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? 
> 
> 
> 



Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
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.


