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\title{Dissertation}
\date{January 21, 2025}
\author{D. Zack Garza}
\address{Department of Mathematics, University of Georgia, Athens GA 30602, USA}
\email{zack@uga.edu}

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\begin{document}

\begin{abstract}
Rough draft of my dissertation
\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}\label{introduction}

Enriques surfaces \(Y\) are minimal algebraic surfaces of Kodaira dimension zero satisfying \(h^1(\cO_Y)=h^2(\cO_Y)=0\) and \(K_Y\neq 0\) but \(2K_Y = 0\). A fundamental property of an Enriques surfaces \(Y\) is that its universal cover \(X\) is isomorphic to a \(\mathrm{K}3\) surface\footnote{A \(\mathrm{K}3\) surface is a smooth projective surface \(X\) with \(K_X = 0\) and \(h^1({\mathcal{O}}_X) = 0\)}. For both \(\mathrm{K}3\) and Enriques surfaces, the theory of compactifications is very rich: once a polarization \(L\) is fixed, there is a Hodge-theoretic period domain parametrizing isomorphism classes of polarized \(\mathrm{K}3\) or Enriques surfaces. These are bounded Hermitian symmetric domains, and thus for appropriate choices of arithmetic subgroups, the resulting arithmetic quotients admit Baily--Borel \cite{BB66}, toroidal \cite{AMRT75}, and Looijenga semitoroidal compactifications \cite{Loo03}.

It is then natural to ask about geometric compactifications such as stable pair compactifications and how they relate to these Hodge theoretic compactifications. In a series of recent papers, Alexeev--Engel--Thompson have made breakthroughs for \(\mathrm{K}3\) surfaces (see \cite{ABE22,AE21,AE22nonsympinv,AEH21,AET23}). In particular, there are explicit and effective answers to this question for \(\mathrm{K}3\) surfaces equipped with a non-symplectic involution whose fixed locus is a curve.

Stable pair compactifications of the moduli space of Enriques surfaces are less well-studied. In \cite{Sch22} the second author studied the stable pair compactification of the moduli space of Enriques surfaces with a degree \(6\) polarization (Enriques' original construction) and give a full description for a \(4{\hbox{-}}\)dimensional subfamily of the moduli space. One of the obstructions to extending similar results to the entire \(10{\hbox{-}}\)dimensional family is the high degree \(d\) of the polarization needed. It is thus natural to consider instead the lowest value possible, which is \(d=2\). In this situation, \(X\) is naturally equipped with a non-symplectic involution, the \textbf{Enriques involution}, whose fixed locus is a curve, and thus the theory developed in \cite{AE22nonsympinv} is applicable.

However, the theory in \cite{AE22nonsympinv} does not immediately apply in this situation -- one must account for the fact that there are certain natural geometric automorphisms. Horikawa gives a construction of a K3 surface \(X\) as a degree 2 cover \(\rho: X\to {\mathbf{P}}^1\times{\mathbf{P}}^1\) branched over a divisor \(D\in -2K_{{\mathbf{P}}^1\times{\mathbf{P}}^1}\) of bidegree \((4, 4)\). It can be shown that \(\rho\) is symplectic and that \(\rho\) commutes with the Enriques involution, and thus the theory in \cite{AE22nonsympinv} must be modified to keep track of this extra symmetry.

\section{Preliminaries}\label{preliminaries}

We follow closely the exposition in \cite[\S\,2]{AE22nonsympinv}.

\subsection{Lattices}\label{lattices}

\begin{remark}

By a \emph{lattice} we mean a finitely generated free abelian group \(L\) of finite rank equipped with a nondegenerate symmetric bilinear form \(b\colon L\times L\rightarrow {\mathbf{Z}}\). In particular, two lattices are \emph{isometric} there exists an isomorphism of the underlying abelian groups which preserves the bilinear forms. Given a set of generators \(e_1,\ldots,e_r\) of \(L\), we can associate a \emph{Gram} matrix given by \((b(e_i,e_j))_{i,j}\). The lattice \(L\) is called \emph{unimodular} provided the determinant of a Gram matrix is \(\pm1\). The lattice \(L\) is called \emph{even} provided \(b(v,v)\in2 {\mathbf{Z}}\) for all \(v\in L\). Given a lattice \(L\) we denote by \(L^*\) its dual \(\mathrm{Hom}_{{\mathbf{Z}}}(L,{\mathbf{Z}})\). As the bilinear form is nondegenerate, we have an inclusion \(L\hookrightarrow L^*\) and the quotient \(A_L=L^*/L\) is a finite abelian group called the \emph{discriminant group} of \(L\). The discriminant group \(A_L\) comes equipped with a quadratic form \(q_L\colon A_L\rightarrow {\mathbf{Q}}/{\mathbf{Z}}\) by sending \(v+L\mapsto b(v,v)~\textrm{mod}~{\mathbf{Z}}\). If \(S\subseteq L\) is a sublattice and \(T\) is its orthogonal complement, we have that \(A_L\cong A_T\) and \(q_S=-q_T\) under this correspondence. The lattices for which \(A_L\cong {\mathbf{Z}}_2^a\) for some positive integer \(a\) are called \textbf{\(2{\hbox{-}}\)elementary}.

\end{remark}

\subsection{K3 surfaces, and nonsymplectic involutions}\label{k3-surfaces-and-nonsymplectic-involutions}

\begin{remark}

A lot of the geometry and moduli theory of K3 surfaces is regulated by lattice theory. For a K3 surface \(X\) it is well-known that \(H^2(X,{\mathbf{Z}})\), endowed with the cup product, is an even, unimodular lattice of signature \((3,19)\). It follows that \(H^2(X,{\mathbf{Z}})\) is isometric to the so-called \emph{K3 lattice} \(\lkt:=U^{\oplus3}\oplus E_8^{\oplus 2}\), where \(U\) is the hyperbolic plane \(\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\) and \(E_8\) is the negative definite root lattice associated to the corresponding Dynkin diagram. In particular, symmetries of the surface \(X\) translate into symmetries of the K3 lattice \(\lkt\).

\end{remark}

\begin{remark}

A particularly rich setting is provided by \emph{nonsymplectic involutions}, i.e.~order \(2\) automorphisms \(\iota\colon X\rightarrow X\) such that the induced map \(\iota^*\colon H^{2,0}(X)\rightarrow H^{2,0}(X)\) satisfies \(\iota^*\omega_X=-\omega_X\). Then we can look at the action of \(\iota^*\) on \(H^2(X,{\mathbf{Z}})\) and we denote by \(S\) its \((+1){\hbox{-}}\)eigenspace. It turns out that \(S\) is a hyperbolic \(2{\hbox{-}}\)elementary lattice, and all the possibilities for \(S\) up to isometries were classified by Nikuln. More precisely, there are \(75\) cases which correspond bijectively to the triples of invariants \((r,a,\delta)\), where \(r\) is the rank of \(S\), \(A_S\cong {\mathbf{Z}}_2^a\), and \(\delta\) is the so-called \emph{coparity} of \(L\): \(\delta=0\) provided \(q_L(v)\equiv0~\mathrm{mod}~{\mathbf{Z}}\), and \(\delta=1\) otherwise.

