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The answer is yes, at least over $\mathbb{C}$, since $2$-dimensional (cyclic) quotient singularities are taut (starr, in German), namely, they are uniquely characterized, up to biholomorphisms, by their resolution graph. In other words, every $2$-dimensional normal singularity, having the same resolution graph of a (cyclic) quotient singularity, is itself a ...

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