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.



What is known about the Brauer group of an arithmetic surface?
As Jason says you should read "Dix Exposes". Part 3 of "Le Groupe de Brauer III" uses a result of Artin to show that $R^i\pi\_\*G\_m =0$ for $i\geq2$ for surfaces such as you are interested in. The issue, of course, is dealing with $p$ torsion when the residue field of a valuation ring has characteristic $p$. So the Leray spectral sequence reduces to a long exact sequence involving cohomology of $\pi\_\*G\_m$ and $R^1\pi\_\*G\_m$. This immediately shows that $Br(P^1\_{ Z})=0$. In general you need to use Neron models to carry out a full calculation since $\pi$ cannot be smooth over ${ Z}$ if the genus is positive. So I don't think there is a simple answer. In the case of a number field $K$ you might be reduced to calculating $H^1({ O\_K}, A)$ where $A$ is an abelian scheme over the ring of integers $O\_K$ if $\pi$ is smooth. 


