Hartshorne Problem I.3.5

Question

The problem asks to show that if $H$ is a hypersurface of $\mathbb{P}^n$ of degree $d$, then $\mathbb{P}^n - H$ is isomorphic to an affine variety. Following the hint, if we apply the $d$-Uple embedding $v_d : \mathbb{P}^n \rightarrow \mathbb{P}^N$, the hypersurface $H$ is embedded in a hyperplane $h$ of $\mathbb{P}^N$. Then $\mathbb{P}^n - H$ is isomorphic to $v_d(\mathbb{P}^n) \cap (\mathbb{P}^N -h)$. Indeed, $\mathbb{P}^N -h$ is an affine variety, but why does that mean that $v_d(\mathbb{P}^n) \cap (\mathbb{P}^N -h)$ is an affine variety? What is the precise argument?

Answer 1

If $X$ is an affine variety, and $Z$ is a closed subvariety of $X$ in the Zariski topology, then $Z$ is affine. In your case, $v_d(\mathbb{P}^n)$ is closed inside of $\mathbb{P}^N$, hence $v_d(\mathbb{P}^n) \cap (\mathbb{P}^N - h)$ is a closed subvariety of $\mathbb{P}^N - h$.