Brauer Groups on K3 Surfaces and Arithmetic Applications

McKinnie, Kelly; Sawon, Justin; Tanimoto, Sho; Várilly-Alvarado, Anthony

doi:10.1007/978-3-319-46852-5_9

Cited by 21 publications

(19 citation statements)

References 41 publications

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“…Some, for example, are parametrized by points on K 2p 2 through a dominant morphism K 2p 2 → K 2 by work of Mukai (see [Muk87], [MSTVA14, §2.6]). However, the results of [GHS07] show that K 2p 2 is also of general type for p ≥ 11.…”

Section: Introductionmentioning

confidence: 99%

Kodaira dimension of moduli of special cubic fourfolds

Tanimoto

Várilly-Alvarado

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C d in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the "low-weight cusp form trick" of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of C d . For example, if d = 6n + 2, then we show that C d is of general type for n > 18, n / ∈ {20, 21, 25}; it has nonnegative Kodaira dimension if n > 13 and n = 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only 20 values of d for which no information on the Kodaira dimension of C d is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

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Section: Introductionmentioning

confidence: 99%

Kodaira dimension of moduli of special cubic fourfolds

Tanimoto

Várilly-Alvarado

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Skorobogatov has conjectured that the Brauer-Manin obstruction is the only obstruction to both Hasse principle and weak approximation for K3 surfaces over number fields [27]. In light of recent work on diagonal quartics ( [5,16]) and various Kummer surfaces ( [1,11,19,28]), it seems natural to analyze one of the other main types of K3 surfaces, namely double covers of P 2 branched over a sextic curve, which are degree 2 K3 surfaces and were also studied in [13,14,20]. In this paper, we study the geometry of one of the simplest such families of surfaces:…”

Section: Computational Evidencementioning

confidence: 99%

“…In fact, one can show that M = P, and also abstractly compute the full Picard group without using the divisor d 20 .…”

Section: An Alternate Approachmentioning

confidence: 99%

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Brauer–Manin obstructions on degree 2 K3 surfaces

Corn

Nakahara

2018

Res. number theory

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We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P 2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

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“…For instance, is the Hasse/Brauer-Manin formalism over global fields compatible with (twisted) derived equivalence? See [HVAV11,HVA13,MSTVA14] for concrete applications to rational points problems.…”

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confidence: 99%

Rational Points on K3 Surfaces and Derived Equivalence

Hassett

Tschinkel

2017

Progress in Mathematics

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The geometry of vector bundles and derived categories on complex K3 surfaces has developed rapidly since Mukai's seminal work [Muk87]. Many foundational questions have been answered:• the existence of vector bundles and twisted sheaves with prescribed invariants; , and other authors. Our focus in this note is on rational points over non-closed fields of arithmetic interest. We seek to relate the notion of derived equivalence to arithmetic problems over various fields. Our guiding questions are: Question 1. Let X and Y be K3 surfaces, derived equivalent over a field F . Does the existence/density of rational points of X imply the same for Y ? Given α ∈ Br(X), let (X, α) denote the twisted K3 surface associated with α, i.e., if P → X is an étale projective bundle representing α, of relative dimension r − 1 then (X, α) = [P/ SL r ].Question 2. Suppose that (X, α) and (Y, β) are derived equivalent over F . Does the existence of a rational point on the former imply the same for the latter?Date: September 9, 2015. Note that an F -rational point of (X, α) corresponds to an x ∈ X(F ) such that α|x = 0 ∈ Br(F ). After this paper was released, Ascher, Dasaratha, Perry, and Zhou [ADPZ15] found that Question 2 has a negative answer, even over local fields.We shall consider these questions for F finite, p-adic, real, and local with algebraically closed residue field. These will serve as a foundation for studying how the geometry of K3 surfaces interacts with Diophantine questions over local and global fields. For instance, is the Hasse/Brauer-Manin formalism over global fields compatible with (twisted) derived equivalence? See [HVAV11, HVA13, MSTVA14] for concrete applications to rational points problems.In this paper, we first review general properties of derived equivalence over arbitrary base fields. We then offer examples which illuminate some of the challenges in applying derived category techniques. The case of finite and real fields is presented first-here the picture is well developed. Local fields of equicharacteristic zero are also fairly well understood, at least for K3 surfaces with semistable or other mild reduction. The analogous questions in mixed characteristic remain largely open, but comparison with the geometric case suggests a number of avenues for future investigation. Acknowledgments:We are grateful to Jean-Louis Colliot-Thélène,

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Brauer Groups on K3 Surfaces and Arithmetic Applications

Cited by 21 publications

References 41 publications

Kodaira dimension of moduli of special cubic fourfolds

Kodaira dimension of moduli of special cubic fourfolds

Brauer–Manin obstructions on degree 2 K3 surfaces

Rational Points on K3 Surfaces and Derived Equivalence

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