\renewcommand{\thesubsection}{\thesection.\arabic{subsection}}
\renewcommand{\thesubsection}{\thesection\Alph{subsection}} 

\makeatletter
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima
      \ifcase #1
      %\or\or \hskip 1em \or \hskip 2em \else \hskip 3em \fi%
       \or\or \hskip 2em \or \hskip 3em \else \hskip 4em \fi%
      #6\nobreak\relax
    \hfill\hbox to\@pnumwidth{\@tocpagenum{#7}}\par% <---- \dotfill -> \hfill
    \nobreak
    \endgroup
  \fi}
\makeatother

\newcommand\cA{{\mathcal A}}
\newcommand\cB{{\mathcal B}}
\newcommand\cC{{\mathcal C}}
\newcommand\cD{{\mathcal D}}
\newcommand\cE{{\mathcal E}}
\newcommand\cF{{\mathcal F}}
\newcommand\cG{{\mathcal G}}
\newcommand\cH{{\mathcal H}}
\newcommand\cI{{\mathcal I}}
\newcommand\cJ{{\mathcal J}}
\newcommand\cK{{\mathcal K}}
\newcommand\cL{{\mathcal L}}
\newcommand\cM{{\mathcal M}}
\newcommand\cN{{\mathcal N}}
\newcommand\cO{{\mathcal O}}
\newcommand\cP{{\mathcal P}}
\newcommand\cQ{{\mathcal Q}}
\newcommand\cR{{\mathcal R}}
\newcommand\cS{{\mathcal S}}
\newcommand\cT{{\mathcal T}}
\newcommand\cU{{\mathcal U}}
\newcommand\cV{{\mathcal V}}
\newcommand\cW{{\mathcal W}}
\newcommand\cX{{\mathcal X}}
\newcommand\cY{{\mathcal Y}}
\newcommand\cZ{{\mathcal Z}}


\newcommand\bA{{\mathbf A}}
\newcommand\bB{{\mathbf B}}
\newcommand\bK{{\mathbf K}}
\newcommand\bL{{\mathbf L}}
\newcommand\bX{{\mathbf X}}

\newcommand{\BB}[0]{\mathbb{B}}
\newcommand{\DD}[0]{\mathbb{D}}
\newcommand{\EE}[0]{\mathbb{E}}
\newcommand{\FF}[0]{\mathbb{F}}
\newcommand{\ZZ}[0]{\mathbb{Z}}
\newcommand{\QQ}[0]{\mathbb{Q}}
\newcommand{\RR}[0]{\mathbb{R}}
\newcommand{\CC}[0]{\mathbb{C}}
\newcommand{\PP}[0]{\mathbb{P}}
\newcommand{\RP}[0]{\mathbb{RP}}
\newcommand{\HH}[0]{\mathbb{H}}
\newcommand{\OO}[0]{\mathcal{O}}
\newcommand{\mcx}[0]{\mathcal{X}}
\newcommand{\mcy}[0]{\mathcal{Y}}
\newcommand{\mcH}[0]{\mathcal{Y}}

\newcommand{\injects}[0]{\hookrightarrow}
\newcommand{\embeds}[0]{\hookrightarrow}
\newcommand{\sing}[0]{{\mathrm{sing}}}
\newcommand{\Sing}[0]{{\mathrm{Sing}}}
\newcommand{\smooth}[0]{{\mathrm{smooth}}}

\newcommand{\union}[0]{\cup}
\newcommand{\Union}[0]{\bigcup}
\newcommand{\intersect}[0]{\cap}
\newcommand{\Intersect}[0]{\bigcap}
\newcommand{\disjoint}[0]{\amalg}
\newcommand{\semidirect}[0]{\rtimes}
\newcommand{\transverse}[0]{\pitchfork}


\newcommand{\abs}[1]{{\left\lvert #1 \right\rvert}}
\newcommand{\qty}[1]{\left( {#1} \right)}
\newcommand{\ts}[1]{\left\{ {#1} \right\} }
\newcommand{\gens}[1]{\left\langle {#1} \right\rangle}
\newcommand\sumpower[1]{{ {}^{ \oplus {#1} }  }}

\newcommand{\da}[0]{\coloneqq}
\newcommand{\lkt}[0]{ {{\rm II}_{3, 19} } }
\newcommand{\lEn}[0]{ {\rm II}_{1, 9} }

\newcommand{\bbcpt}[1]{\overline{ #1 }^{\mathrm{BB}}}
\newcommand{\torcpt}[1]{\overline{ #1 }^{\mathrm{Tor}}}
\newcommand{\torcptf}[1]{\overline{ #1 }^{\mathfrak{F}}}
\newcommand{\semitorcpt}[1]{\overline{ #1 }^{\mathrm{SemiTor}}}
\newcommand{\ksbacpt}[1]{\overline{ #1 }^{\mathrm{KSBA}}}
\newcommand{\rc}[1]{{ #1 }^{\mathrm{rc}}}
\newcommand{\Mpol}[0]{ \bbcpt{ \mathcal{E}_{2} }  }
\newcommand{\Mpolop}[0]{ \mathcal{E}_{2}  }
\newcommand{\Munpol}{ \bbcpt{ \mathcal{E}_{ \emptyset }} } 
\newcommand{\Munpolop}{  \mathcal{E}_{\emptyset } } 
\newcommand{\Mhe}{ \bbcpt{ F_{4, \mathrm{ h.e.} } }  }
\newcommand{\Mheop}{ F_{4, \mathrm{ h.e.} } }

