The Noether-Lefschetz conjecture and generalizations

Bergeron, Nicolas; Li, Zhiyuan; Millson, John J.; Mœglin, Colette

doi:10.1007/s00222-016-0695-z

Cited by 30 publications

(64 citation statements)

References 56 publications

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“…[Bru02]). By [BLMM17] this is in fact an isomorphism. A formula for computing the dimension of S 8,T is given by Bruinier [Bru02] By Proposition 5.1 the cubic pairs (X, H) with X at worst nodal (i.e.…”

Section: Definition 45mentioning

confidence: 89%

“…Remark 4.7. The work of Borcherds, Bruinier (see [Bru02] and references therein) and the refinement given in [BLMM17] allows us to compute the rank of the Picard group of D M /O * (T ) (more generally, the Picard rank of certain modular varieties of type IV). More specifically, let S k,T denote the space of (vector-valued) cusp forms of weight k with values in T .…”

Section: Definition 45mentioning

confidence: 99%

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On the moduli space of pairs consisting of a cubic threefold and a hyperplane

Laza

Pearlstein

Zhang

2018

Advances in Mathematics

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We study the moduli space of pairs (X, H) consisting of a cubic threefold X and a hyperplane H in P 4 . The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of K3 type and, on the other hand, the study of the singularity O 16 (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating (X, H) to certain "lattice polarized" cubic fourfolds Y . A period map for the pairs (X, H) is then defined using the periods of the cubic fourfolds Y . The main result is that the period map induces an isomorphism between a GIT model for the pairs (X, H) and the Baily-Borel compactification of some locally symmetric domain of type IV.but also admits an automorphism of order 3; the corresponding Mumford-Tate subdomain is a 7-dimensional ball.Remark 0.5. We point out that after the appearance of our manuscript, Chenglong Yu and Zhiwei Zheng [YZ18] have made a systematic study of the moduli space of symmetric cubic fourfolds (i.e. cubics with specified automorphism group). Since cubics with Eckardt points can be characterized as cubics admitting an involution that fixes a hyperplane, they fit into the loc. cit. framework. While some overlap between our results and those of Yu-Zheng exist (see esp. [YZ18, Prop. 6.5]), the focus of the two papers is essentially complementary.

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Section: Definition 45mentioning

confidence: 89%

Section: Definition 45mentioning

confidence: 99%

On the moduli space of pairs consisting of a cubic threefold and a hyperplane

Laza

Pearlstein

Zhang

2018

Advances in Mathematics

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“…The Noether-Lefschetz locus is the inverse image by the period map in M (γ) 2n of the union of all Heegner divisors. Its irreducible components were shown in [BLMM,Theorem 1.5] to generate (over Q) the Picard group of M (γ) 2n . As an application of our Theorem 5.1, we study in Section 7 birational isomorphisms between some of these components.…”

Section: Introductionmentioning

confidence: 99%

On the Period Map for Polarized Hyperkähler Fourfolds

Debarre

Macrì

2017

International Mathematics Research Notices

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We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy's Torelli theorem implies that their period map is an open embedding.Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW sextics.In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.

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“…(The cases of degrees 10 to 18 are due to Mukai. ) Group G ss Representation V Degree SL 3 Sym 6 (3) 2 SL 4 Sym 4 (4) 4 SL 5 Sym 2 (5) ⊕ Sym 3 (5) 6 SL 3 × SL 6 3 ⊗ Sym 2 (6) 8 SL 2 Sym 8 (2) ⊕ Sym 12 (2) 10 SL 8 × SO 10 8 ⊗ S + (16) 12 SL 6 × SL 6 6 ⊗ ∧ 2 (6) 14 SL 4 × Sp 6 4 ⊗ ∧ 3 0 (6) 16 SL 3 × G 2 3 ⊗ (14) 18 unordered set of six points in the plane, or via duality, a set of six lines; the double cover of the plane branched along the six lines is a K3 surface of Picard number 16, so over F (but not F), there is a correspondence between these orbits and such K3 surfaces. Such K3 surfaces have been extensively studied in the past; for example, see [46,48,56].…”

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confidence: 99%

“…For example, these moduli spaces are related to Noether-Lefschetz divisors, which are special cycles on moduli spaces of polarized K3 surfaces (see, for example, [45]). There has also been a great deal of recent activity surrounding the Noether-Lefschetz conjecture [6,49], and it would be interesting to extend the work of Greer et al [37] on the GIT stability of the Mukai models to these spaces of lattice-polarized K3s.…”

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confidence: 99%

Orbit Parametrizations for K3 Surfaces

Bhargava

Kumar

2016

Forum of Mathematics, Sigma

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We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

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The Noether-Lefschetz conjecture and generalizations

Cited by 30 publications

References 56 publications

On the moduli space of pairs consisting of a cubic threefold and a hyperplane

On the moduli space of pairs consisting of a cubic threefold and a hyperplane

On the Period Map for Polarized Hyperkähler Fourfolds

Orbit Parametrizations for K3 Surfaces

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