(i) Let $R$ and $S$ be compact connected Riemann surfaces and $f: R \rightarrow S$ a non-constant holomorphic map. Define the branching order $v_{f}(p)$ at $p \in R$ showing that it is well defined. Prove that the set of ramification points $\left\{p \in R: v_{f}(p)>1\right\}$ is finite. State the Riemann-Hurwitz formula.

Now suppose that $R$ and $S$ have the same genus $g$. Prove that, if $g>1$, then $f$ is biholomorphic. In the case when $g=1$, write down an example where $f$ is not biholomorphic.

[The inverse mapping theorem for holomorphic functions on domains in $\mathbb{C}$ may be assumed without proof if accurately stated.]

(ii) Let $Y$ be a non-singular algebraic curve in $\mathbb{C}^{2}$. Describe, without detailed proofs, a family of charts for $Y$, so that the restrictions to $Y$ of the first and second projections $\mathbb{C}^{2} \rightarrow \mathbb{C}$ are holomorphic maps. Show that the algebraic curve

$$Y=\left\{(s, t) \in \mathbb{C}^{2}: t^{4}=\left(s^{2}-1\right)(s-4)\right\}$$

is non-singular. Find all the ramification points of the $\operatorname{map} f: Y \rightarrow \mathbb{C} ;(s, t) \mapsto s$.