\end{remark}

\section{Moduli via period domains}\label{moduli-via-period-domains}

\subsection{A general construction}\label{a-general-construction}

\label{sec:period_domain}

\begin{remark}

We describe here a construction common to the construction of many Hodge-theoretic moduli spaces. Let \(\Lambda\) be an ambient lattice, \(S \leq \Lambda\) a primitive sublattice, and \(T \coloneqq S^{\perp \Lambda}\) its orthogonal complement in \(\Lambda\). Define the \textbf{period domain associated to \(S\)} to be \[
\Omega_S^{\pm} \coloneqq\left\{{[v] \in {\mathbf{P}}(S \otimes_{\mathbf{Z}}{\mathbf{C}})  \mathrel{\Big|}v^2=0 ~\text{and}~ v\overline{v} = 0}\right\}
.\] As a matter of notation, we also set \[
\Omega^S \coloneqq\Omega_{S^\perp} \coloneqq\Omega_T
.\] In cases of interest, we have a decomposition \(\Omega^\pm_S = \Omega^+_S \amalg \Omega^-_S\) into irreducible components, both of which are type IV bounded Hermitian symmetric domains which are permuted by \(\operatorname{Gal}({\mathbf{C}}/{\mathbf{R}})\). We fix a choice of component \(\Omega^+_S\), and let \({\operatorname{O}}(S)^+ \leq {\operatorname{O}}(S)\) be the subgroup fixing this component. We then form a locally symmetric space and a corresponding Baily-Borel compactification \[
F(S) \coloneqq\leftquotient{
{\operatorname{O}}^+(S)
}{
\Omega^+_S
}
\qquad
\bbcpt{F(S)} \coloneqq\bbcpt{\leftquotient{
{\operatorname{O}}^+(S)
}{
\Omega^+_S
}}
.\] More generally, one can let \(\Gamma\) be any neat arithmetic group that acts properly discontinuously on \(S\Omega^+_S\). One can then similarly form \[
F(S, \Gamma) \coloneqq\leftquotient{\Gamma}{\Omega^+_S}, \qquad \bbcpt{ F(S, \Gamma) } \coloneqq\bbcpt{
\leftquotient{\Gamma}{\Omega^+_S}
}
.\] Specific choices of \(S\) are used throughout our work to construct various coarse moduli spaces. In some instances, we must remove a hyperplane arrangement to form the correct moduli space. Let \[
\mathcal{H}_{-2} \coloneqq
\qty{\bigcup_{\delta\in \Phi_{N}} \delta^\perp}\cap\Omega^+_N
= \bigcup_{\substack{
\delta\in N,\\\delta^2=-2}}
\left\{{ [v]\in\Omega_N^+ \mathrel{\Big|}v \cdot \delta = 0}\right\}
.\] and define \[
F(S, \Gamma, \cH_{-2})
\coloneqq
\leftquotient{\Gamma}{
\qty{\Omega^+_S \setminus \cH_{-2}}}, 
\qquad 
\bbcpt{ F(S, \Gamma, \cH_{-2} ) } 
\coloneqq
\bbcpt{
\leftquotient{\Gamma}{
\qty{\Omega^+_S \setminus \cH_{-2} } }
}
\]

\end{remark}

\subsection{Generally finding cusps}\label{generally-finding-cusps}

\begin{remark}

We now discuss how \(\partial F(S, \Gamma, \cH_{-2})\) can be described lattice-theoretically. Let \(\isoGr(S)\) be the isotropic Grassmannian of the lattice \(S\), and write \(\partial F(S, \Gamma, \cH_{-2}) = \bigcup_{i\geq 0} \partial F(S, \Gamma, \cH_{-2})_i\) for a stratification of the boundary by \(i{\hbox{-}}\)dimensional components. One can show that there are bijections \[
\isoGr_1(L)/\Gamma \cong \partial F(S, \Gamma, \cH_{-2})_0, \qquad 
\isoGr_2(L)/\Gamma \cong \partial F(S, \Gamma, \cH_{-2})_1
,\] and so 0-cusps correspond to \(\Gamma{\hbox{-}}\)orbits of primitive isotropic lines and 1-cusps to orbits of isotropic planes.

\end{remark}

\subsection{Moduli of K3 surfaces with nonsymplectic involution}\label{moduli-of-k3-surfaces-with-nonsymplectic-involution}

\begin{remark}[Constructing moduli of quasi-polarized K3 surfaces lattice-theoretically]

The coarse moduli space \(F_{2d}\) of polarized K3 surfaces \((X, L)\) can be realized using the construction described in \autoref{sec:period_domain}. Recall that \(H^2(X; {\mathbf{Z}}) \cong\lkt\). Fix a marking \(\varphi: H^2(X; {\mathbf{Z}})\to \lkt\) and a polarization \(L\) of degree \(2d\), and let \(h \coloneqq\varphi([L]) \in \lkt\). One can then show that \(h^\perp\cong L_{2d}\). Let \[
{\operatorname{Stab}}_{{\operatorname{O}}(\lkt)}(h) \coloneqq\left\{{\gamma \in {\operatorname{O}}(\lkt) \mathrel{\Big|}\gamma(h) = h}\right\}
\] be the stabilizer of \(h\) in \(\lkt\) and define \[
\Gamma_h \coloneqq{{\operatorname{Stab}}}_{{\operatorname{O}}(\lkt)}(h)^+
\] to be the finite index subgroup fixing \(\Omega_{\Lambda_{2d}}^+\). Letting \({\mathcal{F}}_{2d}^{\mathrm{qp}}\) be the moduli stack of quasi-polarized \(\mathrm{K}3\) surfaces of degree \(2d\), there is an analytic isomorphism at the level of coarse spaces \[
F_{2d}^{\mathrm{qp}} \cong
\leftquotient{\Gamma_h}{\Omega_{\Lambda_{2d}}^+}
.\] However, \({\mathcal{F}}_{2d}^{\mathrm{qp}}\) is generally not a separated stack. We can instead use the stack \({\mathcal{F}}_{2d}^{\mathrm{ADE}}\) of polarized \(\mathrm{K}3\)s with ADE singularities, since there is an isomorphism \(F_{2d}^{\mathrm{ ADE}} \cong F_{2d}^{\mathrm{qp}}\) at the level of coarse spaces.

\end{remark}

\begin{definition}

The theory of moduli of pairs \((X, \iota)\) with \(X\) a K3 surface and \(\iota\) a nonsymplectic involution can be approached using the construction in \autoref{sec:period_domain} as well. Let \(S\subseteq \lkt\) be a primitive hyperbolic \(2{\hbox{-}}\)elementary sublattice which is the \((+1){\hbox{-}}\)eigenspace of an involution \(\rho\) of \(\lkt\). A \emph{$\rho{\hbox{-}}$marking} of \((X,\iota)\) is an isometry \(\varphi\colon H^2(X,{\mathbf{Z}})\rightarrow \lkt\) such that \(\iota^*=\varphi^{-1}\circ\rho\circ\varphi\). Fix such a marking \(\rho\). We have a \textbf{period domain} \(\Omega_S^+\) associated to \(S\), and we define the change-of-marking group associated to \(\rho\) to be \[
\Gamma_\rho=\left\{{ \gamma\in {\operatorname{O}}(\lkt)\mathrel{\Big|}\gamma\circ\rho=\rho\circ\gamma}\right\}
.\] One can then show that the coarse moduli space of \(\rho{\hbox{-}}\)markable K3 surfaces is analytically isomorphic to the locally symmetric space \[
F_S \coloneqq F(S^\perp, \Gamma_\rho, \cH_{-2}) \coloneqq
\leftquotient{\Gamma_\rho}{\qty{\Omega_{S^\perp}\setminus\cH_{-2}}}
.\] In particular, the point corresponding to \((X,\iota)\) is \([\varphi(\mathbb{C}\omega_X)]\).

\end{definition}

\subsection{Hodge theoretic compactifications}\label{hodge-theoretic-compactifications}

\begin{remark}

Hodge theory provides different ways to compactify \(\Omega_{S^\perp}/\Gamma\) for any finite index subgroup \(\Gamma\subseteq {\operatorname{O}}(S^\perp)\). A standard way that involves no choices is provided by the \emph{Baily--Borel compactification} \(\overline{\Omega_{S^\perp}/\Gamma}^{\mathrm{bb}}\). This is a projective normal compactification whose boundary is stratified into \(0{\hbox{-}}\)cusps and \(1{\hbox{-}}\)cusps which correspond to \(\Gamma{\hbox{-}}\)orbits of isotropic vectors \(I\subseteq T\) and isotropic planes \(J\subseteq T\). \emph{Toroidal compactifications} \(\overline{\Omega_{S^\perp}/\Gamma}^{\mathfrak{F}}\) are blow-ups of \(\overline{\Omega_{S^\perp}/\Gamma}^{\mathrm{bb}}\) which depend on the choice of a compatible system of admissible fans \(\mathfrak{F}=\{\mathfrak{F}_K\}\) for each isotropic vector \(I\) or plane \(J\). The fan \(\mathfrak{F}_K\) is a rational polyhedral decomposition of the rational closure \(C_{K,{\mathbf{Q}}}\) of the positive cone \(C_K \subseteq K^\perp/K \otimes\mathbb{R}\). It is required to satisfy the usual fan axioms, and additionally be \(\Gamma{\hbox{-}}\)invariant with only finitely many orbits of cones. As this datum is trivial for isotropic planes, it is sufficient to provide the fan only for the isotropic vectors \(I\), hence \(\mathfrak{F}=\{\mathfrak{F}_I\}\). Lastly, \emph{Semitoroidal compactifications} are due to Looijenga and simultaneously generalize the Baily--Borel and toroidal compactifications by allowing the fans \(\mathfrak{F}_I\) to be not necessarily finitely generated.

