•   Page Personnelle Professionnelle 
    Beijing, Beida, BICMR November 2015


     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).


    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.


    Saturday 21st at 11.30 at BICMR
    Beijing Algebraic Geometry Colloquum
      Title of lecture : Chow groups and the third unramified cohomology.

      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.


      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).