This paper discusses the Brauer-Manin obstruction on double covers of the projective plane branched along a plane section of a Kummer surface from both the practical and the theoretical points of view. Theoretical highlights include the determination of a complete set of generators for the Brauer group; on the practical side, we give several surfaces with a Brauer-Manin obstruction and verify that the Brauer-Manin obstruction is the only one for a collection of several thousand surfaces of this type.
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