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We study the Brauer-Manin obstruction to the existence of zero-cycles ofdegree $d$ on certain classes of varieties over number fields. We generaliseexisting results in the literature and prove some results about fibrations overthe projective line, where the geometric Brauer group of the generic fibre isnot assumed to be finite. The idea is to assume that the Brauer-Maninobstruction to the Hasse principle is the only one for certain fibres and thendeduce analogous results for zero-cycles.

Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties