On the algebraic Brauer classes on open degree four del Pezzo surfaces

Jahnel, Jörg; Schindler, Damaris

doi:10.1016/j.jnt.2019.01.010

Cited by 1 publication

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References 31 publications

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“…Theorem 2.2 (Corollary 3.11 in [JS2]). Let S A,B be a smooth del Pezzo surface of degree 4, as defined in (1.1), and H ⊂ P 4 a hyperplane such that H ∩ S A,B is geometrically integral.…”

Section: Preparationsmentioning

confidence: 98%

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On the Frequency of Algebraic Brauer Classes on Certain Log K3 Surfaces

Jahnel¹,

Schindler²

2018

Can. Math. Bull.

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Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer-Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauergroup is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.defines a del Pezzo surface of degree four in P 4 Q . It is well known that the Hasse principle and weak approximation may fail for such surfaces (see for example [BSD]). In all known examples, for del Pezzo surfaces of degree four, the failure of the Hasse principle can be explained by a Brauer-Manin obstruction. I.e., in all of these situations one has adelic points on S A,B but the Brauer-Manin set S A,B (A Q ) Br , that is the subset of adelic points that are in the kernel of the Brauer-Manin pairing with the Brauer group Br (S A,B ), is empty.More recently, Colliot-Thélène and Xu [CX], initiated the study of integral Brauer-Manin obstructions. In [CX], they studied integral points on homogeneous spaces and representation problems by integral quadratic forms. In another direction, the concept of Brauer-Manin obstructions for affine varieties has been pursued in [CW] for families of affine cubic surfaces, such as representation problems of an integer by sums of three cubes. Moreover, see work of Bright and Lyczak [BL] for certain log K3 surfaces and Berg [Be] on the description of the Brauer-Manin obstruction for affine Châtelet surfaces.

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Section: Preparationsmentioning

confidence: 98%

“…The latter group is analysed in detail in [JS2] for an open degree four del Pezzo surface, and we now recall a result from that paper which forms one of the key inputs for our estimates for N 2 (P ).…”

Section: Preparationsmentioning

confidence: 99%