\end{remark}

\section{\texorpdfstring{Hyperelliptic \(\mathrm{K}3\)s}{Hyperelliptic \textbackslash mathrm\{K\}3s}}\label{hyperelliptic-mathrmk3s}

\begin{definition}

Let \(X\) be a \(\mathrm{K}3\) surface and let \(L\in \operatorname{Pic}(X)\) be a line bundle with \(L^2 > 0\) where the linear system \({\left\lvert {L} \right\rvert}\) has no fixed components. We say that \({\left\lvert {L} \right\rvert}\) is a \textbf{hyperelliptic linear system on \(X\)} and \(X\) is a \textbf{hyperelliptic \(\mathrm{K}3\) surface} if \({\left\lvert {L} \right\rvert}\) contains a hyperelliptic curve.

\end{definition}

\begin{remark}

The induced morphism \(\varphi_{{\left\lvert {L} \right\rvert}}: X\to {\mathbf{P}}^g\) where \(L^2 = 2g-2\) in this case is a generally 2-to-1 morphism onto a surface \(F\) of degree \(g-1\) in \({\mathbf{P}}^g\). By the classification of surfaces, either \(F \cong {\mathbf{P}}^2\) or \({ \mathbf{F} }_n\)\footnote{A Hirzebruch surface \({ \mathbf{F} }_n \coloneqq\mathop{\mathrm{Proj}}_{{\mathbf{P}}^1} ({\mathcal{O}}_{{\mathbf{P}}^1}(-n) \oplus {\mathcal{O}}_{{\mathbf{P}}^1})\).} with \(n\in \left\{{0,1,2,3,4}\right\}\) ramified over a curve \(C \in {\left\lvert {-2 K_F} \right\rvert}\).

\end{remark}

\begin{remark}

The open locus of \(\Mhe\) can be realized using the standard construction of \(L{\hbox{-}}\)polarized \(\mathrm{K}3\) surfaces, taking \(L = U{ {}^{ \oplus{2} }  } \oplus D_{16}\). More generally, degree \(n\) hyperelliptic \(\mathrm{K}3\) surfaces can be constructed by taking \(L = U{ {}^{ \oplus{2} }  } \oplus D_{n-2}\).

\end{remark}

\subsection{Hyperelliptic quartic K3s}\label{hyperelliptic-quartic-k3s}

\begin{remark}

We now focus back on our main case of interest: hyperelliptic quartic K3s, i.e.~hyperelliptic K3 surfaces of degree \(4\). In this case, the hyperbolic \(2{\hbox{-}}\)elementary even lattice \(S\) is given by \(U(2)\), which corresponds to the invariants \((r,a,\delta)=(2,2,0)\).

The Baily--Borel compactification \(\overline{\Omega_{S^\perp}/\Gamma}^{\mathrm{bb}}\) \ldots{} for which \(\Gamma\)? Was studied by Laza--O'Grady.

Now relate \(\overline{\mathbf{K}}_{\mathrm{h}}\) with \(\overline{\Omega_{S^\perp}/\Gamma}\) and an appropriate Looijenga semitoroidal. Where is this in Valery and Phil's work? Give appropriate references.

\end{remark}

\begin{remark}

Following \cite{LO21}, consider the period domain construction described in \autoref{sec:period_domain} using the lattice \(\Lambda_N \coloneqq U{ {}^{ \oplus{2} }  }\oplus D_{N-2}\) and \(\Gamma = {\operatorname{O}}(\Lambda_N)^+\).\footnote{For moduli-theoretic purposes, if \(N\equiv 6\operatorname{mod}8\), one then instead passes to a finite index subgroup as detailed in \cite{LO21}.} We then obtain a sequence of locally symmetric spaces \[
\mathcal{F}(N) \coloneqq F(\Lambda_N, {\operatorname{O}}(\Lambda_N)^+) \coloneqq\leftquotient{{\operatorname{O}}(\Lambda_{N})^+ }{\Omega_{\Lambda_{N}}^+ }
.\] In particular, taking \(N=19\) yields the \(F_4\), the coarse moduli space of standard polarized K3 surfaces of degree 4, and taking \(N=18\) yields a coarse moduli space \(\Mheop\) of quartic (i.e.~degree 4) hyperelliptic K3 surfaces. The lattice embedding \(\Lambda_{18} \hookrightarrow\Lambda_{19}\) induced by \(D_{16} \hookrightarrow D_{17}\) produces an inclusion \(\Mheop \subseteq F_4\) realizing \(\Mheop\) as a normal Heegner divisor in \(F_4\). This in turn induces a morphism \(\Mhe \to \bbcpt{F_4}\). The Baily-Borel compactification \(\Mhe\) was studied in LO16

and \cite{LO21}, where in the latter they show \[
\Mhe \cong \operatorname{Chow}_{2,4}{ \mathbin{/\mkern-6mu/}}{\operatorname{SL}}_4
,\] a GIT quotient of the Chow variety of \((2, 4)\) curves in \({\mathbf{P}}^3\).

\begin{quote}
See section 2.1 here https://arxiv.org/pdf/1801.04845.pdf\#page=6\&zoom=auto,-87,319
\end{quote}

\end{remark}

\begin{theorem}[{\cite[Theorem~2.3]{LO21}}]

The Baily--Borel compactification \[
\Mhe \cong
\bbcpt{ 
\leftquotient{ {\operatorname{O}}(\Lambda_{18})^+ }{ \Omega_{\Lambda_{18} }^+ }
}
\] has two \(0{\hbox{-}}\)cusps (type III boundary components) and eight \(1{\hbox{-}}\)cusps (type II boundary components). The incidences between \(0{\hbox{-}}\)cusps and \(1{\hbox{-}}\)cusps are represented in \Cref{fig:cusp-diagram-bb-deg-4-hyper-K3}.

\begin{figure}[H]
\centering
\input{tikz/cusp-diagram-bb-deg-4-hyper-K3}
\caption{Cusp diagram for degree 4 hyperelliptic $\mathrm{K}3$ surfaces $\Mhe$ .}
\label{fig:cusp-diagram-bb-deg-4-hyper-K3}
\end{figure}

\end{theorem}

\begin{remark}

\begin{quote}
See https://arxiv.org/pdf/2006.06816.pdf\#page=1\&zoom=100,-274,431
\end{quote}

If \(C \subseteq {\mathbf{P}}^1\times{\mathbf{P}}^1\) is a smooth curve of bidegree \((4, 4)\) and \(\pi: X_C: \to {\mathbf{P}}^1\times{\mathbf{P}}^1\) is the double cover branched along \(C\), then \(X_C\) is a smooth hyperelliptic polarized \(\mathrm{K}3\) surface of degree 4 and thus \(X_C\in \Mhe\). Letting \(M \coloneqq{\left\lvert {{\mathcal{O}}_{{\mathbf{P}}^1\times{\mathbf{P}}^1}(4, 4)} \right\rvert} { \mathbin{/\mkern-6mu/}}\mathop{\mathrm{Aut}}({\mathbf{P}}^1\times{\mathbf{P}}^1)\) be the GIT quotient, LO21 describes a birational period map \(M\dashrightarrow\Mhe\).

\end{remark}

\begin{remark}

The K3 surfaces parameterized by \(\overline{\mathbf{K}}\) are double covers of \({\mathbf{P}}^1\times{\mathbf{P}}^1\) branched along curves of class \((4,4)\) in the monomials listed in \eqref{eq:monomials-4-4-Enriques}. More in general, the double covers of \({\mathbf{P}}^1\times{\mathbf{P}}^1\) branched along general curves of class \((4,4)\) give rise to K3 surfaces known as \emph{hyperelliptic} K3 surfaces. Let us construct their family and the KSBA compactification.

