Confusion: line bundles over $\mathbb{P}^1$ and sheaves

Question

I understand the notion of a line bundle $L \to X$ where $X$ is the base. Locally at some $U \subset X$ we have that $L = U \times \mathbb{C}$ (if we work over the complex numbers). Over the projective space it is clear that at every point we have an additional structure, that of a line bundle, where each point of $\mathbb{P}(X)$ has as a fiber the line that it represents.

On the other hand I also understand the notion of the structure sheaf $\mathcal{O}_X$ and Serre's twisting sheaf $\mathcal{O}_X(d)$ in terms of regular functions over $U$ and in terms of rational functions of degree $d$ over $U$ respectively.

What I struggle to understand is how to understand the connection between the sections of the line bundle (for $\mathbb{P}^1$ what are those section?) with the sections of sheaves. In specific how are polynomials of degree $d$ of $\mathcal{O}(d)$ related to the line over a point of the projective space, say $\mathbb{P}^1$?

Answer 1

$\newcommand{\Proj}{\mathbf{P}}\newcommand{\Ocal}{\mathcal{O}}$If $d \geq 0$ is an integer, the sections of $\Ocal(d) \to \Proj^{1}$ are homogeneous polynomials of degree $d$ in the homogeneous coordinates. Each such polynomial $$ \sum_{j=0}^{d} a_{j} z_{0}^{d - j} z_{1}^{j} $$ defines a convergent power series at each point, i.e., a germ of the sheaf of local holomorphic sections. The collection of all these power series is a global section of the sheaf.

Speaking of "the line over a point of the projective space" describes the tautological bundle $\Ocal(-1)$, which has no non-trivial global holomorphic section (but which does have non-trivial meromorphic sections).