Hartshorne Exercise I.3.5.: $\mathbb{P}^n - H$ (a degree $d$ hyperplane) is affine

Question

This is Hartshorne Exercise I.3.5.:

By abuse of language, we will say that a variety "is affine" if it is isomorphic to an affine variety. If $H \subseteq \mathbb{P}^n$ is any hyper surface, show that $\mathbb{P}^n - H$ is affine.

The hint says we can let $H$ be degree $d$. Then consider the $d$-uple embedding of $\mathbb{P}^n \to \mathbb{P}^N$ and use the fact that $\mathbb{P}^N$ minus a hyperplane is affine.

My questions are: (1) What does it mean to say that a hyperplane have degree $d$? (2) What happens to the hyperplane after it is embedded in the larger projective space? From the hint, it seems like it remains a hyperplane? Or at least is embedded in a hyperplane? (3) Can you give me some intuition why the $d$-uple embedding works in this case? (I'm having a hard time picturing what is going on)

The question here in MSE did not address my particular confusions.

Thanks for any help!

Answer 1

Let $v: \mathbb{P}^n \to \mathbb{P}^N$ be the $d$-uple embedding. Let $[x_0,...,x_n]$ be homogeneous coordinates on $\mathbb{P}^n$ and $[y_0,...,y_N]$ be homogeneous coordinates on $\mathbb{P}^N$. Then $v$ is described by taking a point $[x_0,...,x_n]$ to the tuple $[M_0, M_1,...,M_N]$, where the $M_i$ are all degree $d$ monomials in the coordinates $x_i$. So, for instance, we could start with $M_0 = x_0^d$, $M_1 = x_0^{d-1} x_1$, and so on.

Suppose that the hypersurface $H$ is defined by the polynomial $f = \sum_{i=0}^{N} a_i M_i$, so $H = \{f=0\}$. Then $f$ is the pullback of a linear polynomial from $\mathbb{P}^N$, namely $L=\sum_{i=0}^N a_i y_i$. Indeed, $v^*L$ can be described informally as substituting the $M_i$ for each $y_i$. From this we can glean that $v(H) = \{L=0\} \cap v(\mathbb{P}^n)$, i.e. that the image of $H$ is a hyperplane section of $v(\mathbb{P}^n)$.

So, to answer each of your questions:

1. A hyperplane of degree $d$ makes no sense; we should say "hypersurface" of degree $d$.
2. From the description I gave in paragraph 2, the image of the hypersurface is a hyperplane section of $v(\mathbb{P}^n)$. In particular, what I imagine is $H\subset \mathbb{P}^n$ is a wiggly thing sitting inside a straight-looking space, and after applying $v$, $H$ is made more "uniform" (since it is a hyperplane section) and the space becomes a bit wiggly-looking.
3. The reason the $d$-uple embedding works is because a hypersurface corresponds to some homogeneous degree $d$ polynomial, which (as seen above) can be described as the pullback of a linear polynomial from $\mathbb{P}^N$.