*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 $\mathbb R^{n+1}$?



Examples of the moduli space of X giving facts about a certain X
The easiest example I can think of is the natural incidence correspondence between $\mathbb{P}^3$ and the parameter space of cubic surfaces. This can be used to show that every cubic surface contains a line; from this [it follows easily](https://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates/20261#20261) that every smooth cubic surface contains exactly $27$ lines.


Another example is the moduli space of stable maps constructed by Kontsevich; this parametrizes certain maps from curves to (to stick with a simple case) $\mathbb{P}^2$. It can be used to answer the following question: given $3d-1$ points in $\mathbb{P}^2$ in general position, compute the number $N\_d$ of *rational* curves of degree $d$ passing through these points. It turns out that the values $N\_d$ satisfy a certain recursive relation which allows you to compute all these numbers starting from the obvious $N\_1 = 1$ (through $2$ points passes exactly one line). You can find the formula at the entry [Kontsevich's formula](http://rigtriv.wordpress.com/2008/12/23/kontsevichs-formula/) on [Rigorous trivialities](https://rigtriv.wordpress.com); it yields for instance $N\_2 = 1$ and $N\_3 = 12$.


Yet another example, again more elementary is the following. The Grassmannian $G = \operatorname{Gr}(1, \mathbb{P}^3)$ parametrizes lines in $\mathbb{P}^3$. The computation of the cohomology of $G$ allows you to compute the number of lines which are incident to $4$ fixed lines in general position (it turns out this number is $2$).


