= Solution
Use the <toric scheme over a base ring> construction over $\mathbb Z$. Take the <cocharacter lattice of an algebraic torus> $N=\mathbb Z^2$ and dual <character lattice of an algebraic torus> $M=\operatorname{Hom}(N,\mathbb Z)$. For every $i\in\mathbb Z$, set
$$
v_i=(i,1),\qquad \rho_i=\mathbb R_{\geq0}v_i,\qquad
\sigma_i=\mathbb R_{\geq0}v_i+\mathbb R_{\geq0}v_{i+1}.
$$
These <cones in toric geometry>, together with their rays and the zero cone, form an <infinite fan in toric geometry>. Adjacent two-dimensional cones meet in their common ray; all other pairs meet only at zero. Thus the face and intersection axioms hold even though there are infinitely many charts.
The consecutive vectors $v_i,v_{i+1}$ are a <basis> of $N$, since their <determinant> is $-1$. The dual cone has integral generators
$$
a_i=(1,-i),\qquad b_i=(-1,i+1),
$$
with $\langle a_i,v_i\rangle=0$, $\langle a_i,v_{i+1}\rangle=1$, $\langle b_i,v_i\rangle=1$, $\langle b_i,v_{i+1}\rangle=0$. Its integral points form the <free commutative monoid> $\mathbb N a_i+\mathbb N b_i$. Hence the associated <semigroup algebra> chart is simply
$$
U_i=\operatorname{Spec}\mathbb Z[\sigma_i^\vee\cap M]
=\operatorname{Spec}\mathbb Z[x_i,y_i].
$$
On the dense <algebraic torus>, write $t=\chi^{(1,0)}$ and $q=\chi^{(0,1)}$. The chart coordinates are
$$
\boxed{x_i=tq^{-i},\qquad y_i=t^{-1}q^{i+1},\qquad q=x_i y_i.}
$$
Although these expressions use inverses on the torus, $x_i,y_i,q$ are regular on the indicated charts.
The common-ray chart of $U_i$ and $U_{i+1}$ is $D(y_i)\subset U_i$, identified with $D(x_{i+1})\subset U_{i+1}$ by
$$
\boxed{x_{i+1}=y_i^{-1},\qquad y_{i+1}=x_i y_i^2.}
$$
The inverse is $y_i=x_{i+1}^{-1}$, $x_i=y_{i+1}x_{i+1}^2$. Nonadjacent charts meet in the common dense torus, and all transition maps agree there with the same <characters of an algebraic torus> $t,q$, so their cocycle identities follow. By <gluing of schemes along open subschemes>, these charts construct a <scheme> $X$.
The lattice projection $(a,b)\mapsto b$ carries every $\sigma_i$ into the nonnegative ray of the fan of the <affine line>. It therefore defines a <toric morphism>
$$
f:X\longrightarrow\operatorname{Spec}\mathbb Z[q].
$$
Explicitly its pullback on each chart sends $q$ to $x_i y_i$. The adjacent-chart formula preserves that product, so the local morphisms glue.
Compute its <scheme-theoretic fibre> over $q=0$:
$$
Y\cap U_i=\operatorname{Spec}\mathbb Z[x_i,y_i]/(x_i y_i).
$$
This is the union of the two coordinate <affine lines>, each with multiplicity one. Let $D_i$ denote the <toric divisor> corresponding to $\rho_i$. Its part in $U_i$ is $y_i=0$, with coordinate $x_i$; its part in $U_{i-1}$ is $x_{i-1}=0$, with coordinate $y_{i-1}$. On the overlap these coordinates satisfy $x_i=y_{i-1}^{-1}$. Consequently these two <affine lines> glue to
$$
D_i\cong\mathbb P^1_{\mathbb Z}.
$$
The neighbouring components $D_i,D_{i+1}$ meet at the section $p_i=(x_i,y_i)=(0,0)$, isomorphic to $\operatorname{Spec}\mathbb Z$. No other components meet. Every special-fibre chart has exactly these two branches and no embedded or extra components. Thus
$$
\boxed{Y=\cdots\cup D_{-1}\cup D_0\cup D_1\cup\cdots,
\quad D_i\cong\mathbb P^1_{\mathbb Z},\quad
D_i\cap D_j=\varnothing\ \text{if }|i-j|>1.}
$$
The union is an infinite chain, not a finite-type curve disguised as one: its infinitely many components account for the infinitely many toric charts.
Over the <principal open subset> $D(q)$, both $x_i$ and $y_i$ are <units> because their product is $q$. Every chart then becomes the same torus chart:
$$
\begin{aligned}
\mathbb Z[x_i,y_i,(x_i y_i)^{-1}]
&=\mathbb Z[q,q^{-1},x_i,x_i^{-1}]\\
&=\mathbb Z[q,q^{-1},t,t^{-1}],\qquad t=x_i q^i.
\end{aligned}
$$
The transition maps identify the coordinate $t$ exactly, so
$$
\boxed{X\times_{\mathbb Z[q]}\mathbb Z[q,q^{-1}]
\cong\mathbb G_{m,\mathbb Z[q,q^{-1}]}.}
$$
This is the generic region described in the question; over the actual generic point of $\operatorname{Spec}\mathbb Z[q]$ its fibre is $\mathbb G_{m,\mathbb Q(q)}$.
The construction is also flat. Indeed, each node chart has a free $\mathbb Z[q]$-module <basis>
$$
1,\quad x_i^a\ (a\geq1),\quad y_i^b\ (b\geq1).
$$
Every monomial $x_i^u y_i^v$ reduces uniquely by taking out $q^{\min(u,v)}$, and the resulting monomials are independent. Thus the chart morphisms are <flat morphisms>. This verifies directly that the <semistable infinite chain from a toric fan> is a degeneration of the <multiplicative group scheme> rather than an unrelated special fibre.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-17-toric-chain.png]
{title=The infinite toric fan, its chain of projective lines, and the period-three quotient fibre}
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