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2017 | Buch

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Brauer Groups and Obstruction Problems

Moduli Spaces and Arithmetic

herausgegeben von: Asher Auel, Brendan Hassett, Anthony Várilly-Alvarado, Bianca Viray

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory.

Contributors:

· Nicolas Addington

· Benjamin Antieau

· Kenneth Ascher

· Asher Auel

· Fedor Bogomolov

· Jean-Louis Colliot-Thélène

· Krishna Dasaratha

· Brendan Hassett

· Colin Ingalls

· Martí Lahoz

· Emanuele Macrì

· Kelly McKinnie

· Andrew Obus

· Ekin Ozman

· Raman Parimala

· Alexander Perry

· Alena Pirutka

· Justin Sawon

· Alexei N. Skorobogatov

· Paolo Stellari

· Sho Tanimoto

· Hugh Thomas

· Yuri Tschinkel

· Anthony Várilly-Alvarado

· Bianca Viray

· Rong Zhou

Inhaltsverzeichnis

Frontmatter

The Brauer Group Is Not a Derived Invariant

Abstract

In this short note we observe that the recent examples of derived-equivalent Calabi–Yau 3-folds with diffierent fundamental groups also have diffierent Brauer groups, using a little topological K-theory.

Nicolas Addington

Twisted Derived Equivalences for Affine Schemes

Abstract

We show how work of Rickard and Toën completely resolves the question of when two twisted affine schemes are derived equivalent.

Benjamin Antieau

Rational Points on Twisted K3 Surfaces and Derived Equivalences

Abstract

Using a construction of Hassett and Vèrilly-Alvarado, we produce derived equivalent twisted K3 surfaces over Q, Q ₂, and R, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.

Kenneth Ascher, Krishna Dasaratha, Alexander Perry, Rong Zhou

Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane

Abstract

We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.

Asher Auel, Jean-Louis Colliot-Thélène, Raman Parimala

Universal Spaces for Unramified Galois Cohomology

Abstract

We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.

Fedor Bogomolov, Yuri Tschinkel

Rational Points on K3 Surfaces and Derived Equivalence

Abstract

We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.

Brendan Hassett, Yuri Tschinkel

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

Abstract

Let \( {X} \rightarrow \mathbb{P}^{2}\) be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut\( ({X}/\mathbb{P}^{2})\) arise as pull-backs of certain Brauer classes on \( {\rm{k}}(\mathbb{P}^{2})\) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on \( {\rm{k}}(\mathbb{P}^{2})\) and relate the kernel of the pullback map to the Picard group of X.

If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses Clifiord algebras arising from symmetric resolutions of line bundles on C to yield Azumaya represen- tatives for the 2-torsion Brauer classes on X. We show that when \( \sqrt{-1}\) is in our base field, both constructions give the same result.

Colin Ingalls, Andrew Obus, Ekin Ozman, Bianca Viray, Hugh Thomas

Arithmetically Cohen–Macaulay Bundles on Cubic Fourfolds Containing a Plane

Abstract

We study ACM bundles on cubic fourfolds containing a plane exploiting the geometry of the associated quadric fibration and Kuznetsov’s treatment of their bounded derived categories of coherent sheaves. More precisely, we recover the K3 surface naturally associated to the fourfold as a moduli space of Gieseker stable ACM bundles of rank four.

Martí Lahoz, Emanuele Macrì, Paolo Stellari

Brauer Groups on K3 Surfaces and Arithmetic Applications

Abstract

For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice T _S of S; we classify these lattices up to isomorphism using Nikulin’s discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two, accounted for by transcendental Brauer-Manin obstructions.

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

On a Local-Global Principle for H 3 of Function Fields of Surfaces over a Finite Field

Abstract

Let K be the function field of a smooth projective surface S over a finite field \( \mathbb{F}\). In this article, following the work of Parimala and Suresh, we establish a local-global principle for the divisibility of elements in \( {H}^{3}(K, \mathbb{Z}/\ell)\) by elements in \( {H}^{2}(K, \mathbb{Z}/\ell), {l} \neq car.K \).

Alena Pirutka

Cohomology and the Brauer Group of Double Covers

Abstract

We calculate the 2-torsion subgroup of the Brauer group of a ramified double covering of a rational surface over an algebraically closed field. This and more general results are obtained by working with cohomology with mod 2 coefficients. Applications to the Brauer groups of K3 surfaces are discussed.

Alexei N. Skorobogatov

Titel: Brauer Groups and Obstruction Problems
herausgegeben von: Asher Auel
Brendan Hassett
Anthony Várilly-Alvarado
Bianca Viray
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-46852-5
Print ISBN: 978-3-319-46851-8
DOI: https://doi.org/10.1007/978-3-319-46852-5