Relation between tautological line bundle and blow up at the origin We can define the projective $n$-space $\mathbb{P}^n$ as the quotient of $\mathbb{C}^{n+1}\setminus \{0\}$ by the action of $\mathbb{C}^*$ with all weights equal to $1$.
Moreover we can define the tautological line bundle of $\mathbb{P}^n$ as
$$\mathcal{O}_{\mathbb{P}^n}(-1)=\{(u,v)\in\mathbb{P}^n\times \mathbb{C}^{n+1}\mid v=tu \text{ for some $t\in\mathbb{C}$} \}.$$
But this definition is equal to the definition of the blow-up at the origin $0$ of $\mathbb{C}^{n+1}$.
I'd like to understand how this definitions talk to each other: at the moment I've a naive idea of what a blow-up is and I've just started talking about line bundles, but I ddon't have a clear idea in mind of the connection between these two objects (and moreover if $\mathcal{O}_{\mathbb{P}^n}(-a)
$ can still has a similar interpretation, or is it just a peculiar property of $\mathcal{O}_{\mathbb{P}^n}(-1)$).
I know this question may sound too vague, but I hope someone can give me some advice on how to see the tautological line bundle as a blow-up.
 A: The way these definitions "talk to each other" has to do with the two projections onto the factors. If you map to $\mathbb C^{n+1}$, you get the blowup. If you map to $\mathbb P^n$, the same variety is realized as the tautological line bundle. The intuitive connection is that blowing up the origin of a vector space separates all of the one-dimensional linear subspaces so that they no longer intersect. These linear spaces correspond precisely to points of $\mathbb P^n$.
Another way to say the same thing is that the projection $\mathbb C^{n+1} \setminus 0 \to \mathbb P^n$ can be thought of as a rational map $\mathbb C^{n+1} \dashrightarrow \mathbb P^n$ with indeterminacy at the origin, so you have a copy of $\mathbb C^*$ as the fiber over each point of $\mathbb P^n$. Blowing up turns the origin (playing the role of vector space origin for all of these $\mathbb C^*$'s) into a copy of $\mathbb P^n$ (the zero section) that now has one point that is the origin of each fiber, so the fibers are now $\mathbb C$ and you can honestly call this thing a line bundle (instead of just a $\mathbb C^*$-bundle).
To see a similar interpretation of the other line bundles on projective space, I would take a look at Daniel Huybrechts' book Complex Geometry.
