I would like to understand how blowing down affects (first order) deformations.
For a concrete example, set $D=k[\epsilon]/(\epsilon^2)$ and consider in $\mathbb{A}^2_D=\operatorname{Spec} D[x,y]$ the following deformation of the two coordinate axis and the line $y=1$:
$$X=\operatorname{Spec} D[x,y]/(xy-a\epsilon)\cap(x(y-1)-b\epsilon)$$ for $a,b\in k$.
Now blow down the $y$-axis. In other words, consider the map to $\text k[u,v]/(uv)$ given by $$(t,0)\mapsto (t,0), (t,1)\mapsto (0,t), (0,t)\mapsto (0,0)$$ which, if I am correct, should be given by $$u\mapsto x(y-1),v\mapsto xy$$ (or maybe the other way around, doesn't matter).
Now it is clear that $uv-\epsilon(au+bv)$ (again, maybe I switched $a$ and $b$) maps to $0$, so the structure at the node is $$D[u,v]/(uv-\epsilon(au+bv))$$ whose completion is $D[[u,v]]/(uv)$ (only constant term matters, and here it is $0$), so it is a trivial deformation.
I would have expected something like $D[[u,v]]/(uv-\epsilon(a+b))$... what did I do wrong? Do really all deformations blow down to the trivial one?
Note: I carried out the same computation in the case of the disjoint union of two nodes mapping to a node collapsing one component from each node. The result seems to still be the same.