Divisor associated to a rational map

Question

Let $C$ be a smooth projective curve, and let $\phi:C\rightarrow\mathbb{P}^1$ be a non-constant rational map. How to recover a divisor $D$ of the curve associated to this rational map? (I understand how to get a rational map out of a divisor, but not the other way round)

Answer 1

I'll interpret this in two different ways:

1.) If you're trying to find a divisor corresponding to a map to projective space, you can look at the pullback $\phi^*\mathcal{O}(1)$. The linear system will be base point free and correspond to the given map.

2.) I think what you probably are asking, though, is how to get a divisor from a rational function. If what you're given is a map to $\mathbb{P}^1$ with no choice of coordinates, you can't recover a divisor. However, if you are given a coordinate system on $\mathbb{P}^1$, the pullback $\phi^*(0-\infty)$ will give you the divisor you are looking for.