---
title: "AGGITATE"
subtitle: "2024"
author-note: "Author note"
abstract: "Some rough notes from the AGGITATE 2024 summer school on moduli theory."
---
























# 2024-07-22

## Valuations

Let \( k={\mathbf{C}} \) and \( X \) be a proper variety over \( k \). Write \( k(X) \) for its function field. Recall that a valuation is a group morphism \( v: k(X)^{\times}\to ({\mathbf{R}}, +) \) where

-   \( v(fg) = v(f) + v(g) \)
-   \( v(f+g) \geq \max\left\{{v(f), v(g)}\right\} \)
-   \( v(k) = 0 \)

We generalize this slightly to *quasi-monomial valuations*. For \( X \) of dimension \( n \), let \( (Y, E) \) be a (log smooth) minimal resolution. Write \( E = \sum E_i \), and e.g. \( p = \bigcap_i E_i \), and write \( \widehat{{\mathcal{O}}_{Y, p}} = k{\llbracket y_1, \cdots y_r \rrbracket  } \). Then any \( f\in {\mathcal{O}}_{Y, p} \) can be written as \( f= \sum_{\alpha} f_ \alpha y^{ \alpha} \). A quasi-monomial valuation \( v_m \) is given by \( v_m(f) = \min\left\{{ m\cdot \alpha }\right\} \) ranging over \( \alpha \). Write \( QM(Y, E) \) for the set of quasi-monomials depending on the choice of \( (Y, E) \).

For \( Y \xrightarrow{\varphi} X \) a resolution with exceptional divisor \( E \), write \( A_X(E) = {\operatorname{Ord}}_E(K_{Y/X}) + 1 \) for the log discrepancy. For \( v_ \beta \in QM_p(Y, E) \) with \( p = \bigcap E_i \), define \( A_X(v_ \beta) \coloneqq\sum \beta_i A_X(E_i) \).

# 2024-07-23-14-30-14

## Review

::: remark
Recall from yesterday that an algebraic stack is a stack \( {\mathcal{X}} \) over the big etale site \( {\mathsf{Sch}}_\text{Ét} \) such that there exists a scheme \( U \) and a representable, smooth, surjective map \( U\to {\mathcal{X}} \), i.e. a smooth presentation. A DM stack replaces smooth with étale, and an algebraic space is an algebraic stack where all stabilizers are trivial. Fact: the diagonal is representable.

\[\begin{tikzcd}
    {\mathcal{X}}&& {{\mathcal{X}}\times {\mathcal{X}}} \\
    \\
    {\mathop{\mathrm{Isom}}(a, b)} && T
    \arrow[from=1-1, to=1-3]
    \arrow[from=3-1, to=1-1]
    \arrow[from=3-1, to=3-3]
    \arrow["{a,b}"', from=3-3, to=1-3]
\end{tikzcd}\]
\> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNYIl0sWzIsMCwiXFxjWCBcXHRpbWVzIFxcY1giXSxbMCwyLCJcXElzb20oYSwgYikiXSxbMiwyLCJUIl0sWzMsMSwiYSxiIiwyXSxbMiwzXSxbMiwwXSxbMCwxXV0=)

The stabilizer of \( x: \operatorname{Spec}k \to {\mathcal{X}} \) is \( G_x = \mathop{\mathrm{Isom}}_k(x, x) \coloneqq\mathop{\mathrm{Aut}}_k(x) \).
:::

::: proposition
If \( {\mathcal{X}} \) is a Noetherian algebraic stack and \( x\in {\left\lvert {{\mathcal{X}}} \right\rvert} \) is finite type, then \( \exists {\mathcal{G}}_x \hookrightarrow{\mathcal{X}} \) a locally closed substack with \( {\left\lvert {{\mathcal{G}}_x} \right\rvert} = \left\{{x}\right\} \) and \( {\mathcal{G}}_x \) representable.
:::

::: remark
See *residual gerbes* \( {\mathcal{G}}_x \). Letting \( [C]\in {\mathcal{M}_g} \) be a curve class, the residual gerbe is the classifying stack \( {\mathcal{G}}_{[C]} = {\mathbf{B}}\mathop{\mathrm{Aut}}(C) \coloneqq[\operatorname{Spec}k / \mathop{\mathrm{Aut}}(C)] \).
:::

::: {.proposition title="Minimal presentations"}
Let \( {\mathcal{X}} \) be a Noetherian algebraic stack and \( x\in {\left\lvert {{\mathcal{X}}} \right\rvert} \) a finite type point with smooth stabilizer. Then there exists a diagram

\[\begin{tikzcd}
    {\operatorname{Spec}k(u)} && {U\in{\mathsf{Sch}}} \\
    \\
    {{\mathcal{G}}_x} && {\mathcal{X}}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-1, to=3-3]
\end{tikzcd}\]
\> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHNwZWMgayh1KSJdLFsyLDAsIlVcXGluXFxTY2giXSxbMCwyLCJcXGNHX3giXSxbMiwyLCJcXGNYIl0sWzAsMV0sWzAsMl0sWzIsM10sWzEsM10sWzAsMywiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d)

Here \( U\to {\mathcal{X}} \) is smooth of relative dimension \( \dim G_x \).
:::

::: corollary
\( {\mathcal{X}} \) is DM iff stabilizers are finite and reduced.
:::

::: proof
Set the top-left corner to an orbit \( {\mathcal{O}}(u) \), show \( {\mathcal{O}}(u) \to {\mathcal{G}}_x \) is smooth, and use the flat slicing criterion.
:::

