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Dichotomy for the Brauer-Manin obstruction in characteristic p
Let X be a smooth, projective, geometrically integral variety over a global function field k of characteristic p. We show that if the unipotent Brauer group of X is zero, and the Picard scheme of X is torsion free, then only finitely many places of k can be potentially relevant to the Brauer--Manin
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