Let $C \subset \mathbb{P}^{2}$ be the plane curve given by the polynomial

$$X_{0}^{n}-X_{1}^{n}-X_{2}^{n}$$

over the field of complex numbers, where $n \geqslant 3$.

(i) Show that $C$ is nonsingular.

(ii) Compute the divisors of the rational functions

$$x=\frac{X_{1}}{X_{0}}, \quad y=\frac{X_{2}}{X_{0}}$$

on $C$.

(iii) Consider the morphism $\phi=\left(X_{0}: X_{1}\right): C \rightarrow \mathbb{P}^{1}$. Compute its ramification points and degree.

(iv) Show that a basis for the space of regular differentials on $C$ is

$$\left\{x^{i} y^{j} \omega_{0} \mid i, j \geqslant 0, i+j \leqslant n-3\right\}$$

where $\omega_{0}=d x / y^{n-1} .$