Moduli functors are contravariant functors from certain category of base spaces to the category of sets, we imagine them as the functors mapping a base space \(B\) to the set of families \(F(B)\) over \(B\). This concept is of fundamental importance in the moduli theory.
Moduli Functors
For a category \(\mathcal{C}\) and an object \(X\in \mathcal{C}\), there is a functor of points \(h_X\) that somehow implements the idea of “points of \(X\)”.
\[ h_X : \mathcal{C}^{op} \to {\bf Set}, \quad T \mapsto \mathrm{Hom}_\mathcal{C}(T, X) \]
For a base space \(B\), the set \(h_X(B)\) is the set of morphisms \(B\to X\) in \(\mathcal{C}\). For example \(B = \mathrm{Spec}k\) for a field \(k\), then \(h_X(B)\) is the set of \(k\)-points of \(X\).
The category \(\mathrm{Fun}(\mathcal{C}^{op}, {\bf Set})\) is usually called the category of presheaves on \(\mathcal{C}\).
Yoneda Lemma
For any contravariant functor \(F: \mathcal{C}^{op} \to {\bf Set}\), there is a natural isomorphism
\[ \mathrm{Hom}(h_X, F) \cong F(X), \quad \eta\mapsto \eta_X(1_X) \]
where \(\eta_X : h_X(X)\to F(X)\) is the component of \(\eta: h_X\to F\) at \(X\).
The functor \(h_\bullet : \mathcal{C}\to \mathrm{Fun}(\mathcal{C}^{op}, {\bf Set})\) is fully faithful.
In fact, simply applying Yoneda Lemma to \(h_Y\)
\[\mathrm{Hom}(h_X, h_Y) = h_Y(X) = \mathrm{Hom}(X, Y).\]
Representable Functors
A presheaf \(F: \mathcal{C}^{op} \to {\bf Set}\) is called representable if there is an isomorphism \(\xi: F\to h_X\) for some \(X\in \mathcal{C}\). The object \(X\) is called a representing object of \(F\).
Usually in moduli problem, \(F(T)=\{\text{families over } T\}/\sim\) for some equivalence relation \(\sim\) with functor maps being pullbacks of families, and we want to know if \(F\) is representable by \(X\), if so, \(X\) is the fine moduli space of the problem because \(T\)-families are just \(T\)-points of \(X\).
Fine Moduli Space
When a data for fine moduli space \(\xi: F\to h_X\) is given, we can define a universal family \(U\to X\) as family in \(F(X)\) that maps to \(1_X\in h_X(X)\) under \(\xi_X\).
Now any family over \(T\)
\[f:V\to T\in F(T) \simeq h_X(T)\]
corresponds to a map \(\xi_T(f) : T\to X\). The functor map \(F(\xi_T(f)) : F(X)\to F(T)\) maps the universal family \(u:U\to X\) to \(f:V\to T\), as shown in the naturality diagram
As a result, the family \(f:V\to T\) is the functor image, and thus the pullback of the universal family \(U\to X\) along the map \(\xi_T(f) : T\to X\).
Coarse Moduli Space
The condition of being fine moduli space is pretty strong and our schemes are not strong enough to handle the singularities and non-separatedness that may arise. So if you want to stay in the usual algebraic geometry world, we often consider a weaker notion of moduli space called coarse moduli space.
A coarse moduli space is a natural transformation (not requiring isomorphism) \(\xi :F\to h_M\), if
It encodes the objects on geometric points correctly, i.e. for any algebraic closed field \(k\), \(\xi_{\mathrm{Spec}k} : F(\mathrm{Spec}k)\to h_M(\mathrm{Spec}k)\) is a bijection.
It is universal, or somehow of minimal degree of freedom, i.e. for any other natural transformation \(\eta : F\to h_{M'}\) factor through \(\xi\).
