Global sections of the line bundle $\mathcal{O}(D)$

For $D$ a divisor on a smooth variety $X$ over $\mathbb{C}$, we define as usual the subsheaf $\mathcal{O} (D)=\mathcal{O}_X(D)$ of the sheaf of rational functions $\mathcal{K}_X$ as follows:

$$\mathcal{O}_X(D)(U):=\{ f\in\mathcal{K}_X(U)\;|\; (f)\geq - D|_U \}$$

for every open $U\subseteq X$, where $(f)$ denotes the divisor attached to the rational function $f$.

From now on, suppose $D\geq 0$.

Then $\mathcal{O}(-D)$ is a subsheaf of $\mathcal{O}$ whose local sections are those regular functions which vanish along $D$ with multiplicity at least as indicated by the coefficients of $D$.

$\mathcal{O}(D)$ has as local sections those rational functions that are regular outiside of $D$ and are allowed to have poles at most along $D$, of order at most as indicated by the coefficients of $D$.

Let $\{U_\alpha\}$ be an open cover of $X$ such that $D\cap U_\alpha$ is defined by the regular function $\eta_\alpha\in\mathcal{O}(U_{\alpha\beta})$, that is: $D|_{U_{\alpha}}=(\eta_\alpha)$.

The multiplication by $\eta_\alpha$ induces local isomorphisms of sheaves:

$$\eta_\alpha\cdot:\mathcal{O}|_{U_\alpha}\to\mathcal{O}(-D)|_{U_\alpha}$$

showing that $\mathcal{O}(-D)$ is locally free of rank one. So, setting $U_{\alpha\beta}$, we get transition functions $\psi_{\alpha\beta}:=\eta_\alpha\cdot\eta_\beta^{-1}\in\mathcal{O}^{\;*}(U_{\alpha\beta})$ which give a cocycle defining the (corresponding) line bundle $\mathcal{O}(-D)$.

Analogously, local multiplication by the rational function $\eta_\alpha^{-1}$ gives local trivializations for $\mathcal{O}(D)$, and the cocycle given by the reciprocal $\psi_{\alpha\beta}^\vee:=\psi_{\alpha\beta}^{-1}=\eta_\alpha^{-1}\cdot\eta_\beta$ defines the line bundle $\mathcal{O}(D)$, showing that the line bundles $\mathcal{O}(D)$ and $\mathcal{O}(-D)$ are dual to each other.

Note that both line bundles are trivial when restricted to $X\setminus D$ because $\psi_{\alpha\beta}$ becomes a coboundary out of $D$.

Given a line bundle $\mathcal{L}$ with cocycle $\{g_{\alpha\beta}\}$, a global section $s\in \Gamma(X,\mathcal{L})$ is given by a bunch of local regular functions $s_\alpha\in\mathcal{O}(U_{\alpha\beta})$ such that $s_\beta=g_{\alpha\beta}\cdot s_\alpha$ on $U_{\alpha\beta}$.

More generally, if the local functions are allowed to be rational, $s_\alpha\in\mathcal{K}_X(U_\alpha)$, then this gives a global rational section $\sigma\in \Gamma (X,\mathcal{L}\otimes\mathcal{K}_X)$.

So, since clearly $\eta_\beta=\psi^\vee_{\alpha\beta}\cdot\eta_\alpha$ and $\eta_\beta^{-1}=\psi_{\alpha\beta}\cdot\eta_\alpha^{-1}$, we get a global section

$$s_D:=\{\eta_\alpha\}\in\Gamma(X,\mathcal{O}(D))$$

such that $(s_D)=D$, and a rational global section

$$s_D^\vee:=\{\eta_\alpha^{-1}\}\in\Gamma(X,\mathcal{O}(-D)\otimes\mathcal{K}_X).$$

Questions:

Question 1. By definition of $\mathcal{O}(D)$ as a subsheaf of $\mathcal{K}_X$, we have $\Gamma(X,\mathcal{O}(D))=\mathcal{O}(D)(X)=\{f\in\mathcal{K}_X(X)\;|\;(f)\geq -D\}$. So $s_D\in\Gamma(X,\mathcal{O}(D))$ must be a global rational function on $X$ such that its associated divisor is $\geq -D$. How can this global rational function $s_D$ be described? To be more precise, for example I would be content with a description of $s_D|_{U_\alpha}$ in terms of $\eta_\alpha, \eta_\alpha^{-1}\in\mathcal{K}_X(U_\alpha)$. Note that even if $\{\eta_\alpha\}$ is the expression of $s_D$ in "local trivializations", it's certainly not true that $s_D|_{U_\alpha}=\eta_\alpha$. Likewise, it's not true that $s_D|_{U_\alpha}=\eta_\alpha^{-1}$, as the identity $s_D|_{U_\alpha}=s_D|_{U_\beta}$ on $U_{\alpha\beta}$ must hold untwisted by any cocycle.

By duality, the natural sheaf inclusion $j_D=s_D^\vee:\mathcal{O}_X(-D)\to\mathcal{O}_X$ gets reversed to the sheaf surjection $s_D:\mathcal{O}_X\to\mathcal{O}_X(D)$, $h\mapsto h\cdot s_D$. In general, there is an isomorphism

$$\mathrm{Hom}(\mathcal{O}(-D),\mathcal{O})\cong\mathrm{Hom}(\mathcal{O},\mathcal{O}(D))=\Gamma(X,\mathcal{O}(D)),$$ where we go from right to left via the map on local sections $s\mapsto j_s$, $j_s(f):=\frac{s}{s_D}\cdot f$ ($f$ local section of $\mathcal{O}(D)$).

Question 2. How do we go from left to right in the above isomorphism, $j\mapsto s_j$, in terms of $j,s_D, s_D^\vee, \eta_\alpha$ etc.?

Amazingly, the canonical section of $\mathcal{O}_X(D)$ is the constant 1: $$s_D=1 \in\Gamma(X,\mathcal{O}(D))\subset K(X)$$ Indeed we have isomorphisms of sheaves (=trivializations) of $\mathcal O_{U_\alpha}$-modules $$g _\alpha:\mathcal{O}_X(D)|U_\alpha=\frac {1}{\eta_\alpha}\mathcal O_{U_\alpha}\stackrel {\cong}{\to} \mathcal O_{U_\alpha}:t\mapsto t\cdot \eta_\alpha$$ and $g_{\alpha\beta}=g_\alpha\circ g_\beta^{-1}=\frac{\eta_\alpha}{\eta_\beta}$ is the transition cocycle associated to the covering $(U_\alpha)$ for the line bundle $\mathcal{O}(D)$.
[my convention is the inverse of yours because usually trivializations go from the restriction of the bundle to the trivial bundle while you go in the opposite direction]

So the section $1|U_\alpha=s_D|U_\alpha \in \Gamma(U_\alpha,\mathcal{O}(D))$ is sent by $g_\alpha$ to $1\cdot \eta_\alpha=\eta_\alpha\in \Gamma(U_\alpha,\mathcal{O})$ and of course we have $\eta_\alpha=g_{\alpha\beta}\cdot\eta_\beta$ on $U_{\alpha\beta}$ by the definition of $g_{\alpha\beta}$.

To sum up, the canonical section $s_D=1 \in\Gamma(X,\mathcal{O}(D))\subset K(X)$ is represented in the given trivializations $g_\alpha$ of $\mathcal O(D)$ over the $U_\alpha$'s by the family of regular functions $\eta_\alpha\in \Gamma(U_\alpha,\mathcal{O}) $