Periods of double EPW-sextics

O’Grady, Kieran G.

doi:10.1007/s00209-015-1434-7

Cited by 20 publications

(26 citation statements)

References 30 publications

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“…Since our constructions are inductive, and since the geometric behavior (for hyperelliptic quartics) identified in [LO18a] matches what we predicted in [LO18b] (for quartics), we have no doubt of the validity of our conjecture for quartic surfaces. Similarly, earlier work on EPW sextics [O’Gr15, O’Gr16] seems compatible with our conjectures; presumably our predictions (for ) can be checked inductively.…”

Section: Introductionsupporting

confidence: 90%

“…As is well known, the period space for quartic surfaces is , and the latter is equal to by Proposition 1.2.3. We will prove that is the period space of hyperelliptic quartic surfaces, see § 2.2, and that is the period space of desingularized EPW sextics (a quotient of the period space of double EPW sextics by the natural duality involution; see [O’Gr15]). Lastly, in § 2.3, we will establish a relation between and the period space of hyperkähler -folds of Type OG10.…”

Section: Locally Symmetric Spaces As Period Spaces Formentioning

confidence: 99%

“…cubic threefolds, del Pezzo surfaces, etc.). The nominal purpose of the present paper is to investigate the analogous cases of degree- surfaces [Sha81] and double Eisenbud–Popescu–Walter (EPW) sextics [O’Gr15, O’Gr16]. While studying these cases, we uncovered a rich and intriguing picture.…”

Section: Introductionmentioning

confidence: 99%

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Birational geometry of the moduli space of quartic surfaces

Laza¹,

O’Grady²

2019

Compositio Math.

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By work of Looijenga and others, one has a good understanding of the relationship between GIT and Baily-Borel compactifications for the moduli spaces of degree 2 K3 surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree 4 K3 surfaces and double EPW sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga to allow us to handle these cases. Specifically, in analogy with the so-called Hassett-Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to D lattices. In particular, for the dimension 19 case, we conjecturally obtain a continuous one-parameter interpolation (via a directed MMP) between the GIT and Baily-Borel compactifications for the moduli of degree 4 K3 surfaces. The predictions for the similar 18 dimensional case, corresponding geometrically to hyperelliptic degree 4 K3 surfaces, can be verified by means of VGIT -this will be discussed elsewhere.

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Section: Introductionsupporting

confidence: 90%

Section: Locally Symmetric Spaces As Period Spaces Formentioning

confidence: 99%

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Birational geometry of the moduli space of quartic surfaces

Laza¹,

O’Grady²

2019

Compositio Math.

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“…In Section 3, we discuss a relation between complex GM varieties and Eisenbud-Popescu-Walter (EPW) sextics [EPW01,OGr06]. An EPW sextic is a hypersurface of degree 6 in the projectivization P(V 6 ) of a 6-dimensional vector space V 6 which depends on the choice of a Lagrangian subspace A ⊂ 3 V 6 .…”

Section: Introductionmentioning

confidence: 99%

Gushel-Mukai varieties: Classification and birationalities

Debarre

Kuznetsov²

2018

Alg. Geom.

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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 genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieties and their automorphism groups in terms of linear-algebraic data naturally associated with these varieties. We carefully develop the relation between Gushel-Mukai varieties and Eisenbud-Popescu-Walter (EPW) sextics introduced earlier by Iliev-Manivel and O'Grady. We describe explicitly all Gushel-Mukai varieties whose associated EPW sextics are isomorphic or dual (we call them period partners or dual varieties, respectively). Finally, we show that in dimension 3 and higher, period partners or dual varieties are always birationally isomorphic.

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“…We say has no decomposable vectors if does not intersect . O’Grady [O’Gr06, O’Gr08, O’Gr16, O’Gr12, O’Gr13, O’Gr15] extensively investigated the geometry of EPW sextics, and proved in particular that (see also [DK18a, Theorem B.2]) if has no decomposable vectors, then:…”

Section: Conjectures On Duality and Rationalitymentioning

confidence: 99%

Derived categories of Gushel–Mukai varieties

Gennad'evich

Perry²

2018

Compositio Math.

115

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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 category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group.We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel-Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.

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Periods of double EPW-sextics

Cited by 20 publications

References 30 publications

Birational geometry of the moduli space of quartic surfaces

Birational geometry of the moduli space of quartic surfaces

Gushel-Mukai varieties: Classification and birationalities

Derived categories of Gushel–Mukai varieties

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