ARITHMETIC SURJECTIVITY OF ZERO-CYCLES REVISITED

Given a morphism f : X → Y between varieties over a number field k, if the morphism induces surjective maps on kv rational points for all but finitely many places of k, the morphism is called arithmetically surjective. Various criteria for arithmetically surjectivity have been found in the literature, in particular, resulting in a new proof of Ax-Kochen theorem. Arithmetic surjectivity has since been generalized to zero-cycles, and following the more geometric approach to arithmetic surjectivity, Gvirtz obtained a similar criterion for arithmetic surjectivity of zero-cycles. In this paper, we obtain a different criterion folloing Pop, and we prove being arithmetic surjective on zero-cycles is a birational property when the varieties considered are proper and smooth.