I am looking for a proof or a reference for the following fact:

Every automorphism of any non-singular algebraic curve in $\mathbb{P}^2(\mathbb{C})$ of genus $g\geq 2$ is linear, i.e., can be viewed as an element of $\mbox{PGL}_3(\mathbb{C})$.

I read the proof for higher dimensional hypersurfaces (Matsumura-Monsky) but they use the fact that the Picard group of the variety is torsion-free (and in this case is false, since $\mbox{Pic}^0(C)$ is an abelian variety of positive dimension).

Thanks!