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

　　Publication: 

　　Journal für die reine und angewandte Mathematik, Volume 2022, Issue 791, pp: 89-133. 

　　Author: 

　　Yonatan Harpaz 

　　Institut Galilee, Universite Sorbonne Paris Nord, 99 avenue Jean-Baptiste Clement, 93430 Villetaneuse, France 

　　Dasheng Wei 

　　Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and System Science, CAS, Beijing 100190, P. R. China & School of mathematical Sciences, University of CAS, Beijing 100049, P. R. China 

　　Email: dshwei@amss.ac.cn 

　　Olivier Wittenberg 

　　Institut Galilee, Universite Sorbonne Paris Nord, 99 avenue Jean-Baptiste Clement, 93430 Villetaneuse, France