Given a variety with a suitable Brauer class, we present a general pullback construction that produces varieties that has Brauer–Manin obstruction to the existence of rational points. We then study Severi–Brauer fibrations and their Brauer groups without relying on explicit defining equations. As a key application, we show that there exist Severi–Brauer fibrations with index one that fails Hasse principle.
Biswas et al. (Fri,) studied this question.