Evaluating Azumaya algebras on cubic surfaces

Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k) -> Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. We show that a suitably generic cubic surface over a number field, whose reduction at some prime is a cone, has no Brauer-Manin obstruction. This extends results of Colliot-Thélène, Kanevsky and Sansuc.

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