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More formal: Let a normal algebraic singularity be given by a local ring $R$ of finite type. Is there always an affine variety $X$ with a point $x$, so that

  1. $\widehat{\mathcal{O}}_{X,x} \cong \widehat{R}$
  2. $CL(X) \cong CL(R)$ or equivalently $Pic(X)=0$

where $CL$ and $Pic$ denote the (Weil) divisor class group and the Picard group respectively. This seems to be a very natural question to me, but I found nothing on it. I would be very happy about any ideas or references.

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  • $\begingroup$ Did you intend to write that $\widehat{\mathcal{O}}_{X,x} \cong \widehat{R}$, or perhaps that $\mathcal{O}^{\text{Hens}}_{X,x} \cong R^{\text{Hens}}$? Already if $R$ is the local ring of a smooth point on a curve of genus $g\geq 1$, then there is no affine variety $X$ with $\mathcal{O}_{X,x}\cong R$ yet $\text{Pic}(X)$ vanishes. $\endgroup$ – Jason Starr Feb 19 '19 at 14:26
  • $\begingroup$ @JasonStarr you are of course right, I edited the question. Now 1. adresses the completions. $\endgroup$ – Lukas Braun Feb 19 '19 at 14:36

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