I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples.

The basic question I would like to understand is how to prove something is a versal deformation of the simple node. To be specific, let $$ X_0 := Spec(\mathbb{C}[x,y]/xy) $$ and let $$ X_{ver} := Spec(\mathbb{C}[x,y,t]/xy-t) $$ Note that there is a natural map $X_{ver}\rightarrow \mathbb{A}^1$ which sends a point to its $t$-coordinate, whose fiber over zero is $X_0$.

I want to claim that $X_{ver}$ is a versal deformation of $X_0$. What I mean by this is as follows. Consider a faithfully flat morphism $\pi:X \rightarrow B$, together with a distinguished point $p\in B$ such that $\pi^{-1}(p)$ is isomorphic to $X_0$. Then there is

- an open neighborhood $B'\subset B$ of $p$,
- a (non-unique) map $f_0:B'\rightarrow \mathbb{A}^1$ which takes $p$ to $0$,
- and a (non-unique) map $f_1:\pi^{-1}(B')\rightarrow X_{ver}$

such that the natural diagram $$\pi^{-1}(B') \rightarrow X_{ver} $$ $$\downarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow $$ $$ B' \;\;\;\;\; \rightarrow \;\;\;\;\; B $$ is commutative, and is a pullback diagram (ie, a fibered product). Conceptually, this says that any (sufficiently small) faithfully flat family which contains a copy of $X_0$ must be gotten from pullback from the family $X_{ver}\rightarrow \mathbb{A}^1$. (Note that when I refer to preimages and fibers, I mean the scheme-theoretic ones)

However, I am having trouble showing this. I have been approaching this from the algebraic perspective, by considering formal deformations of the algebra $\mathbb{C}[x,y]/xy$. These aren't hard to understand, and it is straight-forward to construct maps from $\mathbb{C}[x,y,t]/xy-t$ to a given formal deformation. However, I can't figure out how to make these maps surjective; ie, to keep the corresponding scheme maps from being multi-sheeted covering maps. In any event, I suspect this algebraic approach is wrong anyway, since it will, at best, only prove its a formal versal deformation.

I should also mention I am not sure if `faithfully flat' is the idea I want here. I am bad at knowing what modifiers to a flat family prevent pathological fibers, so if this is the wrong notion, please correct me.