\newcommand{\En}[0]{\mathrm{En}}
\newcommand{\Co}[0]{\mathrm{Co}}
\newcommand{\dP}[0]{\mathrm{dP}}
\newcommand{\Nik}[0]{\mathrm{Nik}}
\newcommand{\Nod}[0]{\mathrm{Nod}}
\newcommand{\opp}[0]{\mathrm{op}}

\DeclareMathOperator{\KSBA}{KSBA}
\DeclareMathOperator{\GIT}{GIT}
\DeclareMathOperator{\IAS}{IAS}
\DeclareMathOperator{\IARP}{IARP}
\DeclareMathOperator{\IA}{IA}
\DeclareMathOperator{\Orth}{O}
\DeclareMathOperator{\Num}{Num}
\newcommand{\isoGr}[0]{\operatorname{Gr}^{\mathrm{iso}}}

\newcommand{\torational}[0]{\dashrightarrow}
\newcommand{\rational}[0]{\torational}
\newcommand{\birational}[0]{\overset{\sim}{\torational}}
\newcommand{\birationaliso}[0]{\overset{\sim}{\torational}}

% environments
\newtheorem{theorem}{Theorem}[section]
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{metatheorem}[theorem]{Meta-Theorem}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem*{acknowledgements}{Acknowledgements}
\newtheorem{addendum}[theorem]{Addendum}
\newtheorem{answer}[theorem]{Answer}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{construction}[theorem]{Construction}
\newtheorem{convention}[theorem]{Convention}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{examples}[theorem]{Examples}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{notations}[theorem]{Notations}
\newtheorem{num}[theorem]{}
\newtheorem{numl}{}[theorem]
\newtheorem{observation}[theorem]{Observation}
\newtheorem{project}[theorem]{Project}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposal}[theorem]{Proposal}
\newtheorem{question}[theorem]{Question}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{setup}[theorem]{Setup}
\newtheorem{warning}[theorem]{Warning}
\newtheorem{sketch}[theorem]{Sketch}
\newtheorem{strategy}[theorem]{Strategy}


\newcommand{\leftquotient}[2]{
  {\textstyle #1}
  \mkern-4mu\scalebox{1.5}{$\diagdown$}\mkern-5mu^{\textstyle #2}
}

\newcommand\dzg[1]{\todo[color=blue!40]{#1}} 
\newcommand\dzginline[1]{\todo[inline,color=blue!40]{#1}}

\newcommand{\diverge}[0]{
\,
\begin{tikzpicture}[baseline]
\draw (0, 0) to[in=270+15, out=90-15] (0,0.3);
\draw (0.15,0) to[in=270-15, out=90+15] (0.15,0.3);
\end{tikzpicture}
\,
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Diagrams

\newcommand{\coxeterA}{\begin{dynkinDiagram}[label, arrows=false, edge length=.75cm]A{***}
\dynkinTripleEdge 12
\end{dynkinDiagram}}


\newcommand{\coxeterHthree}{\begin{dynkinDiagram}[label, arrows=false, edge length=.75cm]A{***}
\dynkinTripleEdge 12
\end{dynkinDiagram}}

\newcommand{\coxeterHfour}{\begin{dynkinDiagram}[label, arrows=false, edge length=.75cm]A{****}
\dynkinTripleEdge 12
\end{dynkinDiagram}}

\newcommand{\coxeterGtwo}{
\begin{dynkinDiagram}[label, arrows=false, edge length=1.75cm]A{**}
\dynkinQuadrupleEdge 12
\end{dynkinDiagram}
}

\newcommand{\coxeterGtwoAffine}{
\begin{dynkinDiagram}[label, arrows=false, edge length=1.75cm, labels={0,1,2}]A{***}
\dynkinQuadrupleEdge 23
\end{dynkinDiagram}
}

\newcommand{\HalphenInvariants}{(10, 10, 1)}
\newcommand{\EnriquesInvariants}{(10, 10, 0)}

\newcommand{\dcoset}[3]{
    \scriptscriptstyle
    {\textstyle #1}
    \mkern-4mu\scalebox{1.5}{$\diagdown$}\mkern-5mu^{\textstyle #2}
    \mkern-4mu\scalebox{1.5}{$\diagup$}\mkern-5mu{\textstyle #3} 
}
\newcommand{\dcosetl}[2]{{
    \scriptscriptstyle
    {\textstyle #1}
    \mkern-4mu\scalebox{1.5}{$\diagdown$}\mkern-5mu^{\textstyle #2}
}}
\newcommand{\dcosetr}[2]{
    \scriptscriptstyle
    {\textstyle #1}
    \mkern-4mu\scalebox{1.5}{$\diagup$}\mkern-5mu{\textstyle #2} 
}


\newcommand{\Poincare}{Poincar\'e}


\newcommand{\weylgroup}[1]{W_{ #1 }}