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Let $(X,\mathcal O_X)$ (here $\mathcal O_X$ is the sheaf of holomorphic functions) be a compact Riemann surface and let $D=\sum_{x\in X} a_x[x]$ be a divisor on $X$. The invertible sheaf associated to $D$ is $$\mathcal O_X(D)(U)=\{f\in\mathscr M_X(U)^\ast: \operatorname{ord}_x(f)+a_x\ge 0 \quad\text{for any $x\in U$}\}\cup\{0\}$$ where $\mathscr M_X$ is the sheaf of meromorphic functions. Now let $\mathcal L$ be an invertible sheaf on $X$ an let $s\in\mathcal (\mathcal L\otimes_{\mathcal O_X}\mathcal M_X)(X)$ be a non-zero meromorphic section of $\mathcal L$. Then we can associate a divisor to $s$ that we denote with $\operatorname{div}(s)$.

Informally $s$ is given as a colletion of "compatible" meromorphic functions on a trivialization covering for $\mathcal L$ and the order of $\operatorname{div}(s)$ at a point $x\in X$ is defined as the order of the right merormophic function at that point (I can be more precise on the definition of $\operatorname{div}(s)$ if you want).

Problem: I don't understand why we have that $\mathcal O_X(\operatorname{div}(s))\cong \mathcal L$.

Can you help me in proving this?

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Here is just a rough sketch. Let $f \in \mathcal O_X(div(s))(U)$, and map it to $fs \in \mathcal L(U)$. To see this map is surjective, notice if $s'$ is any other nonzero section, then $div(s/s')$ is principal, and the meromorphic function $f$ with $div(s/s')$ will lie in $\mathcal O_X(div(s))$. To see the map is injective, if $fs=0$, then after trivializating, $fs=0$ and since $s$ is a nonzero meromorphic function, $f$ will need to be zero. This argument needs many more details!

By the way, this is part of the bijection between the Picard group (isomorphism classes of line bundles) and the divisor class group (linear equivalence classes of divisors). This is worth learning well, and you can find some notes in the algebraic setting here:

http://www.math.harvard.edu/~amathew/linebund.pdf

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Weil divisor on a Riemann surface is Cartier: there is an open cover $\{U_\alpha\}$ and meromorphic functions $f_\alpha$ on $U_\alpha$ s.t. $D|_{U_\alpha} = (f_\alpha)$. Transition functions of the invertible sheaf $\mathcal{L}$ are $g_{\alpha \beta}= \frac{f_\alpha}{f_\beta}$ on $U_{\alpha \beta}= U_\alpha \cap U_\beta$. Note that $g_{\alpha \beta}$ are holomorphic on $U_{\alpha \beta}$.

If $s$ is a meromorphic section then is a collection of meromorphic functions $\{s_\alpha\}$ s.t. $s_\alpha = g_{\alpha \beta} s_\beta$. But then collection $\frac{f_\alpha}{s_\alpha}$ is a globally defined meromorphic function. So, the divisor of $s$ is equivalent to the given divisor $D$: they only different by the principal divisor of this meromorphic function. It is clear from your definition that equivalent divisors give isomorphic invertible sheaves.

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