I'm looking for an example of the following. Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$ and let $\overline{X}$ be a compactification of $X$. We then have a map $Pic(\overline{X}) \rightarrow Pic(X)$. I want an example where this map is not surjective. It will be surjective if, for example, $\overline{X}$ is smooth, but I don't think it should be surjective in general. I'd also be interested in conditions under which it will be surjective.
Thanks!
EDIT : Here are a couple of thoughts. Since $X$ is smooth, every line bundle on it comes from a Weil divisor. The closure in $\overline{X}$ of a Weil divisor in $X$ is another Weil divisor. The only thing that could go wrong is that this new Weil divisor might not come from a line bundle. Thus we are looking for Weil divisors that don't come from Cartier divisors, but I don't know enough examples of this to get what I'm looking for.