Description:
Abstract We revisit the abstract framework underlying the fibrationmethod for producing rational points on the total space of fibrations overthe projective line. By fine-tuning its dependence on external arithmeticconjectures, we render the method unconditional when the degree of thenon-split locus is ≤ 2 {\leq 2} , as well as in various instances where itis 3. We are also able to obtain improved results in the regime that isconditionally accessible under Schinzel’s hypothesis, by incorporating intoit, for the first time, a technique due to Harari for controlling theBrauer–Manin obstruction in families.