Let \({\mathbf{P}}^{24}\) be the space of coefficients, up to scaling, for a bidegree \((4,4)\) polynomial in \({\mathbf{P}}^1\times{\mathbf{P}}^1\). In this case, a monomial \(X_0^iX_1^jY_0^kY_1^\ell\) is indexed by \[
M_{\mathrm{h}}:=\{(i,j,k,\ell)\in {\mathbf{Z}}_{\geq 0}^4\mathrel{\Big|}i+j=k+\ell=4\}.
\] Let \(\mathbf{U}_{\mathrm{h}}\subseteq{\mathbf{P}}^{24}\) be the dense open subset of coefficients \([\ldots:c_{ijk\ell}:\ldots]\) such that the corresponding \((4,4)\) curve is smooth. We can define a KSBA-stable family \[
\left(\mathcal{X}_{\mathrm{h}}:=\mathbf{U}_{\mathrm{h}}\times({\mathbf{P}}^1\times{\mathbf{P}}^1),\frac{1+\epsilon}{2}\mathcal{B}_{\mathrm{hyp}}\right)\rightarrow\mathbf{U}_{\mathrm{hyp}},
\] where \(\mathcal{B}_{\mathrm{h}}\) is the relative divisor given by \[
\sum_{(i,j,k,\ell)\in M_{\mathrm{h}}}c_{ijk\ell}X_0^iX_1^jY_0^kY_1^\ell=0.
\] We can consider the fiberwise double cover \((\mathcal{T}_{\mathrm{h}},\epsilon\mathcal{R}_{\mathrm{h}})\rightarrow\left(\mathcal{X}_{\mathrm{h}},\frac{1+\epsilon}{2}\mathcal{B}_{\mathrm{h}}\right)\), which gives rise to the family of hyperelliptic K3 surfaces. The automorphism group of \({\mathbf{P}}^1\times{\mathbf{P}}^1\) acts on \(\mathbf{U}_{\mathrm{h}}\) identifying isomorphic fibers. In particular, \(\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}({\mathbf{P}}^1\times{\mathbf{P}}^1)\) is the moduli space of smooth hyperelliptic K3 surfaces. To compactify it, we can consider the stack \(\overline{\mathcal{P}}_{\mathrm{h}}'\) given by the closure of the image of the morphism \(\mathbf{U}_{\mathrm{h}}\rightarrow\mathcal{SP}\left(\frac{1+\epsilon}{2},2,8\epsilon^2\right)\). Let \(\overline{\mathbf{P}}_{\mathrm{h}}'\) be the corresponding coarse moduli space and denote by \(\overline{\mathbf{P}}_{\mathrm{h}}\) its normalization, which gives rise to a compactification of the \(18{\hbox{-}}\)dimensional moduli space \(\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}({\mathbf{P}}^1\times{\mathbf{P}}^1)\). Alternatively, by using the family \((\mathcal{T}_{\mathrm{h}},\epsilon\mathcal{R}_{\mathrm{h}})\rightarrow\mathbf{U}_{\mathrm{h}}\) and the moduli functor \(\mathcal{SP}\left(\epsilon,2,16\epsilon^2\right)\) instead, we obtain the compactifications \(\overline{\mathbf{K}}_{\mathrm{h}}\), which instead parameterize generically the hyperelliptic \(\mathrm{K}3\) surfaces. We have that \(\overline{\mathbf{K}}_{\mathrm{h}}\cong\overline{\mathbf{U}}_{\mathrm{h}}\).

\end{remark}

\begin{remark}

The inclusion \(\mathbf{U}\hookrightarrow\mathbf{U}_{\mathrm{h}}\) induces an inclusion of the stacks \(\overline{\mathcal{P}}'\hookrightarrow\overline{\mathcal{P}}_{\mathrm{h}}'\), and hence an inclusion of the corresponding coarse moduli spaces \(\overline{\mathbf{P}}'\hookrightarrow\overline{\mathbf{P}}_{\mathrm{h}}'\). Therefore, we have an induced morphism \(\overline{\mathbf{P}}\rightarrow\overline{\mathbf{P}}_{\mathrm{h}}\) which is finite and birational onto its image. \{\color{red}Luca: The reason why this morphism exists is nontrivial! The normalization is not functorial, so one has to really prove this.\} \{\color{red} The above is also missing the following. Do we have an embedding of \(\mathbf{U}/G\) into \(\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}({\mathbf{P}}^1\times{\mathbf{P}}^1)\)? Recall \(G=\mathbb{G}_m^2\rtimes({\mathbf{Z}}/2{\mathbf{Z}})\).\}

\end{remark}

\begin{remark}

The compactification \(\overline{\mathbf{P}}_{\mathrm{h}}\) should be fully understood from the work in \cite{AE22nonsympinv}. Moreover, the GIT and Baily--Borel should be understood by \cite{LO21}.

\end{remark}

\section{Enriques surfaces}\label{enriques-surfaces}

\subsection{The unpolarized case}\label{the-unpolarized-case}

\begin{remark}

If \(Y\) is an Enriques surface, it is well known that the universal cover \(\pi\colon X\rightarrow Y\) is a \(\mu_2\) Galois cover where \(X\) is a \(\mathrm{K}3\) surface and \(Y\cong X/\iota\) for \(\iota\) the basepoint-free involution swapping the sheets of the cover. We write \(V_{+1}(\iota^*), V_{-1}(\iota^*) \subseteq H^2(X; {\mathbf{Z}})\) for the \((+1)\) and \((-1){\hbox{-}}\)eigenspaces respectively of the induced involution in cohomology \(\iota^*\colon H^2(X; {\mathbf{Z}})\rightarrow H^2(X; {\mathbf{Z}})\). It is well-known that \(V_{+1}(\iota)^{\perp H^2(X; {\mathbf{Z}})} = V_{-1}(\iota)\). The covering map \(\pi\) induces an embedding of lattices \[
\pi^*\colon H^2(Y; {\mathbf{Z}})\hookrightarrow H^2(X; {\mathbf{Z}}),
\] whose image is \(V_{+1}(\iota^*)\). It is well known that

\begin{itemize}
\tightlist
\item
  \(H^2(X;{\mathbf{Z}}) \cong \lkt \cong U^{\oplus3}\oplus E_8^{\oplus2}\) is the \emph{ $\mathrm{K}3$ lattice};
\item
  \(M \coloneqq H^2(Y,{\mathbf{Z}})/{\operatorname{tors}}\cong \lEn \cong U\oplus E_8\) is the \emph{Enriques lattice};
\item
  \(V_{+1}(\iota^*)\cong U(2)\oplus E_8(2) \cong \lEn(2)\);
\item
  \(V_{-1}(\iota^*)\cong U\oplus U(2)\oplus E_8(2)=U\oplus \lEn(2)\).
\end{itemize}

\end{remark}

\begin{remark}

We will use the decomposition of the K3 lattice into summands involving the Enriques lattice \[
\lkt = \lEn \oplus \lEn \oplus U
,\] and describe a vector in the K3 lattice \(\lkt\) with three coordinates \((x,y,z)\) accordingly. Let \(U={\mathbf{Z}}e \oplus {\mathbf{Z}}f\) with \(\left\{{e, f}\right\}\) the standard hyperbolic basis satisfying \(e^2=f^2=e\cdot f-1=0\).

\end{remark}

\begin{remark}[Period domain for unpolarized Enriques surfaces]

Again following the period domain construction described \autoref{sec:period_domain}, now with the lattice \[
N \coloneqq U\oplus \lEn(2)
.\] The period domain for unpolarized Enriques surfaces is \(\Omega_N^+\), and the correct associated locally symmetric space is \[
\Munpolop \coloneqq F(N, {\operatorname{O}}(N)^+, \cH_{-2} )
\coloneqq\leftquotient{{\operatorname{O}}(N)^+}{\qty{\Omega_N^+ \setminus \cH_{-2} }}
.\]

\end{remark}

\begin{lemma}[{Torelli for Enriques surfaces, Horikawa}]

Points in \(\Munpolop\) correspond to isomorphism classes of unpolarized Enriques surfaces.

