(i) Suppose $f(x, y)=0$ is an affine equation whose projective completion is a smooth projective curve. Give a basis for the vector space of holomorphic differential forms on this curve. [You are not required to prove your assertion.]

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

$$X_{0}^{4}-X_{1}^{4}-X_{2}^{4}=0$$

over the complex numbers.

(ii) Prove that $C$ is nonsingular.

(iii) Let $\ell$ be a line in $\mathbb{P}^{2}$ and define $D$ to be the divisor $\ell \cap C$. Prove that $D$ is a canonical divisor on $C$.

(iv) Calculate the minimum degree $d$ such that there exists a non-constant map

$$C \rightarrow \mathbb{P}^{1}$$

of degree $d$.

[You may use any results from the lectures provided that they are stated clearly.]