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Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

Kacper Grzelakowski, Marco Rampazzo, Shizhuo Zhang

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.

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 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 : 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 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.
Paper Structure (11 sections, 27 theorems, 72 equations)

This paper contains 11 sections, 27 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Birational map to a quadric fibration
  3. The geometric resolution of $X_n$
  4. The quadric fibration
  5. A relative Atiyah flop
  6. The categorical resolution of the Kuznetsov component
  7. Mutations inside $D^b(\widetilde{X}_n)$
  8. Nodal GM varieties and hyperbolic reduction
  9. Hyperbolic reduction in families
  10. The relationship with Verra threefolds and fourfolds
  11. Categorical Torelli theorem for nodal Gushel-Mukai threefolds

Key Result

Theorem 1.1

(Theorem thm:kuznetsov_component_clifford_component). Let $X_n(3\leq n\leq 5)$ be an ordinary Gushel-Mukai $n$-fold with a single node $p\in X_n$. Then the categorical resolution$\widetilde{{\mathcal{K}} u}(X_n)$ is given by where $D^b(\mathbb{P}^2,\mathcal{C}_n)$ is the derived category of coherent sheaves of modules on the even part of the Clifford algebra (see 2.4 in kuznetsov_quadric_fibratio

Theorems & Definitions (50)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 40 more