Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat.

What is known about the Brauer group of $X$?

Is there a relation between the Brauer group of $X$ and the Brauer group of the generic fiber $X_{\eta}$? Or is there a relation with the special fibers?

For example can one find this group in the simplest example: what is $Br(\mathbb{P}^1_{\mathbb{Z}})$? How about if the generic fiber is an elliptic curve?

Any other interesting facts about $Br(X)$ or hints to literature is welcome.