Evaluation of Brauer elements over local fields

Brauer–Manin obstruction for zero-cycles on certain varieties

“…In the case of rationally connected varieties one can use [16,Prop. 7] in order to weaken the hypothesis in Proposition 3.3.…”

“…cit. If v / ∈ T then it follows from [16,Prop. 7] that we can replace l v by any F v -point, call it again Q v , without changing the value of the evaluation map given by any element of Br(X).…”

The Brauer–Manin Obstruction for Zero-Cycles on K3 Surfaces

“…If v ∈ T then we replace l v by some Q v as in the proof of loc.sit. If v / ∈ T then it follows from [17,Prop. 7] that we can replace l v by any F v -point, call it again Q v , without changing the value of the evaluation map given by any element of Br(X).…”

“…(2) In the case of rationally connected varieties one can combine the aforementioned formal argument with [17,Prop. 7] in order to weaken the hypothesis in Proposition 3.3 that (wP 1 ) holds over all completions to the hypothesis that it holds over a finite set of places, see Proposition 3.6 for a precise statement.…”