TITLE: Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
QUESTION [7 upvotes]: Let $V_2$ and $V_3$ be the two hypersurfaces of $\mathbb P^3$ defined by
\begin{equation*}
V_2:={x_2x_3 + r(x_0, \, x_1)=0}, \quad V_3:={x_2^3+x_3^3+s(x_0, \, x_1)=0},
\end{equation*}
where $r, \, s \in \mathbb{C}[x_0, \, x_1]$ are general homogeneous forms of degree $2$ and $3$, respectively. 
Then $C_4:=V_2 \cap V_3$ is a smooth, canonical curve of genus $4$.  Denoting by $\xi$ a primitive third root of unity, then $C_4$  admits a free action of the cyclic group $\langle \xi \rangle \cong \mathbb Z/3 \mathbb Z$, defined by
\begin{equation*}
\xi \cdot [x_0: x_1:x_2:x_3] = [x_0: x_1: \xi x_2: \xi^2 x_3]
\end{equation*}
and the quotient $C_2 := C_4/ \langle \xi \rangle$ is a smooth curve of genus $2$. 
A naive count of parameters shows that the number of moduli on which this construction depends is $$\dim |\mathcal{O}_{\mathbb{P}^1}(2)| + \dim |\mathcal{O}_{\mathbb{P}^1}(3)| - \dim G= 7 - \dim G,$$
where $G \subset \textrm{Aut}(\mathbb{P}^4)$ is the subgroup of projectivities sending both $V_2$ and $V_3$ to hypersurfaces of the same form. If my computations are correct, $G$ has dimension $4$ so the construction actually depends on $3$ parameters. 
On the other hand, $3$ is also the dimension of the moduli space $\mathcal{M}_2$. This shows that the general curve $C_2$ of genus $2$ can be constructed in this way.

Question.  Is it possible to obtain all smooth curves of genus $2$ by means of  this construction?

REPLY [14 votes]: Yes. Start from a genus 2 curve $C_2$, and choose a point of order 3 in $JC_2$, giving rise to  an étale $\mathbb{Z}/3$-covering $C_4\rightarrow C_2$. 
Then $C_4$ cannot be hyperelliptic: a $g^1_2$ on $C_4$ would be stable under the covering automorphism  $\sigma $, hence descend to $C_2$, which is impossible for degree reasons.   $\sigma $ acts on $H^0(C_4,K_{C_4})$ with eigenvalues $(1,1,\xi ,\xi ^2)$: the invariant part comes from $C_2$, and the trace must be 1 by the holomorphic Lefschetz formula. Writing down the  invariant quadric and cubic(s) containing $C_4$ leads to your formulas.