Understanding an application of Riemann-Roch in an article I saw the following in an article:
Let $C$ be an irreducible smooth projective curve over an algebraically closed field $K$ and let $g$ be its genus.
By Riemann-Roch, if N is large enough for every line bundle $\mathcal{L}$ on $C$ of degree $\leq g+1$, $\mathcal{O}(N)|_C\otimes\mathcal{L}^{-1}$ is generated by its global sections. It follows that, for $N$ large enough any function on $C$ with at most $g+1$ poles is the quotient of two homogeneous polynomials of degree $N$.
Why is this true? What is the connection two the fact that it is generated by global sections?
(crossposted from: Understanding an application...)
 A: The poles of your function determine an effective divisor of degree at most $g+1$ which in turn determines a line bundle $\mathscr L$ of degree at most $g+1$ and with an appropriate embedding of $\mathscr L$ into $\mathscr K_C$, the constant sheaf determined by $K(C)$ the original function corresponds to a global section of this $\mathscr L$. 
The statement you would like follows if you can embed $\mathscr L$ into $\mathscr O(N)|_C$, since the global sections of the latter correspond to quotients of homogenous polynomials of degree $N$. (Here I am assuming that $C$ is embedded into a projective space $\mathbb P^n$ and $\mathscr O(1)=\mathscr O_{\mathbb P^n}(1)$, which you never said, but you should have!). Well, actually you need a little more, because you want to do this as subsheaves of $\mathscr K_C$.
An embedding of of $\mathscr L$ into $\mathscr O(N)|_C$ is the same as a global section of $\mathscr O(N)|_C\otimes \mathscr L^{-1}$. In other words, every global section of $\mathscr O(N)|_C\otimes \mathscr L^{-1}$ gives you an embedding of of $\mathscr L$ into $\mathscr O(N)|_C$. You need global generation to guarantee that you can do this inside $\mathscr K_C$.
