Let $X$ be a smooth projective curve of genus $g>0$ over an algebraically closed field of characteristic $\neq 2$, and suppose there is a degree 2 morphism $\pi: X \rightarrow \mathbf{P}^{1}$. How many ramification points of $\pi$ are there?

Suppose $Q$ and $R$ are distinct ramification points of $\pi$. Show that $Q \nsim R$, but $2 Q \sim 2 R$.

Now suppose $g=2$. Show that every divisor of degree 2 on $X$ is linearly equivalent to $P+P^{\prime}$ for some $P, P^{\prime} \in X$, and deduce that every divisor of degree 0 is linearly equivalent to $P_{1}-P_{2}$ for some $P_{1}, P_{2} \in X$.

Show that the subgroup $\left\{[D] \in C l^{0}(X) \mid 2[D]=0\right\}$ of the divisor class group of $X$ has order $16 .$