Local sections of $\mathcal{O}(1)$ Let $\mathcal{O}(-1)$ be the tautological bundle over $\mathbb{P}^n$ and $\mathcal{O}(1)$ its dual bundle, also known as the hyperplane bundle. I know that there is a bijection between $\Gamma(\mathbb{P}^n,\mathcal{O}(1))$ and $\mathbb{C}[z_0,\dots,z_n]_1$, the set of homogeneus degree $1$ polynomials in $n+1$ variables. In fact, I think that given any two sections defined over two different affine open sets of $\mathbb{P}^n$ that glue are also defined by one of these polynomials.
So my question is, what can we say about a section only defined in one of the affine open sets? Can these sections be arbitrary or are they also determined by a linear functional?
Edit: MooS already provided a great answer for the algebraic geometric case. Yet, I forgot to mention in the OP that I'm looking for a holomorphic approach to the problem.
Edit 2: In fact, given any affine open set $U_i$ of $\mathbb{P}^n$ it would be enough for me to know how the sections $\mathcal{O}(1)(U_i)$ look in the holomorphic setting. MooS' answer makes me think that these sections should be of the form $\frac{f(z_0,\dots,z_n)}{z_i^{\operatorname{deg}f-1}}$, for $f$ an homogeneous polynomial, just as in the algebraic setting, yet I don't know how to show it.
For example, in $U_i$ a section $s$ is defined by a function $\tilde{s}:\mathbb{C}^n\to\mathbb{C}$ so that $$s([z])=([z],\tilde{s}(z_0/z_i,\dots,z_n/z_i)).$$
Taking the series expansion of $\tilde{s}$ naturally yields a homogeneous polynomial as long as it's finite. And even in this case we end up with something of the form $\frac{g(z_0,\dots,z_n)}{z_i^{\operatorname{deg g}}}$ for $g$ a homogeneous polynomial. Any ideas on how to proceed from here?
 A: Let $U_i = \{ z_i \neq 0 \} = \{ z_i=1 \}$. By defintion we have that $\Gamma(U_i, \mathcal O_{\mathbb P^n}(1))$ is the degree $1$-part of the localization $k[z_0, \dotsc, z_n]_{z_i}$, i.e. the sections are of the form $$\frac{f(z_0, \dotsc, z_n)}{z_i^{\deg f -1}}.$$
After letting $z_i=1$, this is nothing else but an arbitrary polynomial in the $z_j$'s with $j \neq i$.
If you want to glue such sections to a global section, you need to make sure that there are no denominators, i.e. $\deg f=1$. This - along the lines of your question - explains why the global sections are given by linear polynomials.
A: $\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$If $V$ is a finite-dimensional complex vector space, a section of $\mathcal{O}(1) \to \Proj(V)$ is a linear functional $\lambda:V \to \Cpx$. Fixing an identification $V \simeq \Cpx^{n+1}$, a section of $\mathcal{O}(1) \to \Cpx\Proj^{n}$ is a homogeneous linear polynomial
$$
\lambda(Z^{0}, Z^{1}, \dots, Z^{n})
  = \lambda^{0}Z_{0} + \lambda^{1}Z_{1} + \dots + \lambda^{n}Z_{n}
$$
in the homogeneous coordinates on $\Cpx\Proj^{n}$.
In the affine chart $U_{j} = \{Z^{j} \neq 0\}$ with affine coordinates $z_{j}^{i} = Z^{i}/Z^{j}$, this becomes the complex-valued function
$$
\lambda_{j}(z_{j}^{0}, z_{j}^{1}, \dots, \widehat{z_{j}^{j}}, \dots, z_{j}^{n})
  = \frac{\lambda(Z^{0}, Z^{1}, \dots, Z^{n})}{Z^{j}}.
$$
Particularly, a collection $(\lambda_{j})$ of $n + 1$ affine functions on $\Cpx^{n}$ glues into a global section if and only if it satisfies the transition formula
$$
\lambda_{i} = \frac{Z^{j}}{Z^{i}} \lambda_{j},\quad 0 \leq i, j \leq n.
$$
(To me, the transition function $g_{ij} = Z^{j}/Z^{i}:U_{i} \cap U_{j} \to \Cpx^{\times}$ defines the holomorphic line bundle $\mathcal{O}(1)$.)
