"Fibrations"

These are various examples or counterexamples to the slogan “total space = base $\times$ fiber”.

General linear group over projective space

Here we fix an isomorpism $\mb P^1\cong \proj(k[x, y])$, and similarly fix coordinates for $GL_{2,k}$ after discussing the definition thereof. I believe this example holds in positive characteristic, but I generally have been thinking about the characteristic zero case.

The general linear group is a group scheme by this example on the stacks project, and is given by $GL_2 = \spec(\mb Z[x, y, z, w, 1/(xw-zy)])$. Base changing to a field $k$ gives $GL_{2,k} = \spec(k[x, y, z, w, 1/(xw-zy)])$ (this is effectively the classical definition). Consider the map $\psi_1:GL_{2,k}\to U_1\cong \mb A^1$ given by the ring morphism $k[x]\to k[x, y, z, w, 1/(xw-zy)], x\mapsto (xw-yz)$ and the map $\psi_2:GL_{2,k}\to U_2\cong \mb A^1$ given by the ring morphism $k[x]\to k[x, y, z, w, 1/(xw-zy)], x\mapsto 1/(xw-yz)$. Viewing $U_1\to \mb P^1$ and $U_2\to \mb P^1$ as affine patches, we see that $U_1\times_{\mb P^1}U_2\cong \spec k[x, x^{-1}]$; under this isomorphism, we see that $\psi_1$ and $\psi_2$ agree on $U_1\times_{\mb P^1}U_2$, the overlap. This means that we can glue them to a map $\psi:GL_{2,k}\to \mb P^1$.

We now seek to compute the fiber $\psi^{-1}(p) = GL_{2,k}\times_{\mb P^1}\spec k(p)$ for a point $p\in \mb P^1$. Let $U\cong\spec k[x]$ be an affine patch containing $p$. Then we may instead compute the fiber product $\psi^{-1}(p) = GL_{2,k}\times_{\mb A^1}\spec k(p)$, which is affine. Thus this map is a map who’s source and fiber are affine, but who’s target is not.