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I would like to know if there is an example as follows:

$X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that

  1. The minimal resolution of $x$ is a circle of rational curves (or a single nodal rational curve).
  2. The singularity $x$ is non-$\mathbb{Q}$-factorial.

I think it looks reasonable, but I do not know any explicit example of such a surface. I searched some papers but none of them discuss the $\mathbb{Q}$-factoriality.

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