$\mathbb{P}^1$ a variety?

Question

This question comes from Chapter 5 of Gathmann's notes: how is $\mathbb{P}^1$ a variety? I know we have to show the diagonal $\{(x,x) : x \in \mathbb{P}^1\} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$ is closed, I'm just not sure how to go about doing that.

Answer 1

Showing that the diagonal is closed in $\mathbb{P}^1 \times \mathbb{P}^1$ is a step in showing that $\mathbb{P}^1$ itself is a variety, not that the product is a variety. Once you've shown that $\mathbb{P}^1$ is a variety, then show that the product of any two varieties is again a variety.

If your question is why $\mathbb{P}^1$ is a variety to begin with, can you think of a nice choice of affine open sets covering $\mathbb{P}^1$ on which you could check this?