Joint Number Theory Seminars at Beijing

Monday, April 14, 2008

Liu Qing: 1. "The Brauer group of surfaces" 2. "On the Grothendieck ring of varieties"

Prof. Liu Qing from Institut de Mathématiques de Bordeaux, Université Bordeaux 1
will visit us from Apr.20th to 29th. The following are two talks he will give.

Talk 1:



Time: April 23 (Wed.) 2:00-3:00 pm



Location: MCM 410


Title: "The Brauer group of surfaces"

Abstract:

This is a joint work with Dino Lorenzini and Michel Raynaud.
Let X be a projective smooth surface defined over a finite
field. Let Br(X) be its Brauer group. It is conjectured that
this group is finite. For a long time, people believed it is
order is not always a perfect square because of a computation
mistake by Y. Manin. We prove in this work that actually
the order of Br(X), if finite, is a perfect square. The proof
is based on Kato-Trihan's theorem (on Birch-Swinnerton-Dyer's
conjecture for abelian varieties over a finite field),
Poonen-Stoll's theorem on the order of the Tate-Shafarevich group
of Jacobians, Milnes's solution to Artin-Tate conjecture and
our previous working relating Brauer group and Tate-Shafarevich
group.



Talk 2:

Time:Apr. 25 (Friday) 2:00-3:00 pm



Location: MCM 410




Title: "On the Grothendieck ring of varieties"



Abstract:
This is a joint work with Julien Sebag. Let k be a field.
Let K_0(V_k) be the Grothendieck ring of algebraic varieties
over k (it is the ring of isomorphic classes of algebraic
varieties over k, quotient by the relations [X]=[Y]+[X\Y]
whenever X is an algebraic variety over k and Y is a
closed subvariety of X). It is a kind of universal
motives ring. Our aim is to give some properties of huge ring.
Most results are true only for characteristic zero field k
using desingularization and factorization of birational
morphisms, but in positive characteristics we can also
derive some basic properties.

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