(i) Let $f: X \rightarrow Y$ be a morphism of smooth projective curves. Define the divisor $f^{*}(D)$ if $D$ is a divisor on $Y$, and state the "finiteness theorem".

(ii) Suppose $f: X \rightarrow \mathbb{P}^{1}$ is a morphism of degree 2 , that $X$ is smooth projective, and that $X \neq \mathbb{P}^{1}$. Let $P, Q \in X$ be distinct ramification points for $f$. Show that, as elements of $c l(X)$, we have $[P] \neq[Q]$, but $2[P]=2[Q]$.