What follows is from the book "Mirror Symmetry" by Hori et. al.
From the definition of $\mathbb{P}^n$ we see there is a natural line bundle over $\mathbb{P}^n$ whose fiber over a point $l$ in $\mathbb{P}^n$ is the line it represents in $\mathbb{C}^{n+1}$. Define $J ⊂ \mathbb{P}^n \times \mathbb{C}^{n+1}$ to be $\{(l,v) : v \in l \}$. $J$ is called the “tautological line bundle.” Suppose we have coordinates $X_k$ on $\mathbb{C}^{n+1}$ with which to describe the point $v$. Then $X_k$ is a linear map from the fiber $J_l$ to $\mathbb{C}$. In other words, $X_k$ is a section of $\text{Hom}(J, \mathbb{C})$, the line bundle dual to $J$. Let us call this $H$. Note that the equation $X_k = 0$ makes sense on $\mathbb{P}^n$, and its solution defines a hyperplane (hence the “$H$”).
So here is my confusion:
First of all in a local neighbourhood $U \in \mathbb{P}^n$ the bundle should look as $U \times \mathbb{C}^{*}$, since it is a line bundle the rank of the fiber should be $1$ and not $n+1$. Is this correct? Then the line bundle $J$ should look for some trivialisation $i$ as $J = U_i \times \mathbb{C}^{*}$.
Is there some mistake in the first two lines of the quoted text? According to the text the tautological line bundle is the one given by $J ⊂ \mathbb{P}^n \times \mathbb{C}^{n+1}$ to be $\{(l,v) : v \in l \}$.
Another thing is that the tautological line bundle $\mathcal{O}(-1)$ has sections which are inverses of polynomials of degree $1$. How does one see this from the definition given above? For example if $\{ x_i \}$ are coordinates in $\mathbb{P}^n$ for $i=0,\ldots,n$ then $1/x_i$ is a section and of course when multiplying with a section of the dual bundle gives back a c-number.
Therefore, I conclude that their $X_k$ is a coordinate on $\mathbb{C}^{n+1}$ as they say and as a section of the tautological bundle it is just $1/X_k$. Is this a correct way to think about it at least for projective spaces (and their blow ups possibly?)
A follow up: Sections of the hyperplane bundle $\mathcal{O}(1)$ at $U_i$ look like $X_i$. Then, how exactly do we obtain the hyperplane divisor and the canonical divisor?