Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

2 Answers 2

up vote 8 down vote accepted

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.

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.