Appears in collection : Rational points on irrational varieties

We consider the question of when an empty Brauer set over the ground field gives rise to an empty Brauer set over an extension. We first consider the case of quartic del Pezzo surfaces where a conjecture of Colliot-Thélène and Sansuc predicts that the Brauer-Manin obstruction should persist under any odd degree extension. Our results corroborate their prediction, under mild assumptions on the base field. Our proof also gives strong conditions on the reduction type of quartic del Pezzo surfaces that have a Brauer-Manin obstruction to the Hasse principle.