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As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:

Deformation theory

"Moduli of Curves" defines a versal deformation space as a deformation $\phi: X \rightarrow Y$ such that for any other deformation $\xi: X \rightarrow Z$ and for every point in $Z$ there exists an open set (in the complex topology) $U$ such that the pullback of $\phi$ via $f: U \rightarrow Y$ is $\xi$ restricted to $U$. (I imagine that in general instead of an open set one takes open etale covers - is this true?)

Moduli Spaces

In moduli spaces, versality has always meant a space such that instead of the geometric points being in 1-1 correspondence with the objects we're interested in (over the field of the geo. point), each object is going have several geometric points in the versal space corresponding to it.

Question

Are these two notions related? If so - how?

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  • $\begingroup$ Well, deformation theory can be approached in some way dual to the theory of moduli stacks. That is, you can look at categories cofibered (opfibered?) in groupoids. It doesn't seem too crazy to assume that versality is defined dually in deformation theory as well. Ravi Vakil has notes on deformation theory that explain this approach, but I don't know if they contain this specific definition. $\endgroup$
    – Harry Gindi
    Commented Mar 8, 2010 at 20:38
  • $\begingroup$ This approach extends to a much more general theory using the $(\infty,1)$-categorical approach of left fibrations detailed in Lurie's Higher Topos Theory, which gives us not only categories and higher categories cofibered in groupoids, but also recovers deformation spaces and moduli spaces using $\infty$-groupoids. $\endgroup$
    – Harry Gindi
    Commented Mar 8, 2010 at 20:57

2 Answers 2

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There is an easy answer to your question (without stacks) which has not been given yet: Yes.

The deformation space of a curve $X_0$ is just a local model of a moduli space of all curves near a special member $X_0$.

Your definition of versality for deformations says that, $Y$ is over-parametrizing local deforamtions, i.e. every (germ of) a family of curves with central fiber $X_0$ factorizes non-uniquely over $Y$.

Thus every nearby curve $X_1$ might be represented by several elements $y \in Y$.

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I remember that I was going to give essentially the same answer to this question as Hartmann but apparently I forgot! :) However, it can be given in a more conceptual way (using stacks or groupoids) generalizing the above beyond curves and germs (see the reference below).

Have a look in one of the appendices to the book "Introduction to Singularitites and Deformations" by Greul, Lossen and Shustin (Springer-Verlag). The book is however mainly concerned with "complex space germs" and their deformations but the appendices treats some more general stuff. In particular, there are references to more general treatises.

If you want to go really hi-tech, try the paper "Versal deformations and algebraic stacks" by Michael Artin (hard to read though).

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