Why is $\mathbb{Q}$-factoriality not local in the étale topology?

Question

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?

Answer 1

I think what follows gives an "arithmetic" example of what you're looking for: a variety $X$ over a number field $k$ such that $X$ is $\mathbb{Q}$-factorial but $X_L = X \times_k L$ is not, for some finite extension $L/k$. In other words, $X$ is $\mathbb{Q}$-factorial, but not locally $\mathbb{Q}$-factorial in the étale topology.

Take an elliptic curve $E$ over $k$ which has rank zero over $k$, but positive rank over $L$. Embed $E$ as a plane cubic, and let $X$ be the projective cone over $E$. Then the only Cartier divisor classes on $X$ are multiples of a plane section. Now there is an isomorphism $\mathrm{Cl}_0(X) \to \mathrm{Cl}_0(E) = \mathrm{Pic}_0(E)$ (Hartshorne, II, Ex. 6.3). So the fact that $E(k)$ has rank zero means that $\mathrm{Cl}_0(X)$ is finite, so $\mathrm{Cl}(X)$ has rank 1, so the quotient $\mathrm{Cl}(X)/\mathrm{Pic}(X)$ is finite. Thus $X$ is $\mathbb{Q}$-Cartier. On the other hand, over $L$, $\mathrm{Cl}(X)/\mathrm{Pic}(X)$ has positive rank. Specifically, let $P$ be a point of infinite order on $E(L)$; then the line over $P$ gives a Weil divisor on $X_L$ no multiple of which is Cartier.

I imagine the same argument, starting from a elliptic surface over $\mathbb{P}^1_\mathbb{C}$, could give a "geometric" example.

Answer 2

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.