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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?

Thanks.

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    $\begingroup$ I think for local rings this was Samuel's Conjecture, proved by Grothendieck in SGA 2. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Jun 6, 2012 at 1:48
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    $\begingroup$ Post this as an answer then. $\endgroup$
    – J.C. Ottem
    Commented Jun 9, 2012 at 8:30
  • $\begingroup$ here is a reference (to a reference):math.stackexchange.com/questions/381843/… $\endgroup$
    – roy smith
    Commented Aug 24, 2023 at 20:30

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