\end{lemma}

\begin{theorem}[{\cite[Propositions~4.5 and 4.6]{Ste91}}]

The Baily--Borel compactification \[
\Munpol \coloneqq
\bbcpt{
\leftquotient{{\operatorname{O}}^+(N)}{(\Omega_N^+ \setminus\cH_{-2})}
}
\] has two \(0{\hbox{-}}\)cusps and two \(1{\hbox{-}}\)cusps. The incidences between \(0{\hbox{-}}\)cusps and \(1{\hbox{-}}\)cusps are represented in \Cref{fig:cusp-diagram-bb-unpol-En}.

\begin{figure}[H]
\centering
\input{tikz/cusp-diagram-bb-unpol-En}
\caption{Cusp diagram for the moduli space of unpolarized Enriques surfaces $\Munpol$.}
\label{fig:cusp-diagram-bb-unpol-En}
\end{figure}

\end{theorem}

\subsection{\texorpdfstring{Degree \(2\) polarized Enriques surfaces}{Degree 2 polarized Enriques surfaces}}\label{degree-2-polarized-enriques-surfaces}

For degree \(2\) polarized Enriques surfaces, we consider the same period domain, but we change the arithmetic group acting on it.

\begin{definition}

A \textbf{polarization} on an Enriques surface \(Y\) is a pseudo-ample (i.e.~big and nef) line bundle \(L\) on \(Y\); we call this an \textbf{ample polarization} if \(L\) is ample.

\end{definition}

\begin{definition}

A \textbf{numerical (resp. ample numerical) polarization} on \(Y\) is a choice \(L\) of a numerical equivalence class of pseudo-ample (resp. ample) line bundle \(L\).

\end{definition}

\begin{definition}

A \textbf{numerically polarized Enriques surface} is a pair \((Y, L)\) where \(Y\) is an Enriques surface and \(L\) is a numerical polarization on \(Y\).\footnote{Why introduce \textbf{numerical} polarizations? Recall that \(A\) is a polarized abelian variety if it is equipped with an isogeny \(\lambda: A\to A {}^{ \vee }\). If \(L\) is a numerical polarization on \(A\), it induces a unique isogeny \(\lambda_L\), and every such isogeny comes from such an \(L\), so numerical polarization strictly generalizes this notion to other varieties.}

\end{definition}

\begin{remark}

Let \(L\coloneqq U\oplus \lEn{ {}^{ \oplus{2} }  }\), noting that \(N\leq L\), and define the following involution: \begin{align*} I\colon L &\rightarrow L,\\ (x, y, z) &\mapsto(y, x, -z). \end{align*}

A result of Horikawa shows that there is an isometry \(\mu: H^2(X; {\mathbf{Z}})\to L\) such that \(I\circ \mu = \mu \circ I^*\) and produces an embedding \begin{align*} M &\to L \\ m &\mapsto (m, m, 0) .\end{align*}

\begin{quote}
Reference: https://file.notion.so/f/s/b5171ee5-610c-489f-b347-5839cc0005f0/Sterk.pdf?id=2a417bca-589b-4639-86e6-6901fe36ff30\&table=block\&spaceId=7cb2f7c7-7373-4d11-91ab-284625335dc8\&expirationTimestamp=1686073894799\&signature=AIqn4Lp1sajAsu5XQEbx0IS3DBeNdN6LRJIIrZeKpo0\&downloadName=Ste91.pdf\#page=6\&zoom=auto,-155,749
\end{quote}

Define \[
\Gamma'\coloneqq\{g\in {\operatorname{O}}(L)\mathrel{\Big|}g\circ I=I\circ g~\textrm{and}~g(e+f,e+f,0)=(e+f,e+f,0)\}
,\] automorphisms in the centralizer of \(I\) in \({\operatorname{O}}(L)\) fixing the point \((e+f, e+f, 0)\). If \(g\in \Gamma'\) then \({ \left.{{g}} \right|_{{N}} } \in {\operatorname{O}}(N)\). \dzg{this is what Sterk claims, needs proof.} So define \[
\Gamma\coloneqq\{{ \left.{{g}} \right|_{{N}} } \mathrel{\Big|}g\in\Gamma'\}\leq O(N)
,\] which is the image of \(\Gamma'\) in \({\operatorname{O}}(L_-)\). Again using the construction in \autoref{sec:period_domain}, the moduli space for Enriques surfaces with a polarization of degree \(2\) is given by the locally symmetric space \[
\Mpolop \cong 
F(N, \Gamma) =
\leftquotient{\Gamma}{\Omega_N^+}
.\]

\end{remark}

\begin{theorem}[{\cite[\S 4.3]{Ste91}}]

The Baily--Borel compactification \[
\Mpol \coloneqq
\bbcpt{
\leftquotient{\Gamma}{\Omega_N^+}
}
\] has five \(0{\hbox{-}}\)cusps and nine \(1{\hbox{-}}\)cusps. The incidences between \(0{\hbox{-}}\)cusps and \(1{\hbox{-}}\)cusps are represented in \Cref{fig:cusp-diagram-bb-deg-2-En}.

\begin{figure}[H]
\centering
\input{tikz/cusp-diagram-bb-deg-2-En}
\caption{Cusp diagram for degree 2 polarized Enriques surfaces $\Mpol$.}
\label{fig:cusp-diagram-bb-deg-2-En}
\end{figure}

\end{theorem}

\subsection{Numerically polarized}\label{numerically-polarized}

\cite{GH16}

\subsection{\texorpdfstring{The family of degree \(2\) polarized Enriques surfaces}{The family of degree 2 polarized Enriques surfaces}}\label{the-family-of-degree-2-polarized-enriques-surfaces}

\begin{remark}

We review the construction of degree \(2\) polarized Enriques surfaces following \cite[Chapter~V, \S\,23]{BHPV04}. Let us consider the involution on \({\mathbf{P}}^1\times{\mathbf{P}}^1\) given by \[
\iota\colon([X_0:X_1],[Y_0:Y_1])\mapsto([X_0:-X_1],[Y_0:-Y_1]).
\] We have that \(\iota\) has precisely four isolated fixed points, namely \[
([0:1],[0:1]),~([0:1],[1:0]),~([1:0],[0:1]),~([1:0],[1:0]).
\] Let \(B\subseteq{\mathbf{P}}^1\times{\mathbf{P}}^1\) be a general \(\iota{\hbox{-}}\)invariant curve of class \((4,4)\) not passing through the fixed points of \(\iota\). Then, the bi-homogeneous polynomial giving \(B\) consists of the following monomials: \begin{equation} \label{eq:monomials-4-4-Enriques} \begin{gathered} X_0^4Y_0^4,~X_0^4Y_0^2Y_1^2,~X_0^4Y_1^4,~X_0^3X_1Y_0^3Y_1,~X_0^3X_1Y_0Y_1^3,~X_0^2X_1^2Y_0^4,\\ X_0^2X_1^2Y_0^2Y_1^2,~X_0^2X_1^2Y_0^4,~X_0X_1^3Y_0^3Y_1,~X_0X_1^3Y_0Y_1^3,~X_1^4Y_0^4,~X_1^4Y_0^2Y_1^2,~X_1^4Y_1^4. \end{gathered} \end{equation} The coefficients of \(X_0^4Y_0^4, X_0^4Y_1^4, X_1^4Y_0^4, X_1^4Y_1^4\) must be nonzero to guarantee that \(B\) does not pass through the torus fixed points of \(\iota\).

\end{remark}

\begin{remark}

The double cover \(\pi\colon T\rightarrow{\mathbf{P}}^1\times{\mathbf{P}}^1\) branched along \(B\) is a well known to be a \(\mathrm{K}3\) surface: \(T\) is smooth and minimal, \(K_T\sim\pi^*\left(K_{{\mathbf{P}}^1\times{\mathbf{P}}^1}+\frac{1}{2}B\right)\sim0\), and \(\pi_*{\mathcal{O}}_T={\mathcal{O}}_{{\mathbf{P}}^1\times{\mathbf{P}}^1}\oplus{\mathcal{O}}_{{\mathbf{P}}^1\times{\mathbf{P}}^1}\left(-\frac{1}{2}B\right)\), which gives \(h^1({\mathcal{O}}_T)=0\).

