For any finite abelian groups B0 ⊂ B, there exists a smooth rationally connected variety over a number field whose unramified Brauer group is B and whose Hasse principle obstruction is precisely B0.
Rational points on varieties and the Brauer-Manin obstruction
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These lecture notes give an introduction to the Brauer-Manin obstruction to the existence of rational points, focusing on the interplay between theory and computation.
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Varieties with prescribed finite unramified Brauer groups and subgroups precisely obstructing the Hasse principle
For any finite abelian groups B0 ⊂ B, there exists a smooth rationally connected variety over a number field whose unramified Brauer group is B and whose Hasse principle obstruction is precisely B0.