational class of 1-nodal GM threefolds.

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. 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.