Kodaira dimension of moduli of special cubic fourfolds

Tanimoto, Sho; Várilly-Alvarado, Anthony

doi:10.1515/crelle-2016-0053

Cited by 16 publications

(18 citation statements)

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“…Since the Picard rank of D M /O + (T ) is 2, all we need to do is to express λ(O + (T )) as a linear combination of the Heegner divisors H n and H t (and then pull it back to the moduli side). This is by now standard, we call it a Borcherds' relation, and follows by similar computations to those in [LO16] (see also [Kon99], [GHS07] and [TVA15]).…”

Section: Introductionmentioning

confidence: 75%

“…We introduce some Heegner divisors (cf. We briefly describe the strategy for computing Borcherds' relations which has been used for example in [CML09], [CMJL12] and [LO16] (see also [Kon99], [GHS07] and [TVA15]). The idea is to choose a primitive embedding of T into the even unimodular lattice II 26,2 ( ∼ = U ⊕2 ⊕ E The strategy is carried out as follows (using the above notations).…”

Section: Special Heegner Divisors In the Period Domainmentioning

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: Introductionmentioning

confidence: 75%

Section: Special Heegner Divisors In the Period Domainmentioning

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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“…Our effective bound explicitly depends on the Faltings height of the Jacobian of

C

, so it does not provide any uniform bound as conjectured in However, it is an open question whether the Faltings height in Theorem is needed. If there is a uniform bound for Theorem which does not depend on the Faltings height, then our proof provides a uniform bound for the Brauer group.…”

Section: Introductionmentioning

confidence: 96%

Effective bounds for Brauer groups of Kummer surfaces over number fields

Cantoral-Farfán

Tang

Tanimoto

et al. 2018

J. London Math. Soc.

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We study effective bounds for Brauer groups of Kummer surfaces associated to Jacobians of genus 2 curves defined over number fields.We obtain the following corollary as a consequence of results in [33,49]:Corollary 1.2. Given a smooth projective curve C of genus 2 defined over a number field k, there is an effective description of the setwhere X is the Kummer surface associated to the Jacobian Jac(C) of the curve C.Note that given a curve C of genus 2, the surface Y = Jac(C)/{±1} can be realized as a quartic surface in P 3 (see [20, Section 2]) and the Kummer surface X associated to Jac(C) is the minimal resolution of Y , so one can find defining equations for X explicitly.The quartic surface Y has sixteen nodes, and by considering the projection from one of these nodes, we may realize Y as a double cover of the plane. Thus X can be realized as a degree 2 K3 surface and our Theorem 1.1 follows from [23]. It is remarked in [23] that using the algebraic correspondence between X and Jac(C) it is possible to make [23] into an actual algorithm for Kummer surfaces. However we take a different approach from [23], and instead of using the Kuga-Satake construction we use a result of [52] reducing our problem to the case of abelian surfaces. In particular, our algorithm provides a large, but explicit bound for the Brauer group of X. (See the example we discuss in Section 6.)The method in this paper combines many results from the literature. The first key observation is that the Brauer group Br(X) admits the following stratification:Definition 1.3. Let X denote X × k Spec k where k is a given separable closure of k. Then we write Br 0 (X) = im(Br(k) → Br(X)) and Br 1 (X) = ker(Br(X) → Br(X)).Elements in Br 1 (X) are called algebraic elements; those in the complement Br(X) \ Br 1 (X) are called transcendental elements.Thus to obtain an effective bound for Br(X)/ Br 0 (X), it suffices to study Br 1 (X)/ Br 0 (X) and Br(X)/ Br 1 (X). The group Br 1 (X)/ Br 0 (X) is well-studied, and it admits the following isomorphism:Br 1 (X)/ Br 0 (X) ∼ = H 1 (k, Pic(X)).Note that for a K3 surface X, we have an isomorphism Pic(X) = NS(X). Thus as soon as we compute NS(X) as a Galois module, we are able to compute Br 1 (X)/ Br 0 (X). An algorithm to compute NS(X) is obtained in [49], but we consider another algorithm which is based on [10].To study Br(X)/ Br 1 (X), we use effective versions of Faltings' theorem and combine them with techniques in [23,51]. Namely, we have an injectionwhere Γ is the absolute Galois group of k. As a consequence of [52], we have an isomorphism of Galois modules Br(X) = Br(A), where A = Jac(C) is the Jacobian of C. Thus it suffice to bound the size of Br(A) Γ . To bound the cardinal of this group, we consider the following exact sequence as in [51]:Proposition 6.2 (Skorobogatov-Zarhin). For 3, we have that Br(A) Γ ( ) = 0.Proof. It suffices to show that the assumptions of [54, Proposition 4.2] are satisfied when image of Gal(Q/Q) in Aut(A[ ]) is GSp 4 (F ). This follows from PSp 4 (F ) being a simple nonabelian gr...

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“…Theorem 2. The moduli space M 2k is unirational for k < 11 and for k ∈ {13, 16,17,19,21,25,26,29,31,34,36,37,39,41,43,49,59, 61, 64}.…”

Section: Introductionmentioning

confidence: 99%

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

Fortuna

Mezzedimi

2021

J. London Math. Soc.

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We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U ⊕ −2k -polarized K3 surfaces. Such moduli spaces are proved to be of general type for k 220. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for k < 11 and for 19 other isolated values up to k = 64.

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Kodaira dimension of moduli of special cubic fourfolds

Cited by 16 publications

References 29 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

Effective bounds for Brauer groups of Kummer surfaces over number fields

The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

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