TITLE: Show that a smooth plane quartic is never hyperelliptic
QUESTION [7 upvotes]: I have been asked to show that a smooth plane quartic is never hyperelliptic. I know that
i) The genus of any such curve is 3
ii) The statements of Riemann-Roch and Riemann-Hurwitz
iii) A curve is hyperelliptic if there's a degree 2 map from it to P1.
Can I have a hint about what to do please? Cheers.

REPLY [3 votes]: It suffices to show that the plane quartic curve $C$ is a canonical curve. Using adjunction formula, $K_C=i^*_C(\mathcal{O}_{\mathbb{P}^2}(-3)\otimes\mathcal{O}_{\mathbb{P}^2}(4))=i_C^*\mathcal{O}_{\mathbb{P}^2}(1)$. So $i_C: C\hookrightarrow\mathbb{P}^2$ is a canonical embedding.