On the Brauer group of an arithmetic scheme

For an Enriques surface V over a number field k with a k-rational point we prove that the l-component of Br(V)/Br(k) is finite if and only if l≠2. For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the l-component of Br'(V)/Br(k). Moreover, for an arithmetic model X of V we prove a variant of Artin's conjecture on the finiteness of the Brauer group of X. Applications to the finiteness of the l-components of Shafarevich-Tate groups are given