(a) Let $X \subseteq \mathbb{P}^{2}$ be a smooth projective plane curve, defined by a homogeneous polynomial $F(x, y, z)$ of degree $d$ over the complex numbers $\mathbb{C}$.

(i) Define the divisor $[X \cap H]$, where $H$ is a hyperplane in $\mathbb{P}^{2}$ not contained in $X$, and prove that it has degree $d$.

(ii) Give (without proof) an expression for the degree of $\mathcal{K}_{X}$ in terms of $d$.

(iii) Show that $X$ does not have genus 2 .

(b) Let $X$ be a smooth projective curve of genus $g$ over the complex numbers $\mathbb{C}$. For $p \in X$ let

$G(p)=\left\{n \in \mathbb{N} \mid\right.$ there is no $f \in k(X)$ with $v_{p}(f)=n$, and $v_{q}(f) \leqslant 0$ for all $\left.q \neq p\right\} .$

(i) Define $\ell(D)$, for a divisor $D$.

(ii) Show that for all $p \in X$,

$$\ell(n p)= \begin{cases}\ell((n-1) p) & \text { for } n \in G(p) \\ \ell((n-1) p)+1 & \text { otherwise }\end{cases}$$

(iii) Show that $G(p)$ has exactly $g$ elements. [Hint: What happens for large $n$ ?]

(iv) Now suppose that $X$ has genus 2 . Show that $G(p)=\{1,2\}$ or $G(p)=\{1,3\}$.

[In this question $\mathbb{N}$ denotes the set of positive integers.]