If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$.

Question

I am trying to understand the following statement: If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$. This is stated by Hartshorne while proving Lemma 5.1, that a canonical linear system of a curve of genus greater than $1$ has no base ponts.

A curve is said to be rational if it is isomorphic to $\mathbb{P}^1$. Further, $\dim|P|=0$ is equivalent to $h^0(P)=1$. So suppose that there is some section $f\in H^0(P)$ with a single pole at $P$. Does this section yield an isomorphism to $P^1$? I am having trouble seeing this.

Answer 1

When $\dim |P|>0$ there exists $Q \not = P$ such that $P \sim Q$. Now use Hartshorne II.6.10.1