::: {.theorem title="Local structure"}
Let \( {\mathcal{X}} \) be a separated Noetherian DM stack and \( x\in {\left\lvert {{\mathcal{X}}} \right\rvert} \) a finite type point with geometric stabilizer \( G_x \). Then there exists a nice etale presentation: \( ( [\operatorname{Spec}A/G_x], w) \to ({\mathcal{X}}, x) \) which is etale and affine, a quotient of an affine by a finite group, inducing an isomorphism of stabilizer groups at \( w \).
:::

::: definition
A morphism \( \pi: {\mathcal{X}}\to X \) from an algebraic stack to an algebraic space is a coarse moduli space if

1.  \( \pi \) is universal for maps to algebraic spaces, so \( {\mathcal{X}}\to Y \implies \exists! X\to Y \), and
2.  \( \forall k = { \overline{k} } \), \( {\mathcal{X}}(k){_{\scriptstyle / \sim} }\to X(k) \) is bijective.
:::

::: remark
Idea: remove stabilizers from stack in exchange for giving up a universal family. Under mild conditions, (2) implies a bijection on topological spaces. Condition (1) is analogous to being a categorical quotient, while (2) is analogous to \( X \) being an orbit space.
:::

::: theorem
If \( G\curvearrowright\operatorname{Spec}A \) is a finite group action on an affine scheme, then \( [\operatorname{Spec}A/G] \to \operatorname{Spec}A^G \) is a coarse moduli space.
:::

::: {.theorem title="Keel-Mori"}
Let \( {\mathcal{X}} \) be a separated and finite type DM stack over a Noetherian ring. Then \( \exists \pi: {\mathcal{X}}\to X \) a coarse moduli space such that

1.  \( \pi \) is a proper universal homeomorphism,
2.  \( {\mathcal{O}}_X \cong \pi_* {\mathcal{O}}_{{\mathcal{X}}} \),
3.  stable under flat base change.
:::

::: remark
An application: consider \( \overline{{\mathcal{M}_g}} \). Because \( \mathop{\mathrm{Aut}}(C) \) is finite and reduced, this is a DM stack. Semistable reduction implies it is proper. The Keel-Mori theorem implies existence of a coarse moduli space which is a proper algebraic space. Proving projectivity is substantially more difficult.
:::

## Quasicoherent sheaves

::: remark
In particular for algebraic stacks, one can form \( {\mathsf{QCoh}}({\mathcal{X}}) \) and the standard adjunctions \( f^* \) and \( f_* \) where the latter is proper pushforward. If \( G\curvearrowright\operatorname{Spec}A \) is an algebraic group action, then there is a diagram

\[\begin{tikzcd}
    {\operatorname{Spec}A} && {[\operatorname{Spec}A/G]} && {{\mathbf{B}}G} \\
    \\
    && {\operatorname{Spec}A^G}
    \arrow["p", from=1-1, to=1-3]
    \arrow["q", from=1-3, to=1-5]
    \arrow["\pi", from=1-3, to=3-3]
\end{tikzcd}\]
\> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHNwZWMgQSJdLFsyLDAsIltcXHNwZWMgQS9HXSJdLFs0LDAsIlxcQiBHIl0sWzIsMiwiXFxzcGVjIEFeRyJdLFswLDEsInAiXSxbMSwzLCJcXHBpIl0sWzEsMiwicSJdXQ==)

Then

-   \( q_* M = M \) forgets the \( A{\hbox{-}} \)module structure,
-   \( p^* M = M \) forgets the \( G{\hbox{-}} \)action,
-   \( \pi_* M = M^G \) recovers the invariants.
:::

::: definition
A good moduli space is a map \( \pi: {\mathcal{X}}\to X \) from an algebraic stack to an algebraic space such that

1.  \( \pi_* \) is exact on quasicoherent sheaves, and
2.  \( \pi_* {\mathcal{O}}_{{\mathcal{X}}} \cong {\mathcal{O}}_X \).
:::

::: remark
If \( {\mathcal{X}}= [U/G] \) with \( G \) finite reductive, this is equivalent to \( U\to X \) being a good quotient.
:::

::: example
If \( G\curvearrowright\operatorname{Spec}A \) is linearly reductive, then \( [\operatorname{Spec}A/G]\to \operatorname{Spec}A^G \) is a good moduli space.
:::

::: theorem
If \( {\mathcal{X}} \) is an algebraic space with mild hypotheses, then \( \pi: {\mathcal{X}}\to X \) is a good moduli space iff etale locally it looks like taking invariants, i.e. there are diagrams

\[\begin{tikzcd}
    {[\operatorname{Spec}A/G]} && {\mathcal{X}}\\
    \\
    {\operatorname{Spec}A^G} && X
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-1, to=3-3]
\end{tikzcd}\]
\> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJbXFxzcGVjIEEvR10iXSxbMiwwLCJcXGNYIl0sWzIsMiwiWCJdLFswLDIsIlxcc3BlYyBBXkciXSxbMCwxXSxbMywyXSxbMSwyXSxbMCwzXSxbMCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)
:::

::: theorem
Let \( \pi: {\mathcal{X}}\to X \) be a good moduli space. Then

-   \( \pi \) is surjective.
-   If \( Z_i \) are closed and disjoint substacks of \( {\mathcal{X}} \), their images under \( \pi \) are closed and disjoint.
-   Closed points of \( X \) correspond to closed points of \( {\mathcal{X}} \). Two orbits are identified iff their closures intersect.
-   If \( {\mathcal{X}} \) is finite type over a Noetherian scheme then \( X \) is finite type.
-   \( \pi \) is universal for maps to algebraic spaces.
:::