Example of a divisor of a function

Question

I'm studying Fulton's algebraic curves book and on page 97 Fulton defines the divisor of the rational function $z\in k(C)$:

I'm looking for an example of a divisor like this one.

Thanks

Answer 1

Take your curve to be $\mathbb{P}^1$. And take any rational function $z(x) :=p(x)/q(x)$; where $p,q \in \mathbb{C}[x]$.

Then the zeroes of $z(x)$ are exactly the zeroes of the polynomial $p$ and the poles are exactly at the zeroes of $q$.

Depending on the degree of $z$, defined as $deg(p) - deg(q)$ the behaviour at $\infty$ changes.

If you are looking for a particular example then take $z(x) = \frac{x}{1-x}$. This has a zero at $x=0$ and a pole at $x=1$. This is a degree $0$ rational function whose divisor is $div(z) = 1\cdot[0] - 1\cdot[1]$, which is also of degree 0.