Does Cohen-Macaulay + nonsingular in codimension one imply normality?

Question

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?

Thx

Answer 1

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 http://people.fas.harvard.edu/~amathew/chhomologicallocal.pdf