Page Personnelle ProfessionnelleBeijing, 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,
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