# Is there an affine embedding X for every normal singularity, so that Pic(X)=0?

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.

• 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. – Jason Starr Feb 19 '19 at 14:26
• @JasonStarr you are of course right, I edited the question. Now 1. adresses the completions. – Lukas Braun Feb 19 '19 at 14:36