moduli space

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One can say a great deal about the moduli space purely in terms of the moduli functor without even knowing the moduli space exists
- For example, the dual numbers. $\operatorname{Spec}k{ [{\varepsilon}] / {\varepsilon}^2 }$
Prototypical example of a moduli space: the Grassmannian ${\operatorname{Gr}}_k({\mathbf{C}}^n)$.
Common example: the Hilbert scheme.
Apparently a fundamental class exists for closed subvarieties? Maybe just closed subvarieties of a moduli space?
Prominent example when studying elliptic curves over ${\mathbf{Q}}$: the modular curve $X_0(n)$.
- level structure: examples are $\Gamma(N), \Gamma_1(N), \Gamma_0(n)$ for squarefree $n$.

Fine moduli spaces

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$\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{M}\mkern-1.5mu}\mkern 1.5mu_{0, 4}\cong {\mathbf{P}}^1_{/ {{\mathbf{C}}}} $ using the classical cross-ratio.