# 2024-07-24-12-02-54

:::{.remark}
The trichotomy of automorphisms:

- Automorphisms:    
    - None
    - Finite 
    - Reductive (at closed points)
- Type of space:
    - Schemes/algebraic spaces
    - DM stacks
    - Algebraic stacks
- Definition:
    - Zariski/etale locally affine 
    - etale locally affine 
    - smooth locally affine 
- Examples:
    - $\PP^n, \Hilb, \Quot$
    - $\Mg, M_{n, V}^{\KSBA}$
    - $\Bun^\ss(C)$, or $X_{n, V}^{k-\ss}$
- Properties of $[X/G]$:
    - Free action 
    - Finite stabilizers 
    - Reductive stabilizers at closed orbits 
- Local structure
    - Zariski/etale locally $\spec A$
    - etale locally $[\spec A/G_x]$
    - etale locally $[\spec A/G_x]$
- Existence of moduli spaces:
    - Already a scheme, a fine space 
    - Keel-Mori: for $X$ separated, there exists a coarse space
    - Today: existence of good moduli spaces

:::

:::{.definition}
A qcqs map $\pi: \cX\to X$ is a good moduli space (gms) if 

1. $\pi_*: \QCoh(\cX)\to \QCoh(X)$ is exact,
2. $\OO_X \isomorphic \pi_*\OO_X$

:::

:::{.example}
$[\spec A/G]\to \spec A^G$ is a gms.
:::

:::{.theorem}
If $\pi: \cX\to X$ is a gms over a base $S$, then

- $\pi$ is surjective and universally closed,
- For $Z_i \subseteq \cX$ closed, $\pi(Z_1 \intersect Z_2) = \pi(Z_1) \intersect \pi(Z_2)$.
    Thus for all $x,y\in \cX(k)$ where $k=\kbar$, then $\pi(x) = \pi(y) \iff$ their closures don't intersect in $\abs{\cX \fp{S} k}$.
- $\pi$ is universal for maps to algebraic spaces.
- If $\cX$ is finite type over $S$ and $S$ is Noetherian, then $X\to S$ is finite type and $\pi_*$ preserves coherence.
:::

:::{.corollary}
$X = \spec A\to X\modmod G =\spec A^G$ is a good quotient in the sense of GIT.
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:::{.lemma}
If $\pi:\cX\to X$ is a gms and $F\in \QCoh(X)$, then $F\isomorphic \pi_* \pi^* F$.
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:::{.proof}
Apply $\pi_* \pi^*$ to the SES $\OO_X^{\oplus J} \to \OO_X^{\oplus I} \surjects F$ and use that pushforward is exact.
:::

:::{.remark}
Being a gms is stable under base change.
:::

:::{.theorem title="Local structure theorem"}
Let $\cX$ be an algebraic stack, locally of finite type over $k=\kbar$, with affine stabilizers.
If $x\in \cX(k)$ with linearly reductive stabilizer $G_x$, then $\exists \spec[A/G_x]\to \cX$ an etale morphism sending $w\mapsto x$ inducing isomorphisms of stabilizer groups at $w$.
:::

:::{.theorem title="Existence theorem"}
If $\cX$ is an algebraic stack of finite type over $k=\kbar$ where $\characteristic(k) = 0$ with affine diagonal, then $\exists \pi: \cX\to X$ a gms with $X$ separated $\iff$ $\cX$ is $\theta$-complete and $S$-complete.
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:::{.example}
Let $C_2\actson X$ the affine line with a doubled origin by swapping the origins and fixing everything else.
Then $[X/C_2]$ has generic stabilizer $C_2$, except at $x=0$, and admits $\AA^1$ as a gms, but the morphism is not separated.
Thus $X\to [X/C_2]$ is not a gms -- use Serre's affineness criterion and that pushforward is exact to conclude $X$ must be affine, a contradiction.
:::

:::{.remark}
An important construction: $\Theta \da [\AA^1/\GG_m]$.
Maps $\Theta\to [X/G]$ can recover the Hilbert-Mumford criterion, filtrations on vector bundles, and test configurations, yielding a generalized notion of $\Theta$-stability.
:::

:::{.definition title="Theta completeness"}
An algebraic stack is $\Theta$-complete iff for all DVRs $R$,

\[\begin{tikzcd}
	{\Theta_R\setminus U} && \cX \\
	\\
	{\Theta_R}
	\arrow[from=1-1, to=1-3]
	\arrow[hook', from=1-1, to=3-1]
	\arrow["{\exists !}"', dashed, from=3-1, to=1-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXFRoZXRhX1JcXHNldG1pbnVzIFUiXSxbMCwyLCJcXFRoZXRhX1IiXSxbMiwwLCJcXGNYIl0sWzAsMl0sWzAsMSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJib3R0b20ifX19XSxbMSwyLCJcXGV4aXN0cyAhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)


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:::{.definition}
Let $\Phi_R \da [ \spec R[x,y]/(xy-\pi)/\GG_m]$ where $\pi$ is a uniformizer of $R$ a DVR.
An algebraic stack is $S$-complete iff for all DVRs $R$,

\[\begin{tikzcd}
	{\Phi_R\setminus 0} && \cX \\
	\\
	{\Phi_R}
	\arrow[from=1-1, to=1-3]
	\arrow[hook', from=1-1, to=3-1]
	\arrow["{\exists !}"', dashed, from=3-1, to=1-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXFBoaV9SXFxzZXRtaW51cyAwIl0sWzAsMiwiXFxQaGlfUiJdLFsyLDAsIlxcY1giXSxbMCwyXSxbMCwxLCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6ImJvdHRvbSJ9fX1dLFsxLDIsIlxcZXhpc3RzICEiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

:::