Brauer-Manin obstructions requiring arbitrarily many Brauer classes

A fundamental problem in the arithmetic of varieties over global fields is to determine whether they have a rational point.  As a first effective step, one can check that a variety has local points for each place.  However, this is not enough, as many classes of varieties are known to fail this local-global principle.   

The Brauer–Manin obstruction to the local-global principle for rational points is captured by elements of the Brauer group. On a projective variety, any Brauer–Manin obstruction is captured by a finite subgroup of the Brauer group.  I will explain joint work that shows that this subgroup can require arbitrarily many generators.  This is joint with J. Berg, C. Pagano, B. Poonen, M. Stoll, N. Triantafillou and B. Viray.