State the Riemann-Roch theorem for a smooth projective curve $V$, and use it to outline a proof of the Riemann-Hurwitz formula for a non-constant morphism between projective nonsingular curves in characteristic zero.

Let $V \subset \mathbb{P}^{2}$ be a smooth projective plane cubic over an algebraically closed field $k$ of characteristic zero, written in normal form $X_{0} X_{2}^{2}=F\left(X_{0}, X_{1}\right)$ for a homogeneous cubic polynomial $F$, and let $P_{0}=(0: 0: 1)$ be the point at infinity. Taking the group law on $V$ for which $P_{0}$ is the identity element, let $P \in V$ be a point of order 3 . Show that there exists a linear form $H \in k\left[X_{0}, X_{1}, X_{2}\right]$ such that $V \cap V(H)=\{P\}$.

Let $H_{1}, H_{2} \in k\left[X_{0}, X_{1}, X_{2}\right]$ be nonzero linear forms. Suppose the lines $\left\{H_{i}=0\right\}$ are distinct, do not meet at a point of $V$, and are nowhere tangent to $V$. Let $W \subset \mathbb{P}^{3}$ be given by the vanishing of the polynomials

$$X_{0} X_{2}^{2}-F\left(X_{0}, X_{1}\right), \quad X_{3}^{2}-H_{1}\left(X_{0}, X_{1}, X_{2}\right) H_{2}\left(X_{0}, X_{1}, X_{2}\right)$$

Show that $W$ has genus 4 . [You may assume without proof that $W$ is an irreducible smooth curve.]