What does $k(C)$ denote where $C$ is a curve? (link to context included)

Question

On page 5 of http://math.stanford.edu/~vakil/725/bagsrr.pdf is says that "A repartition is an indexed set $\lbrace r _{P}\rbrace_{P\in C}$ where $\lbrace r _{P}\rbrace$ is an element of $k(C)$, and $\lbrace r _{P}\rbrace\in\mathcal{O}_{P}$ for all but finitely many $P$", however, I can't seem to find what $k(C)$ is. Does anyone know?