TITLE: Examples of the moduli space of X giving facts about a certain X
QUESTION [14 upvotes]: What is a good example of a fact about the moduli space of some object telling us something useful about a specific one of the objects?
I am currently learning about moduli spaces (in the context of the moduli space of elliptic curves). While moduli spaces do seem to be fascinating objects in themselves, I am after examples in which facts about a moduli space tell us something interesting about the specific objects that they parametrise. For example, does the study of $\mathbb RP^n$ tell us anything we don't already know about some given line through the origin (say, the $x_1$-axis) in $\mathbb R^{n+1}$?

REPLY [2 votes]: Earlier today I thought of something that may be an example of this. If not, at least it's very elementary and has something of the same spirit. I don't really know what any kind of moduli space is in any technical sense.
Suppose you know the formula for volume of a sphere, but you do not know the formula for surface area. Parameterize the set of all spheres in $ \mathbf{R}^3 $ at the origin by their radius $ r $, forming the space $ X = ( 0, \infty ) $. Assignment of volume $ V(r) $ to the sphere with radius $ r $ defines a differentiable function $ V $ on $ X $. It is clear from geometry that if $ r $ is changed by $ \Delta r $ then its volume should change by approximately $ A ( r ) \cdot \Delta r $, for $ A ( r ) $ the surface area of the sphere with radius $ r $. In other words, $ A ( r ) = V^{\prime} ( r ) = 4 \pi r^2 $.
I guess the point is that even if you are interested in only one specific sphere S, you can compute the number that you want by forming a continuum of spheres and using analysis on that parameter space. Until you have the space, there is no derivative. Does that make any sense?