Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow \text{Cl}(X) \rightarrow \bigoplus\_{x \in X^{\text{sing}}} \text{Cl}(\mathcal{O}\_{X,x})$$ and the injections $\text{Cl}(\mathcal{O}\_{X,x}) \hookrightarrow \text{Cl}(\hat{\mathcal{O}}\_{X,x})$ where $\hat{\mathcal{O}}\_{X,x}$ is the completion of the local ring of $x$.


What criteria are there for the right arrow to be surjective? Or generally, can you say anything specific about the image of $\text{Cl}(X)$?


My situation is pretty specific so I am willing to assume further that $\text{Pic}(X)$ is free and $[\text{Cl}(X) : \text{Pic}(X)] < \infty$ if that helps.



Jaffe's exact sequence
The cokernel of the right arrow is in general $H^2(X, \, \mathcal{O}\_X^\*)$. In many cases, for instance when the strict henselization of every $\mathcal{O}\_{X, \, x}$ is a factorial ring, one can conclude that this group is a torsion group.


See [this](https://mathoverflow.net/questions/137155/exact-sequence-for-divisor-class-groups) MathOverflow question and the related comments and answers.


