\subsection{The curves in the \(S_3\) example}
\label{rw:sec:example-curve}

We carry out the construction of this section for the parameter of
Section~\ref{fl:sec:low-rank-example}, in the notation of
Lemma~\ref{fl:lem:weillocal-reversal} and
Proposition~\ref{fl:prop:geometrization-theta}, and we find the
rational point which was used there as the auxiliary puncture. The
numbers are chosen for simple arithmetic, not to minimize the
genus.

Work over \(R_0=\mathbb Z_5\), with
\(\mathcal X=\mathbb P^1_{R_0}\) and coordinate $t$.
The scheme of ordered roots of \(z^3-z-t\) is an \'etale
\(S_3\)-torsor off \(4-27t^2=0\) and infinity; it is the lift
$\sigma^{\rm lift}$ of the parameter. Take
$\mathcal A=\mathcal O(2)$, so that
\[
 \mathcal L=\Omega_{\mathcal X}\otimes\mathcal A^2=\mathcal O(2),
 \qquad
 \kappa_1=\Omega_{\mathcal X}^{(d+1)/2}\otimes\mathcal A^d
         =\mathcal O(d-1).
\]
For the degree $d$ of the cyclic cover we choose a positive integer
satisfying
\[
 d\equiv-1\pmod6,\qquad d\equiv1\pmod4,\qquad
 d\equiv4\pmod5.
\]
The first congruence gives the property of
Lemma~\ref{fl:lem:weillocal-reversal} with $P_x=1$: the inertia
groups of $\sigma^{\rm lift}$ have orders two and three, and
$N_x=0$ because the image is finite, so that
$\rho_x(\gamma)^{-d}=\rho_x(\gamma)$. The second makes the $d$-th
power map the identity on \(\mathbb F_5^\times\), which makes a
rational point easy to find. The third makes \(2d-3\) divisible by
five, which gives the separable polynomial below. The least
positive solution is \(d=29\).

Set \(c_0=4/27\) and take the section $s$ of
\(\mathcal L^{29}=\mathcal O(58)\) with affine polynomial
\begin{equation}\label{fl:eq:rank-one-section}
 s(t)=(t^2-c_0)(t^{55}+t+1).
\end{equation}
Its affine degree is \(57\), so its divisor \(D\) includes
infinity with multiplicity one. Modulo five the derivative of
\(t^{55}+t+1\) is \(1\); its remainder modulo \(t^2-2\) is
\(4t+1\), coprime to \(t^2-2\). Thus \(D\) is finite \'etale
of degree \(58\), contains all the branch points, and avoids
\(t=0\). The factor \(t^2-c_0\) contains the branch points of the
lift, not only their reductions; the zeros of the other factor
form the divisor $D_{\rm ur}$.

In the total space of \(\mathcal L\), the curves
\eqref{fl:eq:integral-composite} are
\[
 C_2:\ w^{29}=s,\qquad
 C_0:\ w(w^{29}-s)=0.
\]
The latter has components \(\mathcal X\) and \(C_2\),
meeting along \(D\). At each simple zero of \(s\) the
cyclic equation is Eisenstein, so $C_2$ is
geometrically irreducible; since \(29\) is a unit in \(R_0\),
it is smooth. The local system on it is the pullback of
$\sigma^{\rm lift}$. At a point of $D$ the pullback raises the
inertia to the power $29$, and
$\rho_x(\gamma)^{29}=\rho_x(\gamma)^{-1}$: the inertia on the two
branches is inverse to each other, as is needed for gluing.
Riemann--Hurwitz gives
\[
 2g(C_2)-2=29(-2)+58(28)=1566,
 \qquad g(C_2)=784.
\]
There are \(58\) nodes, so the arithmetic genus of the nodal
curve \(C_0\) is \(0+784+58-1=841\).

The theta characteristic is determined by the same degrees. By
\eqref{fl:eq:composite-theta} the dualizing sheaf of $C_0$ is the
pullback of
\(\Omega_{\mathcal X}\otimes\mathcal L^{29}
=\mathcal O(-2)\otimes\mathcal O(2)^{29}=\mathcal O(56)\).
Its square root $\pi_0^*\kappa_1$, the theta characteristic,
is the pullback of \(\mathcal O(28)\); no additional choice is
involved.

For the section of $\mathcal L^{30}=\mathcal O(60)$ in
\eqref{fl:eq:explicit-smoothing}, which is called $t$ there, we
take \(1+t^3s\), where $t$ is the coordinate. It is nonzero
everywhere on \(D\), including infinity, where its leading
coefficient is \(1\). It satisfies the normalization
\eqref{rw:eq:t-normalized}: in the standard frames of $\mathcal L$
along $D$ its restriction to $D$ is $1$, and $1=-b^{30}$ for a
primitive fourth root of unity $b\in\mathbb Z_5$. With $e=6$ the
family \eqref{fl:eq:explicit-smoothing} is
\begin{equation}\label{fl:eq:rank-one-smoothing}
 w(w^{29}-s)=\varepsilon^6(1+t^3s),
\end{equation}
and it smooths every node. The exponent \(6\) is a common
multiple of the two inertia orders; locally the equation is
\(xy=\varepsilon^6\) up to a unit.
Propositions~\ref{fl:lem:balanced-parameter-tower}
and~\ref{fl:prop:geometrized-parameter} give the unramified local
system $\rho$ on the smooth proper generic fibre, and
Proposition~\ref{fl:prop:geometrization-theta} extends the theta
characteristic. The arithmetic and the geometric monodromy group
of $\sigma$ are both \(S_3\), so
Proposition~\ref{fl:prop:geometrized-parameter} gives
$\mathsf{Mon}_\rho=S_3$.

At \(t=0\) the equation \(w^{29}=s(0)=-4/27\) has the solution
\(w=3\) modulo five, and Hensel's lemma gives its unique lift
\(w_0\) in \(\mathbb Z_5\). The derivative of the left side of
\eqref{fl:eq:rank-one-smoothing} with respect to $w$ at this point
is \(29w_0^{29}\), a unit, so the point extends to a section of
the smoothing; this is the divisor $D_{\rm aux}$ of
Lemma~\ref{fl:lem:closed-return-puncture}, with $d_{\rm aux}=1$.
It lies on the auxiliary component, away from \(D\). The torsor of
ordered roots is split at \(t=0\), so the Frobenius element at
this puncture is $g_D=1$, which gives the determinant in
\eqref{fl:eq:rank-one-nonzero-trace}. The legs of the Hecke
modifications on the original component still range over
\(\mathcal X-D\).

The genus \(841\) is the cost of satisfying the three elementary
congruences with the same cyclic cover. The example makes
the inverse inertia, the theta characteristic and the rational
puncture explicit;
only their existence enters the realization theorem.