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Let $X\subset\mathbb{P} _{\mathbb{C}}^N$ be a normal complete intersection.

$X$ is called factorial if every Weil divisor on it is Cartier; equivalently if all local rings $\mathcal{O}_{X,x}$ are unique factorization domains.

Is it true that if $$ \dim(\mathrm{sing}(X))<\dim(X)-3, $$ then $X$ is factorial?


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I think for local rings this was Samuel's Conjecture, proved by Grothendieck in SGA 2. –  Mahdi Majidi-Zolbanin Jun 6 '12 at 1:48
here is reference: books.google.com/… –  roy smith Jun 8 '12 at 1:41
Post this as an answer then. –  J.C. Ottem Jun 9 '12 at 8:30

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