I'm currently trying to understand the blow-up process for 4-manifolds. A step in this journey is to understand, topologically, what happens when you pluck out a 4-ball and replace it with $\mathbb{CP}^2$ (minus a 4-ball). The tautological bundle is defined as $$\tau=\lbrace (l,p):p\in l\in \mathbb{CP}^1\rbrace$$ whose complex line bundle structure is given by the first projection map $\pi_1:\tau\to\mathbb{CP}^1$. Shouldn't we be including the pairs $(l,(0,0))$? Because, for example, if we pull back $l=[0:1]$ by $\pi_1$, we obtain $\lbrace ([0:1],(0,y)):y\in\mathbb{C}\setminus\lbrace 0\rbrace\rbrace$ which is just $\mathbb{C}$ minus a point. It feels like we need to put the origin back if we want a complex line bundle. So, my first question is: Is this correct? Do we need to add $(0,0)$ to each fiber?
My second question is more general. If $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{R}$ and $S$ is a closed orientable surface, then how do we know how many $\mathbb{F}$ bundle structures there are on $S$? There is a proof I've seen recently which shows that $\mathbb{CP}^2-\lbrace P\rbrace\cong \tau$ and the last line is "since the self-intersection of the 0-section is $-1$, these are the same complex line bundles and thus are diffeomorphic."
What results does this conclusion lean on? It seems to imply that complex line bundle structures on the sphere are classified by some integer. I'd like to know what that classification is.