My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the geometry of the objects that the space parametrizes.
As an example, it is known that the dimension of the moduli space of curves of genus $g$ has dimension $3g-3$, which we might think of as $dimension \leftrightarrow genus$. Are there other examples of this type of relationship?
I accept the possibility that maybe this isn't even an interesting question, since it misses the point of moduli spaces somehow, and I would appreciate an explanation if that is the case!
When one considers the moduli space of curves of genus $g$, it is natural to expect its dimension to depend on $g$. In the same way, for instance, the dimension of the space $$\mathcal{M}_{g,n}^{G}=\mathrm{Hom}( \pi_1(C_{g,n}),G )/\sim$$ of representations of the fundamental group of $n$-punctured Riemann surface of genus $g$ into a group $G$ will depend on $g$, $n$, the dimensions of $G$ and of its maximal torus in the case of Lie groups, etc. However, this looks very tautological. The abelian case of the above provides an illustrative example: $\mathcal{M}_{g,n}^{\mathbb{Z}}\cong H^1\left(C_{g,n}\right)$.
More generally, if one can distingush geometric characteristics of different points of a moduli space $\mathcal{M}$, then it becomes natural to refine its definition and to study the corresponding subspaces (e.g. moduli space of curves breaks into components labeled by different genera).