I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O\_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a ~~globally generated~~ line bundle $L$ such that $L$ is torsion in $Pic(X)$ and $H^1(L) \neq 0$. 


Does such surface exist? How can I construct one if it does exist? What if one ask for even nicer surface, such as arithmetically Gorenstein? If not, then I am willing to drop smooth or globally generated, but would like to keep the torsion condition. 


More motivations(thanks Andrew): Such a line bundle would give a cyclic cover of $X$ which is not ACM, which would be of interest to me. I suppose one can think of this as a special counter example to a weaker (CM) version of purity of branch locus. 


To the best of my knowledge this is not a homework question (: But I do not know much geometry, so may be some one can tell me where to find an answer. Thanks.


EDIT: Removed the global generation condition, by Dmitri's answer. I realized I did not really need it that much. 



Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces 
Let us show that a globaly generated torsion line bundle $L$ on a (compact) complex surface is trivial. Ideed, a globally generated line bundle has at least one section, say $s$. Let us take it. If $s$ has no zeros, then $L$ is trivial. But if $s$ vanishes somewhere then any positive power $L^n$ has a section $s^n$ that vanishes at the same points. So any power of $L$ is not trivial, i.e. $L$ is not a torsion bundle, contradiction.


Notice that we did not use the fact that the surface is smooth. And we also did not use the fact that we work with a surface...


