# Tuesday, November 01

## Applications of GIT to Moduli

:::{.remark}
Some major applications of GIT:

- Moduli of sheaves: $\Pic$ and $\Jac$ as varieties/schemes, moduli of (semi)stable vector bundles (Narasimhan, Seshadri, Mumford), and more generally moduli of semistable coherent sheaves (Maruyama, Simpson).
  See Gieseker for moduli of vector bundles over surfaces.
  In this situation, GIT works very well -- this is a "linear" problem.

- Moduli of varieties: stable curves (Mumford), some surfaces (Gieseker).
  GIT works less well here, since this is "nonlinear".

We'll proceed to look at the first case, moduli of sheaves.
Note that for *quasicoherent* sheaves, one instead needs to pass to pro-objects in coherent sheaves.
:::

:::{.remark}
Let $X\in \Proj\kvar$ where $k$ is is not necessarily algebraically closed.
We can define the abstract group $\Pic(X)$ of invertible (i.e. locally free of rank 1) sheaves on $X$ modulo isomorphism.
There is also a Picard scheme $\Pic(X) = \Jac(X)$ which is the fine moduli space of invertible sheaves of fixed degree or fixed Hilbert polynomial, which has the structure of a scheme over $k$ -- if $\characteristic k = 0$ then it is an algebraic variety, but may have nilpotents in positive characteristic.
For appropriate choices, this can be made into a group scheme/variety.
:::

:::{.example title="?"}
Let $C$ be a smooth projective genus $g$ curve over $k=\CC$.
The degree map provides a SES 
\[
\Pic^0(C) = \ker \deg = \Jac(C) \injects \Pic C \surjects \ZZ
.\]
One can realize $\Pic^0(C) \cong \CC^g/\ZZ^g$, giving it the structure of a projective algebraic variety and a complex manifold.
Note that a random choice of lattice $L \cong \ZZ^g$ will yield a Kähler variety, but potentially not an algebraic variety unless $L$ satisfies strict numerical conditions (which it does for $\Pic^0$).
:::

:::{.remark}
Families of invertible sheaves will correspond to moduli functors
\[
\ul{M}: (\Sch^{\ft}_{\spec \CC})\op &\to \Set \\
S &\mapsto \ts{ \text{Invertible sheaves $F$ on $X\fiberprod{\spec \CC} S$} } / \cong
.\]
Such an $F$ should be thought of as a family of invertible sheaves on $X$ parameterized by $S$, i.e. for every $s\in S$ there is a sheaf $F_s \da \ro{F}{X_s}$ where $X_s$ is the fiber over $s$:

![](figures/2022-11-01_13-15-54.png)

For each $f: S\to T$ we obtain $\ul{M}(T) \to \ul{M}(S)$, and pullbacks $X\times S \mapsvia{f} X\times T$ induces $F\mapsto f^* F$.

We also require that each $F\in \ul{M}(S)$ is equipped with a *rigidification*: a fixed trivialization $\ro{F}{p\times S}\cong\OO_S$:

![](figures/2022-11-01_13-29-33.png)

This kills automorphisms and gives a fine moduli space.
Without this, one could twist by anything coming from the base, so one could alternatively define 
\[ 
\ul{M}'(S) = { \ts{\text{$F$ on $X\times S$ }} \over  F\to F\tensor \pi^* L}, \qquad L\in \Pic(S)
\] 
This coincides with the previous notion when $L$ has a section.
:::

:::{.definition title="Hilbert polynomial"}
For $\mcf\in \Coh(X)$, define the **Hilbert polynomial**
\[
p_\mcf(n) \da \chi(X, \mcf(n)) = \sum_{i\geq 0} h^i(X; \mcf(n))
,\]
noting that by Serre vanishing, for $n \gg 0$, $h^{i > 0}(X; \mcf(n)) = 0$.
:::

:::{.example title="?"}
If $L$ is a line bundle on a curve $C$, by RR we have 
\[
\chi(L(n)) = \deg L(n) + 1-g = nd + \deg(L) + 1 - g
\]
where $d \geq \deg \OO_X(1)$ (which is very ample).
Thus $\deg (L)$ defines the Hilbert polynomial $p_L(n)$ uniquely, and we often write $\ul{\Pic}^d_{X/\CC}$.
More generally, if $\mcf$ is a rank $r$ locally free sheaf on a curve $C$, one obtains 
\[
\chi(\mcf(n)) = nd + \deg\mcf + r(1-g)
.\]

> Todo: is this $nd$ or $n+d$?

