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… 

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References

SHOWING 1-10 OF 26 REFERENCES

Derived categories of Gushel–Mukai varieties

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques

Gushel–Mukai varieties: Linear spaces and periods

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: classification and birationalities

We perform a systematic study of Gushel-Mukai varieties---quadratic sections of linear sections of cones over the Grassmannian Gr(2,5). This class of varieties includes Clifford general curves of

Smooth toric Deligne-Mumford stacks

Abstract We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms

On the period map for prime Fano threefolds of degree 10

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We

Gromov-Witten theory of Deligne-Mumford stacks

Given a smooth complex Deligne-Mumford stack ${\cal X}$ with a projective coarse moduli space, we introduce Gromov-Witten invariants of ${\cal X}$ and prove some of their basic properties, including

A "bottom up" characterization of smooth Deligne-Mumford stacks

In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely

Periods of double EPW-sextics

We study the period map for double EPW-sextics, which are varieties making up a locally versal family of polarized hyperkähler fourfolds. Double EPW-sextics are parametrized by Lagrangian subspaces

Double covers of quadratic degeneracy and Lagrangian intersection loci

We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P 5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation