Lecture course : The Brauer group of schemes

3, 10, 17, 24 november from 3pm to 6 pm

Summary : The Brauer group of algebraic varieties features prominently in at least two directions of research: birational problems (the Lüroth problem) and arithmetic geometry (Brauer-Manin obstruction). The November 2015 lectures will be devoted to the algebraic theory of the Brauer group. The first part will be devoted to the general properties of the Brauer group. The second part will be concerned with concrete computations of the Brauer group for various classes of algebraic varieties.

Basic references :

P. Gille and T. Szamuely : Central simple algebras and Galois cohomology.

A. Grothendieck's three lectures on the Brauer group, in Dix exposés sur la cohomologie des schémas.

Milne : Étale cohomology.

J-P. Serre, Galois cohomology.

J-P. Serre, Local fields.

J-P. Serre, Cohomological invariants, Witt invariants and trace forms, in Cohomological invariants in Galois Cohomology (ed. Garibaldi, Serre, Merkurjev).

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Wednesday 18th, 2pm

Distinguished lecture at BICMR

Title of lecture : Birational invariants

Summary : Given a rationally connected variety over the complex field, one may ask whether it is stably rational, i.e. whether

after possibly multiplication by a projective space it becomes birational to a projective space. One classical tool used to disprove such a statement

is the Artin-Mumford invariant (1972). For smooth hypersurfaces of dimension at least three, this invariant vanishes. In 2013, Claire Voisin introduced a degeneration method which also leads to disproof of stable rationality for suitable varieties.

The method was generalized by Alena Pirutka and the speaker in 2014 and it has been applied by several authors to many types of rationally connected varieties, in particular to quartic hypersurfaces. B. Totaro (2015) combined the method with a technique of Koll\'ar (1995) on differentials in positive

characteristic. I shall survey the method and its very concrete applications.

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Saturday 21st at 11.30 at BICMR

Beijing Algebraic Geometry Colloquum

Summary : Algebraic K-theory provides relations between the third unramified cohomology group (with torsion coefficients) of a smooth projective variety and the Chow group of codimension 2 cycles. This is used to study the image of such cycles under various cycle class maps into integral cohomogy. It is also used to investigate rationality questions for Fano hypersurfaces and for homogeneous spaces of connected linear algebraic groups. There are many open questions.

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Wednesday 25th, Capital Normal University

Title of lecture : The set of non-n-th powers is a diophantine set (joint work with J. Van Geel)

Summary : A subset of a number field k is called diophantine if it is the image of the set of rational points of some affine variety under a morphism to the affine line, i.e. if is the set of values of a function on the variety. For n = 2 the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and the speaker (1980), on the Brauer–Manin obstruction for rational points on these varieties. For n = p, p any prime number, A. V\ ́arilly-Alvarado and B. Viray (2012) considered an analogous family of varieties. Replacing this family by its (2p+1)th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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