Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties
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| Format: | Preprint |
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2026
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| author | Grzelakowski, Kacper Rampazzo, Marco Zhang, Shizhuo |
| author_facet | Grzelakowski, Kacper Rampazzo, Marco Zhang, Shizhuo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14109 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties Grzelakowski, Kacper Rampazzo, Marco Zhang, Shizhuo Algebraic Geometry 14J45, 14F08, 14J17 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. |
| title | Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties |
| topic | Algebraic Geometry 14J45, 14F08, 14J17 |
| url | https://arxiv.org/abs/2602.14109 |