the monomials of the form $$t_{\alpha}=(z_{1}z_{2})^{\alpha_{1}}(z_{3}z_{4})^{\alpha_{2}}\cdots(z_{2s-3}z_{2s-2})^{\alpha_{s-1}} z_{2s-1}^{\alpha_{s}},$$ where $\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}$. As $V_{s}$ has cardinality $2^{s-1}-1$, the number of the above set of generators is $N=2^{s-1}+2s-2$.

Using the map $\Phi:D \to {\mathbb C}^{N}$, whose coordinates are $t_{1},\ldots,t_{2s-1}$ and the monomials $t_{\alpha}$, $\alpha \in V_{s}$, one obtain that the Riemann surface induced by $\Phi(D)$ is isomorphic to $S$ and that its equations are given by

$$\Phi(D)=\left\{ \begin{array}{ccl}
t_{1}+t_{2}+t_{3}&=&0\\
\lambda t_{1}+t_{2}+t_{4}&=&0\\
\mu_{1,1}t_{1}+t_{2}+t_{5}&=&0\\
\mu_{1,2}t_{1}+t_{2}+t_{6}&=&0\\
\vdots & \vdots& \vdots\\
\mu_{k,1}t_{1}+t_{2}+t_{2k+3}&=&0\\
\mu_{k,2}t_{1}+t_{2}+t_{2k+4}&=&0\\
\vdots & \vdots& \vdots\\
\mu_{s-2,1}t_{1}+t_{2}+t_{2s-1}&=&0\\
\mu_{s-2,2}t_{1}+t_{2}+1&=&0\\
t_{\alpha}^{2}&=&(t_{3}t_{4})^{\alpha_{2}}\cdots(t_{2s-3}t_{2s-2})^{\alpha_{s-1}} t_{2s-1}^{\alpha_{s}}, \quad \alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s} 
\end{array}
\right\}.$$

The first linear equations permit to write $t_{2},\ldots,t_{2s-1}$ in terms of $t_{1}$ as follows: $$t_{2}=-1-\mu_{s-2,2}t_{1}$$ $$t_{3}=1+(\mu_{s-2,2}-1)t_{1}$$ $$t_{4}=1+(\mu_{s-2,2}-\lambda)t_{1}$$ $$t_{2k+3}=1+(\mu_{s-2,2}-\mu_{k,1})t_{1}, \quad t_{2k+4}=1+(\mu_{s-2,2}-\mu_{s,2})t_{1}, \quad k=1,\ldots, s-3,$$ $$t_{2s-1}=1+(\mu_{s-2,2}-\mu_{s-2,1})t_{1}.$$

We may then eliminate the variables $t_{2},\ldots,t_{2s-1}$ and just keep the variables $t_{1}$ and $t_{\alpha_{1},\ldots,\alpha_{s}}$.

Let us set $t_{1}=z$ and $t_{\alpha}=w_{\alpha}$, for $\alpha \in V_{s}$. In these new $2^{s-1}$ coordinates, the above curve is isomorphic to the one given by

$$C=\left\{ \begin{array}{c}
w_{\alpha}^{2}=z^{\alpha_{1}}(-1-\mu_{s-2,2}z)^{\alpha_{1}}
(1+(\mu_{s-2,2}-1)z)^{\alpha_{2}}(1+(\mu_{s-2,2}-\lambda)z )^{\alpha_{2}}\cdots\\
\cdots  (1+(\mu_{s-2,2}-\mu_{k,1})z)^{\alpha_{k+2}}(1+(\mu_{s-2,2}-\mu_{k,2})z)^{\alpha_{k+2}} \cdots \\
\cdots (1+(\mu_{s-2,2}-\mu_{s-3,1})z)^{\alpha_{s-1}}(1+(\mu_{s-2,2}-\mu_{s-3,2})z)^{\alpha_{s-1}} (1+(\mu_{s-2,2}-\mu_{s-2,1})z)^{\alpha_{s}},\\ 
\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}.\\
\end{array}
\right\}$$

By making the choices as described in the hypothesis of the theorem for $K_{\alpha}$ and the values of $\eta_{0}$, $\eta_{1}$, $\eta_{2}$, $\eta_{3}$ and $\eta_{k,j}$, then the above curve 