Exercise 8.3 Fulton

Question

Let $C=X$ be a nonsingular cubic. a) Let P,Q $\in{C}$. Show that $P \equiv Q$ if and only if $P=Q$. (Hint: Lines are adjoints of degree 1) Where $P \equiv Q$ if and only if $P=Q+div(z)$

Please give me an idea, I was thinking about it a lot of time.

Answer 1

Your $z$ defines a degree 1 map from $C$ to $\mathbb{P}^1$ (with a pole at $Q$ and zero at $P$). Assuming that by cubic you mean plane cubic, this is impossible because smooth plane cubics are of genus 1, and $\mathbb{P}^1$ has genus zero.