Let $F(X, Y, Z)$ be an irreducible homogeneous polynomial of degree $n$, and write $C=\left\{p \in \mathbb{P}^{2} \mid F(p)=0\right\}$ for the curve it defines in $\mathbb{P}^{2}$. Suppose $C$ is smooth. Show that the degree of its canonical class is $n(n-3)$.

Hence, or otherwise, show that a smooth curve of genus 2 does not embed in $\mathbb{P}^{2}$.