Define the branching order $v_{f}(p)$ at a point $p$ and the degree of a non-constant holomorphic map $f$ between compact Riemann surfaces. State the Riemann-Hurwitz formula.

Let $W_{m} \subset \mathbb{C}^{2}$ be an affine curve defined by the equation $s^{m}=t^{m}+1$, where $m \geqslant 2$ is an integer. Show that the projective curve $\bar{W}_{m} \subset \mathbb{P}^{2}$ corresponding to $W_{m}$ is non-singular and identify the points of $\bar{W}_{m} \backslash W_{m}$. Let $F$ be a continuous map from $\bar{W}_{m}$ to the Riemann sphere $S^{2}=\mathbb{C} \cup\{\infty\}$, such that the restriction of $F$ to $W_{m}$ is given by $F(s, t)=s$. Show that $F$ is holomorphic on $\bar{W}_{m}$. Find the degree and the ramification points of $F$ on $\bar{W}_{m}$ and their branching orders. Determine the genus of $\bar{W}_{m}$.

[Basic properties of the complex structure on an algebraic curve may be used without proof if accurately stated.]