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It is well-known that normal curves are smooth. Moreover, a UFD of Krull dimension one is regular. Is there any higher-dimensional analog?

For example, given a normal projective surface $S$ over $\mathbb{C}$ whose local ring at every point is a UFD, then is $S$ always smooth?

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    $\begingroup$ No. For instance if $a,b,c$ are three mutually coprime integers, the ring $\mathbb{C}[x,y,z]/(x^{a}+y^{b}+z^{c})$ is factorial (see e.g. Bourbaki, Commutative Algebra VII, §3, exercise 7). $\endgroup$
    – abx
    Commented Mar 1 at 14:04

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