\end{remark}

\begin{remark}

Let \(\mathcal{L}^{\otimes2}={\mathcal{O}}_{{\mathbf{P}}^1\times{\mathbf{P}}^1}(4,4)\) and let \(p\colon L\rightarrow{\mathbf{P}}^1\times{\mathbf{P}}^1\) be the total space of the line bundle \(\mathcal{L}\). Then the double cover \(T\) of \({\mathbf{P}}^1\times{\mathbf{P}}^1\) branched along \(B\) can be viewed inside \(L\) as the vanishing locus of \(t^2-p^*s=0\), where \(B=V(s)\) and \(t\in\Gamma(L,p^*\mathcal{L})\) is the tautological section. We have that then \(\iota\) lifts to an involution \(\widetilde{\iota}\) of \(T\) with exactly eight fixed points: two over each fixed point of \(\iota\). If \(\tau\) denotes the deck transformation of the cover, i.e.~\(t\mapsto-t\), then we have that \(\widetilde{\iota}\) commutes with \(\tau\) and the composition \(\sigma=\tau\circ\widetilde{\iota}\) is a fixed-point free involution of \(T\). The quotient \(q\colon T\rightarrow T/\sigma=S\) is then an Enriques surface called \emph{Horikawa model}, and comes equipped with a degree \(2\) polarization induced by \({\mathcal{O}}_{{\mathbf{P}}^1\times{\mathbf{P}}^1}(1,1)\).

Let \(R\subseteq T\) be the ramification locus, so that \(2R=\pi^*B\), define \(\overline{R}=q(R)\), and let \(0<\epsilon\ll1\) rational. Then we have the two following covering equalities: \[
K_T+\epsilon R\sim_{{\mathbf{Q}}}\pi^*\left(K_{{\mathbf{P}}^1\times{\mathbf{P}}^1}+\frac{1+\epsilon}{2}B\right),
\]

\[
K_T+\epsilon R\sim_{{\mathbf{Q}}}q^*\left(K_S+\frac{\epsilon}{2}\overline{R}\right).
\]

\end{remark}

\begin{lemma}

With the notation introduced above, we have the following self-intersection numbers:

\begin{itemize}
\item
  \(\left(K_{{\mathbf{P}}^1\times{\mathbf{P}}^1}+\frac{1+\epsilon}{2}B\right)^2=8\epsilon^2\);
\item
  \((K_T+\epsilon R)^2=16\epsilon^2\);
\item
  \(\left(K_S+\frac{\epsilon}{2}\overline{R}\right)^2=8\epsilon^2\).
\end{itemize}

\end{lemma}

\begin{remark}

We now relativize the above construction. Let \({\mathbf{P}}^{12}\) be the space of coefficients, up to scaling, for a bidegree \((4,4)\) polynomial in the monomials in \eqref{eq:monomials-4-4-Enriques}. So, if \(c_{ijk\ell}\) denotes the coefficient of \(X_0^iX_1^jY_0^kY_1^\ell\), then \([\ldots:c_{ijk\ell}:\ldots]\in{\mathbf{P}}^{12}\) with \((i,j,k,\ell)\) within the following set: \[
M\coloneqq\{(i,j,k,\ell)\in {\mathbf{Z}}_{\geq0}^4\mathrel{\Big|}i+j=k+\ell=4,~i+k\equiv j+\ell\equiv0~\mathrm{mod}~2\}.
\] Let \(\mathbf{U}\subseteq{\mathbf{P}}^{12}\) be the dense open subset of coefficients such that the corresponding \(\iota{\hbox{-}}\)invariant \((4,4)\) curve \(B\subseteq{\mathbf{P}}^1\times{\mathbf{P}}^1\) is smooth and does not pass through the torus fixed points of \({\mathbf{P}}^1\times{\mathbf{P}}^1\). Define \(\mathcal{X}\coloneqq\mathbf{U}\times({\mathbf{P}}^1\times{\mathbf{P}}^1)\) and let \(\mathcal{X}\rightarrow\mathbf{U}\) be the projection. Let \[
\mathcal{B}\coloneqq V\left(\sum_{(i,j,k,\ell)\in M}c_{ijk\ell}X_0^{i}X_1^{j}Y_0^{k}Y_1^{\ell}\right)\subseteq\mathcal{X}.
\] Then \(\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}\) is a family of stable pairs with fibers given by \(\left({\mathbf{P}}^1\times{\mathbf{P}}^1,\frac{1+\epsilon}{2}B\right)\) as described above. Additionally, we observe that \(\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}\) is a KSBA-stable as defined in \cite[8.7]{Kol23}.

\end{remark}

\begin{remark}

The family \(\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}\) has isomorphic fibers. To eliminate this redundancy, we consider the action of \(\mathrm{Aut}({\mathbf{P}}^1\times{\mathbf{P}}^1)\cong(\mathrm{PGL}_2\times\mathrm{PGL}_2)\rtimes {\mathbf{Z}}/2 {\mathbf{Z}}\) (see \cite{Dol12}) on \(H^0({\mathcal{O}}(4,4))\). More precisely, we want to look at the subgroup \(G\) which preserves \(\iota{\hbox{-}}\)invariant \((4,4){\hbox{-}}\)curves not passing through the torus fixed points. Note that the \({\mathbf{Z}}_2{\hbox{-}}\)action preserves the set of monomials \(M\) as \((i,j,k,\ell)\in M\) if and only if \((k,\ell,i,j)\in M\). Now consider a generic \(\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]\in\mathrm{PGL}_2\) acting on \([X_0:X_1]\). One can check directly that the action of this matrix preserves the monomials in \(M\) if and only if \(b=c=0\), and the same holds if we consider the action of the second copy of \(\mathrm{PGL}_2\) which acts on \([Y_0:Y_1]\). In particular, we have that \(G\cong\mathbb{G}_m^2\rtimes({\mathbf{Z}}/2{\mathbf{Z}})\). Therefore, we have an action \(G\curvearrowright\mathbf{U}\) which identifies the isomorphic fibers of \(\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}\).

Over \(\mathbf{U}\), we can also consider the cover \(\left(\mathcal{T},\epsilon\mathcal{R}\right)\rightarrow\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\) which gives the family of isomorphism classes of pairs \((T,\epsilon R)\) and the fiberwise quotient by the Enriques involution \(\left(\mathcal{T},\epsilon\mathcal{R}\right)\rightarrow\left(\mathcal{S},\frac{\epsilon}{2}\overline{\mathcal{R}}\right)\) which gives the family of isomorphism classes of Enriques surfaces \(\left(S,\frac{\epsilon}{2}\overline{R}\right)\). Summarizing, we have the following commutative diagram:

\begin{center}
\begin{tikzpicture}[>=angle 90]
\matrix(a)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.25ex]
{(\mathcal{T},\epsilon\mathcal{R})&\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\\
\left(\mathcal{S},\frac{\epsilon}{2}\overline{\mathcal{R}}\right)&\mathbf{U}\\};
\path[->] (a-1-1) edge node[above]{}(a-1-2);
\path[->] (a-1-1) edge node[left]{}(a-2-1);
\path[->] (a-2-1) edge node[below]{}(a-2-2);
\path[->] (a-1-2) edge node[right]{}(a-2-2);
\end{tikzpicture}
\end{center}

\end{remark}

\begin{definition}

Following the notation in \cite[Theorem~8.1]{Kol23}, consider the moduli functors \(\mathcal{SP}(\mathbf{a},d,\nu)\) for \[
(\mathbf{a},d,\nu)=\left(\frac{1+\epsilon}{2},2,8\epsilon^2\right),~\left(\epsilon,2,16\epsilon^2\right),~\left(\frac{\epsilon}{2},2,8\epsilon^2\right).
\] and the corresponding coarse moduli spaces \(\mathrm{SP}(\mathbf{a},d,\nu)\). We now define the following stacks:

Consider the KSBA-stable family \(\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}\). Therefore there is an induced morphism \(\mathbf{U}\rightarrow\mathcal{SP}\left(\frac{1+\epsilon}{2},2,8\epsilon^2\right)\) and denote by \(\mathcal{P}'\) the closure of its image. Let \(\overline{\mathbf{P}}'\) be the coarse moduli space corresponding to \(\overline{\mathcal{P}}'\), and denote by \(\overline{\mathbf{P}}\) its normalization. We have that \(\overline{\mathbf{P}}\) provides a projective compactification of \(\mathbf{U}/G\). By using the families \((\mathcal{T},\epsilon\mathcal{R})\rightarrow\mathbf{U}\) and \(\left(\mathcal{S},\frac{\epsilon}{2}\overline{\mathcal{R}}\right)\rightarrow\mathbf{U}\) instead, we obtain the compactifications \(\overline{\mathbf{K}}\) and \(\overline{\mathbf{E}}\) of \(\mathbf{U}/G\) respectively, which instead parameterize generically the \(\mathrm{K}3\) and Enriques surfaces.

