Automorphisms of non-hyperelliptic curve of genus 3 in $\mathbb{P}^{2}$ I have a question from R. Vakil's exercise 19.7.C which goes as follows: Suppose $C'\subset\mathbb{P}^{2}$ is a smooth plane quartic curve. Show that there is bijection between automorphisms of $C'$ and automorphisms of $\mathbb{P}^{2}$ that preserves $C'$ as a set.
I have no idea how this should be done. The only information I have is this: that a curve of genus $3$ that is non-hyperelliptic is canonically embedded into $\mathbb{P}^{2}$, which means it's canonical bundle $\omega_{C'}$ is very ample.
I suppose that if we have an automorphism of $P^{2}$ that preserves $C'$, say given by $\varphi:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$, then it is obvious that the restriction of $\varphi$ to $C'$ is an automorphism of $C'$. 
But I have trouble with finding the inverse: given an automorphism of $C'$, why is it that we can find an automorphism of $\mathbb{P}^{2}$ that fixes $C'$ as a set? (i.e. why do morphism of curves lifts to morphism of $\mathbb{P}^{2}$?)
I think it has something to do with the canonical bundle on $C'$, but I can't make the link.
Thanks!
 A: The canonical series is cut out by the straight lines. Namely, the sections of the canonical bundle are the coordinates $X,Y,Z$ (and its linear combinations). Then any automorphism of $C'$ induces a linear transformation of the three dimensional $span_{\mathbb{C}} (X,Y,Z)$ i.e. an automorphism of $H^{0}(C', \omega_{C'})$. This automorphism is the automorphism of $\mathbb{P}^2$ you are looking for. 
About your question:  why do morphism of curves lifts to morphism of $\mathbb{P}^2$? This is not true in general. 
For the smooth quartic you can lift the automorphism since the very ample divisor giving the embedding is invariant by automorphism i.e. an automorphism of $C'$ take canonical divisors into canonical divisors. 
I should say that a more general and well known fact is true. Namely, if $C'$ is a non hyperelliptic curve of genus $g$ then a canonical embedding $C' \hookrightarrow \mathbb{P}^{g-1}$ is unique up to automorphism of $\mathbb{P}^{g-1}$ hence any automorphism of $C'$ lift to an automorphism of $\mathbb{P}^{g-1}$.
