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\title{AGGITATE}
\date{July 23, 2024}
\author{D. Zack Garza}
\address{Department of Mathematics, University of Georgia, Athens GA 30602, USA}
\email{zack@uga.edu}

\usepackage[style=alphabetic,sorting=ynt]{biblatex} %Imports biblatex package
\addbibresource{/home/dzack/zotero.bib} %Import the bibliography file

\begin{document}

\begin{abstract}
Some rough notes from the AGGITATE 2024 summer school on moduli theory.
\end{abstract}

\maketitle

\tableofcontents

\newpage

\section{2024-07-22}\label{section}

\subsection{Valuations}\label{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

\begin{itemize}
\tightlist
\item
  \(v(fg) = v(f) + v(g)\)
\item
  \(v(f+g) \geq \max\left\{{v(f), v(g)}\right\}\)
\item
  \(v(k) = 0\)
\end{itemize}

We generalize this slightly to \emph{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 \(A_X(v_ \beta) \coloneqq\sum \beta_i A_X(E_i)\).

\section{2024-07-23-14-30-14}\label{section-1}

\subsection{Review}\label{review}

\begin{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.

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

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

\end{remark}

\begin{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.

\end{proposition}

\begin{remark}

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

\end{remark}

\begin{proposition}[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

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

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

\end{proposition}

\begin{corollary}

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

\end{corollary}

\begin{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.

\end{proof}

\begin{theorem}[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 \(w\).

\end{theorem}

\begin{definition}

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

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\pi\) is universal for maps to algebraic spaces, so \({\mathcal{X}}\to Y \implies \exists! X\to Y\), and
\item
  \(\forall k = { \overline{k} }\), \({\mathcal{X}}(k){_{\scriptstyle / \sim} }\to X(k)\) is bijective.
\end{enumerate}

\end{definition}

\begin{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.

\end{remark}

\begin{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.

\end{theorem}

\begin{theorem}[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

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\pi\) is a proper universal homeomorphism,
\item
  \({\mathcal{O}}_X \cong \pi_* {\mathcal{O}}_{{\mathcal{X}}}\),
\item
  stable under flat base change.
\end{enumerate}

\end{theorem}

\begin{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.

\end{remark}

\subsection{Quasicoherent sheaves}\label{quasicoherent-sheaves}

\begin{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

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

Then

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

\end{remark}

\begin{definition}

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

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\pi_*\) is exact on quasicoherent sheaves, and
\item
  \(\pi_* {\mathcal{O}}_{{\mathcal{X}}} \cong {\mathcal{O}}_X\).
\end{enumerate}

\end{definition}

\begin{remark}

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

\end{remark}

\begin{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.

\end{example}

\begin{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

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

\end{theorem}

\begin{theorem}

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

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

\end{theorem}

\newpage

\printbibliography %Prints bibliography

\end{document}
