Let $X$ be a smooth curve of genus 0 over an algebraically closed field $k$. Show that $k(X)=k\left(\mathbb{P}^{1}\right) .$

Now let $C$ be a plane projective curve defined by an irreducible homogeneous cubic polynomial.

(a) Show that if $C$ is smooth then $C$ is not isomorphic to $\mathbb{P}^{1}$. Standard results on the canonical class may be assumed without proof, provided these are clearly stated.

(b) Show that if $C$ has a singularity then there exists a non-constant morphism from $\mathbb{P}^{1}$ to $C$.