State a theorem which describes the canonical divisor of a smooth plane curve $C$ in terms of the divisor of a hyperplane section. Express the degree of the canonical divisor $K_{C}$ and the genus of $C$ in terms of the degree of $C$. [You need not prove these statements.]

From now on, we work over $\mathbb{C}$. Consider the curve in $\mathbf{A}^{2}$ defined by the equation

$$y+x^{3}+x y^{3}=0$$

Let $C$ be its projective completion. Show that $C$ is smooth.

Compute the genus of $C$ by applying the Riemann-Hurwitz theorem to the morphism $C \rightarrow \mathbf{P}^{1}$ induced from the rational map $(x, y) \mapsto y$. [You may assume that the discriminant of $x^{3}+a x+b$ is $-4 a^{3}-27 b^{2}$.]