Semistable infinite chain from a toric fan ($U_i=\mathrm{Spec}\mathbb{Z}[x_i,y_i],q=x_i{y}_i$)

Take rays $(i,1)$ and cones generated by consecutive rays in ${\mathbb{Z}}^2$. Their dual generators $(1,-i)$ and $(-1,i+1)$ give $x_i=t{q}^{-i}$, $y_i=t^{-1}{q}^{i+1}$. Adjacent charts glue by $x_{i+1}=y_i^{-1}$, $y_{i+1}=x_i{y}_i^2$. The fibre $q=0$ is a chain of projective lines: each component has two affine-line charts joined by reciprocal coordinates, and consecutive components meet at one node. Inverting $q$ identifies all charts with the multiplicative group scheme over $\mathbb{Z}[q,q^{-1}]$. The local node algebra is free over $\mathbb{Z}[q]$, with basis $1,x_i^a,y_i^b$ for $a,b\ge 1$, so the morphism is flat.

// Target: toric-geometry.bigb