:::

:::{.lemma title="Easy"}
For $S$ connected, each $p_{F_s}(n)$ are the same.
:::

:::{.theorem title="Representability of the Picard functor"}
For any field $k$, not necessarily algebraically closed, of any characteristic, and for all projective varieties $X$ over $k$, the rigidified functor $\ul{\Pic}_{X/k, h(n)}$ is represented by a scheme $\Pic_{X/k, h(n)}$, i.e. 
\[
\ul{\Pic}_{X/k, h(n)}(S) \iso \Hom_{\Sch}(S, \Pic_{X/k, h(n) })
.\]
Moreover there exists a universal invertible sheaf $\mcu$ over $\Pic_{X/k, h(n)}$ and the sheaves $F$ on $X\fiberprod{k} S$ are all pullbacks:

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\end{tikzpicture}

:::

:::{.remark}
Note that $\Pic^0_{X/k}$ is a group variety and the other components are torsors over it.
Since $[\OO_X]\in \Pic_{X/k}$, one can compute $\dim \T_{[\OO_X]} \Pic_{X/k} = h^1(\OO_X)$, which is $\dim \Pic_{X/k}$ if $\Pic_{X/k}$ is reduced -- this is automatic in characteristic zero, and necessary since $k[\eps]/\eps^2$ has dimension 0 but tangent space dimension 1.
:::

:::{.remark}
Adapting this moduli problem to vector bundles: take the functor sending $S$ to sheaves $F$ on $X\fiberprod{\spec \CC} S$ which are flat over $S$, noting that there is no way to rigidify in this case.
Without any additional conditions, this leads to something horribly infinite.
Consider $X = \PP^1$ and take $F = \OO(k) \oplus \OO(-k)$, so $\deg F = 0$ and $\rank F = 2$, where $k\in \ZZ$.
This is an unbounded family, parameterized by an infinite discrete set $\ZZ_{\geq 0}$, so we need to restrict to nice vector bundles to exclude this case.
:::

:::{.definition title="(Semi)stability for vector bundles, easy case"}
If $C$ is a curve, if $F$ is a vector bundle (a locally free sheaf of rank $r$) then $F$ **stable** (resp. **semistable**) if for any vector sub-bundle $E\leq F$ there is an inequality
\[
{\deg E\over \rank E} < {\deg F \over \rank F},\qquad \text{resp. }
{\deg E\over \rank E} \leq {\deg F \over \rank F},\qquad 
.\]
These quantities are called **slopes**, and this is sometimes referred to as **slope stability**.
:::

:::{.theorem title="?"}
There is a moduli space $\ts{\text{semistable sheaves}}/S\dash\text{equivalence} \contains \ts{\text{stable sheaves}}/\cong$.
:::

:::{.definition title="(Semi)stability for vector bundles, general case"}
Let $X$ is a projective variety equipped with $\OO_X(1)$ and $F$ is a pure coherent sheaf, i.e. $\supp F$ is pure-dimensional (equidimensional) and there does not exist a subsheaf $0\neq G\leq F$ with $\dim \supp G < \dim \supp F$.[^line_bundle]
Then stability (resp. semistability) is the condition that for every subsheaf $E\leq F$, 
\[
{ p_E(n) \over \rank E} < {p_F(n) \over \rank F},\qquad\text{resp. }
{ p_E(n) \over \rank E} \leq {p_F(n) \over \rank F},
,\]
i.e. the normalized Hilbert polynomials (dividing by the leading coefficients) satisfying this inequality.

Note that this definition still works for $X$ a scheme, potentially non-reduced with many components.
This is sometimes referred to as Seshadri stability.

[^line_bundle]: 
This is not an issue for line bundles, since there are no nonzero subsheaves with different supports since every subsheaf is supported on the entire variety.
This is also automatic if $X$ is irreducible, otherwise a subsheaf could be supported on different components which could have different dimensions.

:::

:::{.remark}
One interesting case: $X$ a curve but not irreducible.
The moduli of invertible sheaves is already nontrivial, since subsheaves may only be defined on some irreducible components and thus not be invertible.
Here $\OO_X(1)$ may have different degrees on different components; as long as they are positive, $\OO_X(1)$ is ample, and different polarizations yield different Jacobians and balancing these leads to interesting combinatorics.
:::