Brauer Groups and Obstruction Problems : Moduli Spaces and by Asher Auel, Brendan Hassett, Anthony Várilly-Alvarado,

By Asher Auel, Brendan Hassett, Anthony Várilly-Alvarado, Bianca Viray

The contributions during this booklet discover a variety of contexts within which the derived class of coherent sheaves on a spread determines a few of its mathematics. This atmosphere offers new geometric instruments for reading components of the Brauer team. With a view in the direction of destiny mathematics purposes, the e-book extends a few robust instruments for studying rational issues on elliptic curves, e.g., isogenies between curves, torsion issues, modular curves, and the ensuing descent thoughts, in addition to higher-dimensional kinds like K3 surfaces. encouraged through the speedy contemporary advances in our figuring out of K3 surfaces, the ebook is meant to foster cross-pollination among the fields of complicated algebraic geometry and quantity 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

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Additional resources for Brauer Groups and Obstruction Problems : Moduli Spaces and Arithmetic

Example text

Adv. , 218(5):1340–1369, 2008. [10] Alexander Kuznetsov. Derived categories of cubic fourfolds. In Cohomological and geometric approaches to rationality problems, volume 282 of Progress in Mathematics, pages 219–243. , Boston, MA, 2010. -P. Serre. A Course in Arithmetic. Springer-Verlag, New York-Heidelberg, 1973. [12] Alexei Skorobogatov. Torsors and Rational Points, volume 144 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001. edu Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane Asher Auel, Jean-Louis Colliot-Th´el`ene and Raman Parimala Abstract.

For any quadric X over any field k of characteristic = 2, we have that A0 (X) is universally trivial (see [57]). However, if X is the 4-dimensional quadric associated to an anisotropic Albert form X over a field 3 k of characteristic = 2, then coker H 3 (F, Q/Z(2)) → Hnr (X/k, Q/Z(2)) ∼ = Z/2Z by [41, Thm. 5]. Hence A0 (X) is universally trivial, while there are nontrivial unramified elements of some cycle module over X. Note that the index of an anisotropic quadric is 2. We remark that if X is a smooth proper variety of dimension n over a field k and X has a unirational parameterization Pnk X of degree N , then A0 (X) is universally N -torsion.

Let P be a fixed 0-cycle of degree 1 on X. By hypothesis, we have ξ = Pk(X) in CH0 (Xk(X) ). The closures of Pk(X) and ξ in X ×k X are P ×k X and the diagonal ∆X , respectively. By the closure in X ×k X of a 0-cycle on Xk(X) , we mean the sum, taken with multiplicity, of the closures of each closed point in the support of the 0-cycle on Xk(X) . Hence the class of ∆X − P ×k X is in the kernel of the contravariant map CHn (X ×k X) → CHn (Xk(X) ). Since CHn (Xk(X) ) is the inductive limit of CHn (X ×k U ) over all dense open subvarieties U of X, we have that ∆X −P ×k X vanishes in some CHn (X×k U ).

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