Automorphism of genus $6$ plane curves

Question

Let $C\subset \mathbf{P}^2$ be a genus $6$ smooth plane curve, $\sigma\colon C\to C$ be an automorphism, is $\sigma$ necessarily induced from an automorphism of $\mathbf{P}^2$?

Answer 1

I think the following argument works over $\mathbb{C}$.

Let $\phi \in \text{Aut}(C)$. Then $\phi^* \mathcal{O}(1) \cong \mathcal{O}(1)$. This follows from M. Noether theorem: a smooth plane curve of degree $d$ has no $g^1_k$ with $k<d-2$, every $g^1_{d-1}$ is induced by the pencil of lines through a point on the curve and linear system $|\mathcal{O}(1)|$ is the unique $g^2_d$ on $C$.

Suppose that $\phi$ acts trivially on $H^0(C, \mathcal{O}(1))$ and therefore induces the trivial action on $\mathbb{P}^2=\mathbb{P}(H^0(C, \mathcal{O}(1))^{\vee})$. Product of sections give a surjective map $$ H^0(C, \mathcal{O}(1)) \otimes H^0(C, \mathcal{O}(1)) \to H^0(C, \mathcal{O}(2)). $$

So, $\phi$ acts trivially on $H^0(C, \mathcal{O}(2))$, but by adjunction formula $K_C \cong \mathcal{O}(2)$, then by Hodge decomposition $\phi$ acts trivially on $H^1_{DR}(C,\mathbb{C})$. Therefore, the Lefschetz number is $$ L(\phi) = \sum_i (-1)^i \text{tr}(\phi_*| H^i_{DR}(C,\mathbb{C}))=2-2g < 0, $$ that is not possible because for complex manifolds all local intersection numbers are positive.

Therefore, we get an injective map $$ i: \text{Aut}(C) \to \mathbb{P}\text{GL}_3(\mathbb{C}). $$