\end{definition}

\begin{remark}

It is a standard observation that the compactifications \(\overline{\mathbf{P}},\overline{\mathbf{K}},\overline{\mathbf{E}}\) are isomorphic to each other (see \cite[\S\,3]{MS21} for an analogous situation). We will mostly focus on \(\overline{\mathbf{P}}\) as it parameterized the simplest objects.

\end{remark}

\section{Morphisms of moduli and cusps}\label{morphisms-of-moduli-and-cusps}

\subsection{Mapping the boundaries: matching cusp diagrams}\label{mapping-the-boundaries-matching-cusp-diagrams}

\begin{remark}

Let \({\mathcal{F}}_{2d}\) be the moduli space of polarized \(\mathrm{K}3\) surfaces of degree \(2d\). How do we match the cusp diagram for the Baily--Borel compactification for the moduli of degree \(2\) Enriques surfaces with the Baily--Borel compactification for degree \(4\) hyperelliptic \(\mathrm{K}3\) surfaces? This is actually quite subtle, and it works as follows:

\begin{itemize}
\item
  \cite[\S\,5]{Ste91} matches cusps for degree \(2\) Enriques surfaces and degree \(4\) \(\mathrm{K}3\) surfaces, \[
  \Mpol \rightleftharpoons\bbcpt{{\mathcal{F}}_4}
  .\]
\item
  Using Scattone's method in \cite[\S\,6]{Sca87}, we can match the cusps for degree \(4\) hyperelliptic \(\mathrm{K}3\) surfaces and degree \(4\) \(\mathrm{K}3\) surfaces, \[
  \Mhe \rightleftharpoons\bbcpt{{\mathcal{F}}_4}
  .\]
\item
  The above two points imply the matching we need.
\end{itemize}

\end{remark}

\begin{remark}

What is Scattone's method uses the following observations:

\begin{itemize}
\tightlist
\item
  \(1{\hbox{-}}\)cusps of \(\bbcpt{ {\mathcal{F}}_4 }\) are in one-to-one correspondence with the orthogonal complements of \(D_8\) in the Niemeier lattices, and
\item
  The \(1{\hbox{-}}\)cusps of \(\Mhe\) are in one-to-one correspondence with the orthogonal complements of \(D_7\) in the Niemeier lattices.
\end{itemize}

\[\begin{tikzcd}
& {\Mpol} \\
{\Munpol} && {\Mhe}
\arrow[from=1-2, to=2-1, "\text{Finite}"']
\arrow[from=1-2, to=2-3, "\text{Finite-to-one}"]
\end{tikzcd}\]

As a result, there are two cusp incidence diagrams to match:

\begin{itemize}
\tightlist
\item
  Polarized \(\left\{{X_i, p_i}\right\}\) to unpolarized \(\left\{{Y_i, q_i}\right\}\):
\end{itemize}

\input{tikz/cusp_correspondence_one}

\begin{itemize}
\tightlist
\item
  Polarized \(\left\{{X_i, p_i}\right\}\) to hyperelliptic \(\left\{{Z_i, r_i}\right\}\):
\end{itemize}

\input{tikz/cusp_correspondence_two}

\end{remark}

\subsection{Type III KPP model at the (18,2,0) odd 0-cusp}\label{type-iii-kpp-model-at-the-1820-odd-0-cusp}

\begin{remark}

At the (18, 2, 0) odd 0-cusp the Type III stable models are of pumpkin type

Stable models vs KPP models, write down the definition and the differences.

Let \(X\) be a \(\mathrm{K}3\) surface with a nonsymplectic involution \(\iota\) with induced involution \(i^*\) on \(H^2(X; {\mathbf{Z}})\). We define \(S\) to be the \((+1){\hbox{-}}\)eigenspace \(\iota^*\); it is a hyperbolic lattice 2-elementary lattice, and all the possibilities for such lattices were classified by Nikulin. We denote by \(T\) the orthogonal complement of \(S\) in \(H^2(X;{\mathbf{Z}})\).

\end{remark}

\begin{definition}

What makes an \textbf{odd} \(0{\hbox{-}}\)cusp different from an \textbf{even} \(0{\hbox{-}}\)cusp?

\end{definition}

\begin{definition}

We define two types of stable models \(\overline{X}_0=\cup\overline{V}_i\):

\begin{enumerate}

\item \emph{Pumpkin}. Each surface $\overline{V}_i$ has two sides $\overline{D}_i=\overline{D}_{i,\mathrm{left}}+\overline{D}_{i,\mathrm{right}}$, they are glued in a circle, all of $D_i$ meeting at the north and south poles.

\item \emph{Smashed pumpkins}. Starting with a surface of the pumpkin type, one short side is contracted to a point, so that the north and south poles are identified.

\end{enumerate}

If the surface \(V_i\), say to the left, is \((\mathbb{F}_1,D_1+D_2)\), where \(D_1\sim f\) is the short side being contracted, \(D_2\sim2s+2f\) is the other side, and \(C_g\sim f\) on \(V\) contract \(V_i\) by the \({\mathbf{P}}^1{\hbox{-}}\)fibration \(V_i\rightarrow{\mathbf{P}}^1\). Then on the next surface \(V_{i-1}\) to the left the long side will fold \(2:1\) to itself, creating a non-normal singularity along that side.

If on \(V_i\) the divisor \(C_g\) has degree \(C_g^2\geq2\), then only the short side is contracted and the resulting surface \(\overline{V}_i\) is normal in codimension \(1\), with only two points in the normalization glued together (the poles).

\end{definition}

\begin{theorem}[{\cite[Theorem~9.9]{AE22nonsympinv}}]

Let \((\overline{X}_0,\cup\overline{V}_i,\epsilon\overline{C}_g)\) be the stable model of a pair \((X_0=\cup V_i,\epsilon C_g)\), where \(X_0\) is the KPP model of a Type III Kulikov surface and \(C_g\) is the component of genus \(g\geq2\) in the ramification divisor \(R\). Then the normalization of each \(\overline{V}_i\) is an ADE surface with an involution from \cite[Table~2]{AT21}. Moreover,

\begin{itemize}
\item
  If \(\overline{T}\) is an odd \(0{\hbox{-}}\)cusp of \(F_S\), then \(\overline{X}_0\) is of pumpkin type.
\item
  If \(\overline{T}\) is an even \(0{\hbox{-}}\)cusp of \(F_S\), then \(\overline{X}_0\) is of smashed pumpkin type. The surfaces \(V_i\) of the last type in definition Definition\textasciitilde9.8, on which \(V_i\rightarrow\overline{V}_i\) contracts one side are surfaces of \cite[Table~2]{AT21} for which one of the sides has length \(0\), i.e.~those with a double prime or a ``\(+\)''.
\end{itemize}

\end{theorem}

\subsection{ADE surfaces}\label{ade-surfaces}

\begin{definition}

An \textbf{ADE surface} is a pair \((Y,C)\), where \(Y\) is a normal surface. \((Y,C)\) has log canonical singularities and the divisor \(-2(K_Y+C)\) is Cartier and ample. \(L\coloneqq-2(K_Y+C)\) is referred to as the \textbf{polarization} of the ADE surface \((Y,C)\).

