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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.

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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 MathOverflow question and the related comments and answers.

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  • $\begingroup$ Thanks for your answer! That's interesting. Do you also know what happens when the right group is torsion in the first place? For example when $X$ has at worst rational double points as singularities. $\endgroup$
    – user269218
    Sep 23, 2015 at 14:41
  • $\begingroup$ This clearly answers my original question so I've accepted it. Thanks again. However this has led me to a follow-up question. $\endgroup$
    – user269218
    Sep 24, 2015 at 12:02

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