Author: Eiko

Tags: algebraic geometry, moduli space, moduli functor, grassmannian

Time: 2024-10-21 22:21:00 - 2024-10-26 07:32:00 (UTC)

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 $X$”.

$$h_X:{\mathcal{C}}^{op}\to \mathbf{S}\mathbf{e}\mathbf{t},T\mapsto {\mathrm{Hom}}_{\mathcal{C}}(T,X)$$

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 $\mathrm{Fun}({\mathcal{C}}^{op},\mathbf{S}\mathbf{e}\mathbf{t})$ is usually called the category of presheaves on $\mathcal{C}$.

Yoneda Lemma

• For any contravariant functor $F:{\mathcal{C}}^{op}\to \mathbf{S}\mathbf{e}\mathbf{t}$, there is a natural isomorphism

  $$\mathrm{Hom}(h_X,F)\cong F(X),\eta \mapsto {\eta }_X(1_X)$$

  where ${\eta }_X:h_X(X)\to F(X)$ is the component of $\eta :h_X\to F$ at $X$.

• The functor $h_{\bullet }:\mathcal{C}\to \mathrm{Fun}({\mathcal{C}}^{op},\mathbf{S}\mathbf{e}\mathbf{t})$ 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 $F:{\mathcal{C}}^{op}\to \mathbf{S}\mathbf{e}\mathbf{t}$ 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 $\eta :F\to h_{M^{\prime }}$ factor through $\xi$.

Examples

• The global section functor $\mathrm{\Gamma }:{\mathbf{S}\mathbf{c}\mathbf{h}}_S^{op}\to \mathbf{S}\mathbf{e}\mathbf{t}$ mapping $X$ to ${\mathcal{O}}_X(X)$ is representable by the affine line (over $S$) ${\mathbb{A}}_S^1$.

  To see this, note that

  $${\mathcal{O}}_X(X)={\mathrm{Hom}}_{{\mathbf{S}\mathbf{c}\mathbf{h}}_S}(X,{\mathbb{A}}_S^1)={\mathrm{Hom}}_{{\mathcal{O}}_S}({\mathcal{O}}_S[t],{\mathcal{O}}_X).$$

  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 ${\mathbb{A}}_S^1$, the moduli space of points. The universal family is the projection ${\mathbb{A}}_S^1\to S$.

• 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 $\mathcal{O}(1)\to {\mathbb{P}}_S^n$ 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

• $\mathrm{Fun}({\mathcal{C}}^{op},\mathbf{S}\mathbf{e}\mathbf{t})$ is closed under limits and colimits.

• $h_{◻}$ 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 ${\mathcal{V}}_k\cong {\mathcal{V}}_k^{\prime }$.

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 ${\mathcal{O}}_S^{\oplus n}$. If you like the classical convention, you can take the dual and the image of ${\alpha }^{\ast }({\mathcal{V}}_k)\subset {\mathcal{O}}_S^{\oplus n}=({\mathcal{O}}_S^{\oplus n}{)}^{\ast }$ 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