Irreducible varieties with easy computable class group

Question

In Hartshorne, the author mentions that computing the class group and Picard group of an irreducible variety is in general a very hard problem. I would like to know some easy examples where these computations are not that hard. Maybe for toric varieties, given their nature, these computations are really easy. Do you know any specific examples?

Answer 1

For normal toric variety $X(\Sigma)$ this can be done provided that you know fan $\Sigma$ and lattice $N$ of one parameter groups. Actually it is one of the basic results in the theory of toric varieties. You can find it for example in Fulton's book Introduction to Toric Varieties.

There are many relative results available.

1. If $X$ is a regular scheme, then you have

$$\mathrm{Cl}\left(\mathbb{A}^1_{X}\right) = \mathrm{Cl}(X)$$

2. If $X$ is a regular scheme and $\mathcal{E}$ is a vector bundle of rank $\geq 2$ on $X$, then

$$\mathrm{Cl}(\mathbb{P}_X(\mathcal{E})) = \mathrm{Cl}(X) \oplus \mathbb{Z}$$

3. For elliptic fibrations there is a paper by Shioda On Elliptic Modular Surfaces, where author computes class group of an elliptic surface in terms of class group of generic fiber, singular fibers and base.

These are really results concerning Picard groups.