Question

As I understand, one of the reasons for "bootstrapping" to the category of algebraic spaces before constructing the category of Artin stacks is that algebraic spaces form a stack in the etale (at least) topology, while schemes do not, even though one frequently has (at least in other contexts) to use the fact that, say, affine or quasi-affine schemes do form a stack even in the fpqc topology (by descent theory for quasi-coherent sheaves). As a result, I'm curious: what is the simplest example of non-gluable (say, etale) descent data for schemes?

To clarify, I'm looking for an example of an fpqc morphism $Y' \to Y$, a scheme $X' \to Y'$ together with the usual patching after pull-back to $Y' \times_Y Y'$ that does not come from a scheme over $Y$.

Answer 1

I'm pretty sure this example works. I do not include any proof that this is the simplest example, and it may not be, but it's not too complicated.

Let $L_1$ and $L_2$ be two rational curves in $\def\P{\mathbb P}\P^3$ which intersect in two points. A standard example of a proper non-projective variety $X$ is obtained by blowing up $L_1$ and $L_2$, but doing it in one order at one intersection point and in the other order at the other intersection point (I think this example is explained at the end of Hartshorne).

There is an involution $\sigma$ of $\P^3$ which switches the two lines and the two intersection points. Let $U\subseteq \P^3$ be the open locus where $\sigma$ acts freely, and let $Y=U\times_{\P^3}X$. Then $Y/\sigma$ is an algebraic space (over the scheme $U/\sigma$) which is not a scheme. It becomes a scheme after the etale base change $U\to U/\sigma$.