Examples
The global section functor \(\Gamma: {\bf Sch}_S^{op} \to {\bf Set}\) mapping \(X\) to \(\mathcal{O}_X(X)\) is representable by the affine line (over \(S\)) \(\mathbb{A}^1_S\).
To see this, note that
\[\mathcal{O}_X(X) = \mathrm{Hom}_{{\bf Sch}_S}(X, \mathbb{A}^1_S) = \mathrm{Hom}_{\mathcal{O}_S}(\mathcal{O}_S[t], \mathcal{O}_X).\]
This can be seen as the family of points in the trivial fibre bundle, and it exactly coincides with the concept of family of points in \(\mathbb{A}^1_S\), the moduli space of points. The universal family is the projection \(\mathbb{A}^1_S\to S\).
The scheme \(\mathbb{P}_S^n\) represents the following moduli problem, the families over \(T\) are line bundles \(L\) on \(T\) and \(n+1\) sections \(s_0,\dots, s_n\), \((L,s_0,\dots,s_n)\) such that these sections do not vanish together, or equivalently, \(\mathcal{O}_X^{n+1}\to L\) is surjective. Two families over \(T\) are equivalent if there is an isomorphism of line bundles that sends one family to the other.
The universal family is the line bundle \(\mathcal{O}(1)\to \mathbb{P}^n_S\) with its sections \(x_0,\dots, x_n\).
Representabilify
A presheaf \(F\) is a sheaf (Zariski) if for any Zariski cover \(\{U_i\}\) of \(T\), the sequence
\[ F(T) \to \prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_i\cap U_j) \]
is exact (equalizer of the two maps). Here we are actually using the Zariski site.
Fact 1. A representable presheaf must be a sheaf in Zariski topology, since \(h_X\) is sheaf by gluing morphisms.
Categorical Interlude
\(\mathrm{Fun}(\mathcal{C}^{op}, {\bf Set})\) is closed under limits and colimits.
\(h_\square\) preserves limits.
Open Subfunctors
A subfunctor \(F\subset G\) is called open if for any scheme \(T\), and any morphism \(h_T\to G\), the pullback \(h_T\times_G F\) (which functions as intersection here) is representable by an open subscheme of \(T\).
Similarly we can define closed subfunctors.
A collection of open subfunctors \(\{F_i\}\) is called an open cover of \(F\) if for any scheme \(T\) and map \(h_T\to F\), the pullbacks \(h_T\times_F F_i\to h_T\) form an open cover of \(T\).
For simplicity we will simply write \(T\) instead of \(h_T\) when we speak of schemes as functors.
Theorem. If a presheaf \(F\) is a Zariski sheaf and has an representable open cover \(\{F_i\}\), then \(F\) is representable.
Grassmannians
Consider the functor given by
\[ S\mapsto \{\alpha: \mathcal{O}_S^{\oplus n}\to \mathcal{V}_k\}/\sim \]
where \(\alpha\) is a surjective map, \(\mathcal{V}_k\) is a rank \(k\) locally free sheaf (vector bundle) on \(S\). The equivalence relation is given by \(\alpha\) compatible automorphisms \(\mathcal{V}_k \cong \mathcal{V}_k'\).
This is the Grassmannian functor \(\mathrm{Gr}(k,n)\). By taking kernel we have that it actually maps \(S\) to the set of rank \(n-k\) subbundles of \(\mathcal{O}^{\oplus n}_S\). If you like the classical convention, you can take the dual and the image of \(\alpha^*(\mathcal{V}_k)\subset \mathcal{O}_S^{\oplus n}=(\mathcal{O}_S^{\oplus n})^*\) is a rank \(k\) subbundle.
Theorem. The Grassmannian functor \(\mathrm{Gr}(k,n)\) is representable by a finite type scheme over \(\mathbb{Z}\).
For each \(k\)-subset \(I\subset [n]\), we can define an open subfunctor \(F_i\) corresponding to the affine stratum where the the \(k\)-subspace