Angelo Vistoli (SNS Pisa)

I am going to report on joint work in progress with Giulio Bresciani and Zinovy Reichstein. Brauer groups are basic invariants of fields, of great importance in non-commutative algebra, number theory and algebraic geometry; the Brauer group Br(K) of K is a torsion abelian group that can be defined in three different ways: cohomological, algebraic, using central simple algebras, and geometric, using Brauer-Severi varieties, which are varieties over K that become isomorphic to projective spaces extending the scalars to the algebraic closure of K. If L/K is a field extension, the relative Brauer group Br(L/K) is the subgroup of Brauer classes in Br(K) which vanish in Br(L). If L is finitely generated over K, we can think of L as the field of rational functions on an algebraic variety M over K. We are going to present some results connecting finite p-subgroups of Br(L/K) and the geometry of M. As a consequence we obtain that Br(L/K) is finite if M is smooth, and the Euler characteristic of its structure sheaf is plus or minus 1. In the first part of my talk I will review the basic theory of Brauer groups; in the second part I will state the main result, and give a brief sketch of proof.