TITLE: smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group
QUESTION [5 upvotes]: I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group.
Related questions: Schemes with trivial brauer group and Brauer group of projective space
The affine plane $\mathbf{A}^2_k$ should be one example. Edit: I am especially interested in cases where the surface is not rational.
Note that it is difficult to pass from projective surfaces to affine surfaces because of purity for Brauer groups: $0 \to \mathrm{Br}(S)(\ell) \to \mathrm{Br}(S - C)(\ell) \to H^1(C,\mathbf{Q}_\ell/\mathbf{Z}_\ell)$. Here, $H^1(C,\mathbf{Q}_\ell/\mathbf{Z}_\ell) = 0$ iff $C$ has genus $g(C) = 0$, is affine and $\bar{C} - C$ (the divisor at infinity) is a one-point set.

REPLY [2 votes]: Here is an approach that should work. We know that $Br(\bf{A}_k ^2)=0$ if $k$ is an algebraically closed field. Now let $G$ be a finite group of order $n$ equipped with an action on $\bf{A}_k ^2$ such that $X$ is the quotient surface and $\bf{A}_k ^2\to X$ is flat. Now if $(\ell,n)=1$ then $Br(X)$ has no $\ell$ torsion since $Br(X)$ is split by a finite covering of order $n$ prime to $\ell$ and so must consist of torsion elements of order dividing $n$.