Projective space minus point

Question

It is not difficult to see that $\mathbb{A}^n-\{\text{point}\}$ is not an affine variety for $n \geq 2$. Is the same true for $\mathbb{P}^n-\{\text{point}\}$? I think it is not difficult to show it is not projective variety but I am not sure if it is affine variety or not. I tried to compute the coordinate ring of $\mathbb{P}^n-\{\text{point}\}$ but did not find what it is.

Answer 1

Let $X$ be a curve in $\mathbb{P}^n_k$ not containing $\{pt\}$ and consider $X\subset\mathbb{P}_k^n-\{pt\}$. Then $X$ is projective over $k$. If $\mathbb{P}_k^n-\{pt\}$ were affine over $k$, $X$ would be affine over $k$. Now, a curve can't be both projective and affine because its dimension is $1$ (see e.g. Which affine schemes are projective?).