In this Wikipedia article I read:
This [the canonical quotient map $\mathbf{C}^{n+1} \setminus \{0\} \twoheadrightarrow \mathbf{CP}^n = (\mathbf{C}^{n+1}\setminus \{0\}) / \mathbf{C}^\times$] quotient realizes $\mathbf{C}^{n+1} \setminus \{0\} $ as a complex line bundle over the base space $\mathbf{CP}^n$. (In fact this is the so-called tautological bundle over $\mathbf{CP}^n$.)
The only interpretation of this sentence I can think of is the claim that this quotient map realizes $\mathbf{C}^{n+1} \setminus \{0\}$ as the total space of the line bundle $\mathscr{O}(-1)$. (I write total space, since I come from algebraic geometry and so I am used to think of $\mathscr{O}(-1)$ as a sheaf rather than a projection with vector space fibers.) But this claim can't really be true, because the above projection map has by construction fibers isomorphic to $\mathbf{C}^\times$ rather than $\mathbf{C}$. Also I can't really imagine that the total space of a vector bundle over a projective variety can be affine, but I might be wrong here. So I have two questions:
- What else could the Wikipedia article mean?
- Is there an easy description of the total space of $\mathscr{O}(-1)$ (or $\mathscr{O}(1)$ for that matter)?
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Sheaf}{\mathscr{O}}$As you suspect, the complement of the origin in $\Cpx^{n+1}$ is not the total space of a line bundle, but the total space of a $\Cpx^{\times}$-bundle.
If we let $\Cpx^{\times}$ act multiplicatively on $\Cpx$ via $(\lambda, z) \mapsto \lambda z$, the induced line bundle is the tautological bundle $\Sheaf(-1)$, whose total space is the blow-up of $\Cpx^{n+1}$ at the origin. (Presumably this is the intent of the cited article.)
If instead we let $\Cpx^{\times}$ act on $\Cpx$ via $(\lambda, z) \mapsto \frac{1}{\lambda}z$, the induced line bundle is the hyperplane bundle $\Sheaf(1)$, i.e., the dual of the tautological bundle, whose total space is the complement of a point $p$ in $\Cpx\mathbf{P}^{n+1}$, whose projection map is projection away from $p$, and whose zero section may be viewed as the hyperplane at infinity in $\Cpx^{n+1}$.