# Gushel–Mukai varieties: Moduli

@article{Debarre2020GushelMukaiVM, title={Gushel–Mukai varieties: Moduli}, author={Olivier Debarre and Alexander Kuznetsov}, journal={International Journal of Mathematics}, year={2020} }

We describe the moduli stack of Gushel–Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of so-called Lagrangian data defined in our previous works; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel–Mukai varieties and construct some complete…

## 21 Citations

### Universal families of Gushel-Mukai fourfolds

- Mathematics
- 2020

We define universal Gushel-Mukai fourfolds over certain $\textit{Noether-Lefschetz}$ loci in the moduli stack of Gushel-Mukai fourfolds $\mathcal{M}^4_{GM}$. Using the relation between these…

### Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties

- Mathematics
- 2019

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these…

### Gushel-Mukai varieties

- Mathematics
- 2020

Gushel-Mukai varieties are smooth (complex) dimensionally transverse intersections of a cone over the Grassmannian Gr(2,5) with a linear space and a quadratic hypersurface. They occur in each…

### Gushel–Mukai varieties: Linear spaces and periods

- MathematicsKyoto Journal of Mathematics
- 2019

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperk\"ahler…

### Gushel--Mukai varieties: intermediate Jacobians

- MathematicsÉpijournal de Géométrie Algébrique
- 2020

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of
dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove
that the intermediate Jacobian of $X$ is isomorphic…

### The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$

- Mathematics
- 2022

. We study Gushel–Mukai (GM) varieties of dimension 4 or 6 in characteristic p . Our main result is the Tate conjecture for all such varieties over ﬁnitely generated ﬁelds of characteristic p ≥ 5 .…

### One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

- Mathematics
- 2022

We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on CH1 of a GushelMukai fourfold. We verify the…

### On some families of Gushel-Mukai fourfolds

- Mathematics
- 2020

We give explicit descriptions of some Noether-Lefschetz divisors in the moduli space of Gushel-Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.

### Explicit computations with cubic fourfolds, Gushel-Mukai fourfolds, and their associated K3 surfaces

- Mathematics
- 2022

. We present some applications of the Macaulay2 software package SpecialFanoFourfolds , a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel– Mukai fourfolds. In…

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Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperk\"ahler…

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