Rational points on fibrations with few non-split fibres

We revisit the abstract framework underlying fibration method for producing rational points on total space of fibrations over projective line. By fine-tuning its dependence external arithmetic conjectures, we render unconditional when degree non-split locus is $\leq 2$, as well in various instances where it $3$. are also able to obtain improved results regime that conditionally accessible under Schinzel's hypothesis, by incorporating into it, first time, a technique due Harari controlling Brauer--Manin obstruction families.