On Brauer groups of double covers of ruled surfaces

Creutz, Brendan; Viray, Bianca

doi:10.1007/s00208-014-1153-0

Cited by 6 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another example of a rationally connected variety X over a number field k such that X(A k ) Br(X) = X(A k ) Br 1 (X) is given in [DLAN17]. Transcendental elements and their influence on the Brauer-Manin set have received a lot of attention for other classes of varieties as well (see [Wit04], [Ier10], [HVAV11], [Pre13], [HVA13], [IS15], [New16], [CV15], [MSTVA16] for examples of K3 and Enriques surfaces for which transcendental elements play a role in the Brauer-Manin set).…”

Section: Over Number Fields: General Contextmentioning

confidence: 99%

Rational points and zero-cycles on rationally connected varieties over number fields

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

View full text Add to dashboard Cite

We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory. p. 174]). Let X be a smooth, proper, geometrically irreducible variety defined over a number field k. If X is rationally connected, then X(k) is dense in X(A k ) Br(X) . Conjecture 2.3By "X is rationally connected", we mean that the variety X ⊗ k k is rationally connected in the sense of Campana, Kollár, Miyaoka and Mori, where k denotes an algebraic closure of k. Equivalently, for any algebraically closed field K containing k, two general K-points of X can be joined by a rational curve defined over K. We recall that examples of rationally connected varieties include geometrically unirational varieties (trivial), Fano varieties (see [Cam92], [KMM92]), and fibrations into rationally connected varieties over a rationally connected base (see [GHS03]).Conjecture 2.3 is wide open even for surfaces.Remarks 2.4. (i) The group Br 0 (X) = Im(Br(k) → Br(X)) does not contribute to the Brauer-Manin set: the global reciprocity law implies the equality X(A k ) B = X(A k ) B+Br 0 (X) for any subgroup B ⊆ Br(X). When Br(X)/Br 0 (X) is finite, the subset X(A k ) Br(X) ⊆ X(A k ) is therefore cut out by finitely many conditions. In particular, in this case, it is an open subset of X(A k ).(ii) When X is rationally connected, or, more generally, when X is simply connected and H 2 (X, O X ) = 0, an analysis of the Hochschild-Serre spectral sequence and of the Brauer group of X ⊗ k k implies that Br(X)/Br 0 (X) is finite (see [CTS13b, Lemma 1.1]). Presumably, this quotient should be finite under the sole assumption that X is simply connected, but this is out of reach of current knowledge. (As follows from [CTS13a] and [CTS13b, §4], the finiteness of the ℓ-primary torsion subgroup of Br(X)/Br 0 (X) is equivalent, when X is simply connected, to the ℓ-adic Tate conjecture for divisors on X.) (iii) Letting v be any archimedean place of k and noting that the image of the projection map X(A k ) Br(X) → X(k v ) is a union of connected components, we see

show abstract

Section: Over Number Fields: General Contextmentioning

confidence: 99%

Rational points and zero-cycles on rationally connected varieties over number fields

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

View full text Add to dashboard Cite

show abstract

“…Q . Other examples have come from surfaces with fibrations; for example, Wittenberg [Wit04] and Ieronymou [Ier10] constructed transcendental classes obstructing weak approximation on elliptic K3 surfaces, and Creutz and Viray [CV15] also used fibrations to construct 2-torsion transcendental classes on double covers of geometrically ruled surfaces. Hassett, Várilly-Alvarado and Varilly [HVAV11] exhibited a transcendental Brauer class on a K3 surface of Picard rank 1 that is associated to a cubic fourfold containing a plane.…”

Section: Introductionmentioning

confidence: 99%

A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

Hashimoto,

Honigs,

Lamarche

et al. 2020

Preprint

View full text Add to dashboard Cite

In this paper we investigate the Q-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation.Hosono and Takagi also showed that over C each of these Calabi-Yau threefolds Y is derived equivalent to a Reye congruence Calabi-Yau threefold X. We show that these derived equivalences may also be constructed over Q and give sufficient conditions for X to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over C.

show abstract

“…in finding an explicit representative for [the nontrivial] Brauer class of [X a ]." Recent work of Creutz and the last author [CV14a,CV14b], and Ingalls, Obus, Ozman, and the last author [IOOV] makes this problem more tractable. Building on techniques from [CV14a, CV14b, IOOV], we prove:…”

Section: Introductionmentioning

confidence: 99%

“…After recalling relevant results from [VAV11], we construct double cover maps π : Y a → S and π : X a → S, where S and S are ruled surfaces, and we study the geometry of these morphisms. These maps allow us to apply the results of [CV14a] to construct, in §4, an explicit central simple k(X a )-algebra representative A of the nontrivial Brauer class of X a . This representative A will necessarily be defined over a number field K 1 , be unramified over an open set U ′ K 1 , and be geometrically unramified.…”

Section: Introductionmentioning

confidence: 99%

Insufficiency of the Brauer–Manin Obstruction for Rational Points on Enriques Surfaces

Balestrieri

Berg

Park

et al. 2016

Association for Women in Mathematics Series

Self Cite

View full text Add to dashboard Cite

In [VAV11], Várilly-Alvarado and the last author constructed an Enriques surface X over Q with anétale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of X Q does not descend to Q. Together with the results of [VAV11], this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces.The methods of this paper build on the ideas in [CV14a,CV14b,IOOV]: we study geometrically unramified Brauer classes on X via pullback of ramified Brauer classes on a rational surface. Notably, we develop techniques which work over fields which are not necessarily separably closed, in particular, over number fields.

show abstract

On Brauer groups of double covers of ruled surfaces

Cited by 6 publications

References 19 publications

Rational points and zero-cycles on rationally connected varieties over number fields

Rational points and zero-cycles on rationally connected varieties over number fields

A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

Insufficiency of the Brauer–Manin Obstruction for Rational Points on Enriques Surfaces

Contact Info

Product

Resources

About