Towards the Brauer-Manin obstruction on varieties fibred over the projective line

TL;DR

This paper extends the understanding of the Brauer-Manin obstruction, showing it is the only obstruction to weak approximation for 0-cycles on certain fibrations over the projective line, generalizing previous results.

Contribution

It proves the exactness of the global-to-local sequence for Chow groups of 0-cycles on these varieties, establishing the Brauer-Manin obstruction as the sole obstruction.

Findings

Brauer-Manin obstruction is the only obstruction to weak approximation for 0-cycles.

Proves the exactness of the global-to-local sequence for Chow groups of 0-cycles.

Generalizes several existing results in the area.

Abstract

Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results.