TITLE: Why is $\mathbb{Q}$-factoriality not local in the étale topology?
QUESTION [10 upvotes]: I was reading Kollár and Mori's book today and stumbled on the following passage:
"The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many proofs easier.  On the other hand it is a rather unstable condition in general.  It is not local in the Euclidean (or étale) topology, and it is very hard to keep track of when we pass from a variety to a divisor in an inductive proof."
(For those curious this is in section 3.7)
My question is, what are some examples that show why $\mathbb{Q}$-factoriality is not local in the étale topology?

REPLY [4 votes]: A different set of examples, closer to Kollár--Mori, comes from projective geometry over $\mathbb C$. For a projective variety $X$ over $\mathbb C$, $\mathbb Q$-factoriality is a global topological property: it depends on the prime divisors lying on $X$, rather than just on the local analytic type of its singular points, the issue being that local $\mathbb Q$-Weil but not $\mathbb Q$-Cartier divisors may fail to glue to a global non-$\mathbb Q$-Cartier divisor. For example, for a quartic threefold $X=X_4\subset {\mathbb P}^4$, a small number of ordinary double points do not effect the factoriality, even though the threefold ordinary double point is manifestly non-factorial. See this paper with a nice introduction which has a fuller discussion, explicit examples and further references.