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?



Versality in deformation theory vs. versality in moduli spaces
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 $y \in Y$.


