Let $V$ be a smooth projective regular variety over the field $k$ of algebraic numbers, and let $X\to Spec(A)$ be its arithmetic model. If there exists a $k$-rational point on $V$, and the Tate conjecture holds for divisors on $V$ (for example, $V$ is a hyperkahler variety, $K3$ surface, etc.), then the $l$-components of $Br(V)/Br(k)$ and $Br(X)$ are finite for a prime number $l$ which does not divide the order of the torsion subgroup of the Neron-Severi group of $V$. It gives a variant of the M.Artin conjecture about the finiteness of the Brauer group of an arithmetic scheme. Many finiteness theorems in Diophantine geometry are closely related to this result.
If $V/k(t)$ is a smooth projective regular variety over purely transcendental extension of a number field $k$ and $X\to P^1$ be its geometric model, then the existence of $k(t)$-rational point on $V$ implies an injection $Br(X)/Br(k)\subset Br(V)/Br(k(t))$. Applications to some finiteness problems are given.
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