Affine open sets of projective space and equations for lines

Question

I am reading Introduction to Algebraic Geometry by Smith et al. and I have some questions about some vocabulary that they use but that is not explicitly defined (I guess it is probably obvious and I am just slow). Note that I am only dealing with classical AG some when I say projective space I don't mean anything fancy.

1) What is an affine open subset of $\mathbb{P}^n$? I am familiar with the cover of projective space by the open sets that are defined by the nonvanishing of the projective coordinates and how these sets are homeomorphic to $\mathbb{A}^n$ - are there other affine open sets?

2) What is the definition of a line in projective space? What do the equations for lines in affine and projective space look like (presumably they are given by zero sets of polynomials and are therefore affine and projective varieties reap.)?

Thank you for your time!

Answer 1

$\newcommand{\Proj}{\mathbf{P}}$An "affine open subset" of $\Proj^{n}$ normally refers to one of the $(n+1)$ sets you mention. That said, the projectivization $\Proj(V)$ of a vector space $V$ doesn't come with coordinates (in the same way $V$ itself doesn't come with a preferred basis), so an "affine open set" could conceivably refer to an arbitrary complement of a projective hyperplane. (The affine sets you mention are, of course, complements of the coordinate hyperplanes.)

A line in a projective space $\Proj(V)$ is the image of a two-dimensional (linear) subspace of $V$. If $F$ is a field and $V = F^{n+1}$, then just as in linear algebra, a line in $\Proj(V)$ can be specified by a system of $(n - 1)$ homogeneous linear equations in $(n + 1)$ variables with coefficients in $F$. In affine coordinates, a line is specified by a system of $(n - 1)$ inhomogeneous equations in $n$ variables.

Answer 2

Adding to the previous answer one might notice that all complements of hypersurfaces $\mathbb{P}^n_k - V(f)$ where $f$ is homogeneous in $S=k[x_0,\ldots,x_n]$ of degree $d > 0$ are affine open subsets. Scheme theoretic this is just the isomorphism $\mathrm{spec}(S_{(f)}) = D_+(f)$ where $D_+(f) = \{\mathfrak{p} \in \mathrm{proj}(S) \mid f \notin \mathfrak{p}\}$.

But one can see it also classical by the Veronese immersion $v_d:X=\mathbb{P}^n \to \mathbb{P}^{N_{n,d}}=X'$. ($N_{n,d} = ({n+d \atop n})-1$). The map $v_d$ gives an isomorphism between $\mathbb{P}^n - V(f)$ and $(\mathbb{P}^{N_{n,d}} - V(F)) \cap v_d(X)$ where $F$ is the linear form in $M= \mathcal{O}_{X'}(1)(X')$ that corresponds to $f$ by substituting for every monomial of $f$ of degree $d$ the corresponding variable of $M$.

See for this also Hartshorne, p.21, ex 3.5.