"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.

The blowup of the cone at a point over the cone itself

Recall the example the blowup of the cone; let $Y$ denote the cone and $\tilde Y$ it’s blowup. It is difficult to look at the scheme

and assert that this is a surface; in particular, it’s homogeneous ideal is generated by four generators and there is no obvious way to reduce that number. However, we know that $\proj(A)$ is a scheme over $\spec A_0$, the degree zero peice of $A$; we examine this map and use it to deduce that $\tilde Y$ is a surface.

Let $f: \tilde X\to X$ be the natural map given by the $\proj$ construction. We seek to compute for a point $P$ the sheaf-theoretic fiber

where $P = V(x-P_1, y-P_2, z-P_3)$ and $P_1^2-P_2^2-P_3^2=0$. We dehomogenize and compute

Recall that $A/(f)\otimes_AA/(g) \cong A/(f, g)$, so

By substitution we obtain

But $P_1^2-P_2^2-P_3^2=0$. Supposing $P_1\neq 0$, it has an inverse, so

Substitution again

If (on the other hand) $P_1=0$ and either $P_2$ or $P_3$ are nonzero, we quotient by the unit ideal and obtain $f^{-1}(P)=\{\}$, the empty set; otherwise, we obtain $f^{-1}(P)\cong\spec( k[Y,Z])$. Similarly, we see

and

When any of the $P_i$ are nonzero, the points glue to a single reduced point; when all are zero, the fiber is a copy of $\mb P^2$.

This shows that the blowup of a singular point is not always a copy of $\mb P^1$, which is interesting; if we blew up a smooth point, the result would be different. I would venture to argue that this demonstrates the intimate connection between the blowup and “tangents”; we can think of this singularity as being a point through which the lines form a two dimensional space, not a one dimensional space.

We’ve also showed that the map is an isomorphism on geometric points away from $V(x, y, z)$, which gives that the dimension of the blowup is 2. I might elaborate more on the many ways we can look at this as a “base $\times$ fiber” argument in the future.