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.

  • 2
    $\begingroup$ Your neighborhoods are Zariski neighborhoods? You should really use the étale topology, with the Zariski topology I don't think this is true. Also, your definition of "versal" is not the standard one, for very good reasons, even if you use the étale topology. I'll explain this tomorrow, if someone else does not do this first. You should read Mike Artin's "Lectures on deformations of singularities". $\endgroup$ – Angelo Sep 2 '10 at 19:49
  • 4
    $\begingroup$ Read the largely self-contained (modulo some knowledge of the etale topology, and accepting Artin approx. as black box if not known already) section 2 of Chapter III (esp. 2.7 and 2.8) of the book "Etale cohomology and the Weil conjecture" by Freitag & Kiehl. That is a really nice discussion of the deformation theory of ordinary double points in any relative dimension over any noetherian base (though they don't use terminology such as "versal deformations", they do prove the central fact for the case of ordinary double points, in fact going far beyond the setting of an artinian base). $\endgroup$ – BCnrd Sep 2 '10 at 20:17

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