smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group

Question

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.

Answer 1

I think the following should work, but I have not checked all the details.

Let $k$ be an algebraically closed field and let $\ell$ be coprime to the characteristic of $k$. Then the Grothendieck purity sequence implies that any non-trivial element of $\mathrm{Br}(k(x,y)) \{\ell\}$ must be ramified along at least two irreducible divisors in $\mathbb{P}^2$ (I think this can also be proved using the Faaddev reciprocity law, see Section 5 of the appendix to Chapter 2 of Serre's book on Galois cohomology, p114).

Hence, again by purtiy, it follows that if $f \in k[x,y]$ is irreducible and we take $U = \mathbb{P}^2 \setminus \{f = 0\}$, then we have $\mathrm{Br}(U) \{\ell\} = 0$.

This generalises the case of $\mathbb{A}^2$ which you already gave.

Edit about the non-rational case: If you want the $\ell$-primary part of the Brauer group to be non-trivial, then certainly you need the $\ell$-primary part of a smooth compactification $S$ to be non-trivial. By a result of Grothendieck, this can happen only if $b_1(S) = p_g(S) = 0$. For minimal surfaces, it seems like $S$ must be either an Enriques surface $(\ell > 2)$ or a surface of general type satisfying these cohomological restrictions. It seems possible that the complement of a hyperplane section of a surface of general type in $\mathbb{P}^3$ might give an example.

Answer 2

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 $n$.