Q:

Complements of hypersurfaces in a projective space is affine.

Suppose $H_0$ is the hypersurface defined by a homogeneous polynomial $H$ in $\mathbb{P}^n(k)$. How do we show its complement $\mathbb{P}^n(k)_H$ is affine? (It is a problem in Mumford's Redbook Ch1.5)

A:

If $H$ has degree $d$, embed $\mathbf{P}^n$ into $\mathbf{P}^N$ via the $d$-th Veronense embedding. (Here $N = {n+d \choose d} -1$.) Then $H_0$ will be the intersection of the image of $\mathbf{P}^n$ with a hyperplane in $\mathbf{P}^N$, and so its complement will be a closed subset of $\mathbf{A}^N$, hence affine. 

