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Consider a variety $X$. As the title suggests I would like to know if the hypotheses

1) $X$ is Cohen-Macaulay

2) the singular locus of $X$ has codimension $>1$

imply that $X$ is normal.

I hope it is true or trivial. Any suggestion or reference? If it was true, are there wild conditions for 1) such that used with 2) give normality for X?


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This is a consequence of Serre's criterion: a ring which is $R_1$ (regular in codimension one) and $S_2$ (which is implied by Cohen-Macaulay) is normal. See Theorem 2.10 in

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