\end{definition}

\begin{remark}

Let \(B\in|L|\) effective divisor such that \((Y,C+\frac{1+\epsilon}{2}B)\) is log canonical for \(0<\epsilon\ll1\), then \((Y,C+\frac{1+\epsilon}{2})\) is called an ADE pair. We can take the double cover \(X\rightarrow Y\) branched along \(B\) and I guess possibly along \(C\). It can happen that \(Y\) is toric and \(C\) is part of the toric boundary.

ADE surfaces admit a combinatorial classification. The classes of ADE surfaces are called shapes. A shape can be \emph{pure} or \emph{primed}. Surfaces of pure shape are fundamental. Surfaces of primed shape are secondary and can be obtained from surfaces of pure shape using an operation called \textbf{priming}.

The ADE surfaces of pure shape are all toric. To construct these we start from a polarized toric surface \((Y, L)\), where \(L=-2(K_Y+C)\). This corresponds to a lattice polytope \(P\) in \(M\otimes_{\mathbf{Z}}{\mathbf{R}}\). Given a surface \((Y, C)\) of pure shape, the irreducible components of \(C\) are called \textbf{sides}. There are two sides with a point in common called left or right. They decompose \(C=C_1+C_2\). A side can be \textbf{long} or \textbf{short} depending on whether a side \(C'\) satisfies \(C'\cdot L=2,4\) or \(C'\cdot L=1,3\) respectively. The ADE surfaces of pure shape are listed in \cite[Table~1]{AT21} (see Figures\textsubscript{1,}2,\textasciitilde3 therein). Here are some basic examples:

\begin{itemize}
\item
  The ADE surface \((Y, C)\) corresponding to \(D_4\) is \(Y={\mathbf{P}}^1\times{\mathbf{P}}^1\) and \(C\) is the sum of two incident torus fixed curves.
\item
  The ADE surface \((Y, C)\) corresponding to \(A_1\) is \(Y={\mathbf{P}}^2\) and \(C\) is the sum of two torus fixed curves. The polarization is \({\mathcal{O}}(2)\).
\end{itemize}

\end{remark}

\begin{remark}

The superscripts minus signs on the left or right denote the location of the short side. Note both sides can be long or short. Do they correspond to the visible length? Not at all! The ADE surface \(A_3\) has two long sides, but one edge is shorter than the other. By the way, in this case, \(Y=\mathbb{F}_2^0\). \(A_2^-\) has a long side on the left and a short side on the right.

Primed shapes. Priming is an operation that produces a new del Pezzo surface \((\overline{Y}',\overline{C}')\) from an old one \((Y,C)\). The priming operation is basically a weighted blow-up given by the composition of two ordinary blow-ups and the contraction of a \((-2){\hbox{-}}\)curve making an \(A_1\) singularity. Weighted blow-ups of this form are the basis of the priming operation. Weighted blow-up with respect to the idea \((y,x^2)\). Priming has the meaning of disconnecting a curve from another. Given an ADE pair \((Y, C+\frac{1+\epsilon}{2})\), then the priming operation is performed on the points of intersection between \(C\) and \(B\), which intersect transversely by \cite[Remark~3.3]{AT21}. Priming may not exist, and there are some necessary and sufficient conditions for priming to exist.

For an ADE shape, we add a prime symbol when priming on a long side. When priming a short side, we change the minus into a plus. All the ADE surfaces, pure or primed are in \cite[Table~2]{AT21}.

\end{remark}

\section{Our new results}\label{our-new-results}

\subsection{Enriques strategy}\label{enriques-strategy}

\begin{remark}

This story suggests the following approach to Enriques surfaces:

\begin{itemize}
\tightlist
\item
  Fully understand the cusps of the Enriques moduli space, possibly in terms of what has been done for K3s already.
\item
  For each cusp, find the Coxeter diagram.
\item
  For each Coxeter diagram, cook up the right \(\mathrm{IAS}\) pair of a manifold and a divisor \(R_{\mathrm{IAS}}\). For us, instead of an \(\mathrm{IAS}^2\) it may be an \(\IARP^2\), and may come from some fusion of known \(\mathrm{IAS}^2\)s for K3s, maybe as simple as quotienting the \(\mathrm{IAS}^2\) by the antipodal map.
\item
  Reverse-engineer the \(\IARP^2\) so that it carries two commuting involutions, and probably take \(R_{\mathrm{IAS}}\) to be the intersection of the two ramification divisors on the \(\IARP^2\).
\item
  Describe all of the ways the \(\IARP^2\) can degenerate, a la Valery's pumpkin-type models.
\end{itemize}

\end{remark}

\subsection{Lemmas/theorems}\label{lemmastheorems}

\begin{lemma}

If \(\operatorname{sig}(L) = (p, q)\) and \(e\in L\) is isotropic, then \(\operatorname{sig}({\mathbf{Z}}e) = (?, ?)\) and \(\operatorname{sig}({\mathbf{Z}}e^\perp) = (?, ?)\).

\end{lemma}

\begin{lemma}

Let \(L\) be a lattice of signature \((p, q)\) and let \(e\in L\) be an isotropic vector. Then \[\operatorname{sig}(e^\perp/e) = (p-1, q-1).\]

\end{lemma}

\begin{proposition}

Let \(\lkt\) be the K3 lattice and let \(h\) be an ample class of degree \(d\). Then \[
L_{2d} \coloneqq h^{\perp \lkt } \cong \left\langle{-2d}\right\rangle \oplus U{ {}^{ \oplus{2} }  }\oplus E_8{ {}^{ \oplus{2} }  }
\] and \(\operatorname{sig}L_{2d} = (2, 19)\). Thus \(F_{2d}\) arises as the Hodge-theoretic moduli space associated with the period domain \(D_L\) for the lattice \(L \coloneqq L_{2d}\).

\end{proposition}

\begin{remark}

The theorem below needs the following notations and conventions (these will all be introduced before as needed).

\(\lkt=U^{\oplus3}\oplus E_8^{\oplus2}=(U\oplus E_8)^{\oplus2}\oplus U\) and \[
I(m,m',h)=(m,m',-h).
\]

\[
L_-=U\oplus U(2)\oplus E_8(2)
\]

\[
\Omega_-=\{[v]\in{\mathbf{P}}(L_-\otimes\mathbb{C})\mathrel{\Big|}v^2=0,~v\cdot\overline{v}>0\}
\]

\end{remark}

\begin{definition}

Consider the following subgroup of \(O(\lkt)\): \[
\Gamma'=\{g\in O(\lkt)\mathrel{\Big|}g\circ I=I\circ g,~g(e+f,e+f,0) = (e+f,e+f,0)\}
\] Note that we have a natural group homomorphism \(\Gamma'\rightarrow O(L_-)\) given by \(g\mapsto g|_{L_-}\). To prove that \(g|_{L_-}\in O(L_-)\) it is enough to observe that \(g(L_-)=L_-\). Let \(x\in L_-\). We have that \(g(x)\in L_-\) if \(I(g(x))=-g(x)\). This holds because \[
I(g(x))=g(I(x))=g(-x)=-g(x).
\] We denote by \(\Gamma\) the image of \(\Gamma'\rightarrow O(L_-)\).

\end{definition}

\begin{remark}

\[
E_2=\Omega_-/\Gamma
\]

\[
\Omega_{4,\mathrm{h}}=\{[v]\in{\mathbf{P}}(\Lambda_{18}\otimes\mathbb{C})\mathrel{\Big|}v^2=0,~v\cdot\overline{v}>0\}
\]

\(\Lambda_{18}=U^{\oplus2}\oplus D_{16}\), \(\Gamma_{4,\mathrm{h}}=O(\Lambda_{18})\).

\begin{figure}[H]
\begin{tikzpicture}

\draw  (0,0)-- (1,0);
\draw  (1,0)-- (2,0);
\draw  (2,0)-- (3,0);
\draw  (3,0)-- (4,0);
\draw  (4,0)-- (5,0);
\draw  (5,0)-- (6,0);
\draw  (2,0)-- (2,-1);

\fill [color=black] (0,0) circle (2.4pt);
\fill [color=black] (1,0) circle (2.4pt);
\fill [color=black] (2,0) circle (2.4pt);
\fill [color=black] (3,0) circle (2.4pt);
\fill [color=black] (4,0) circle (2.4pt);
\fill [color=black] (5,0) circle (2.4pt);
\fill [color=black] (6,0) circle (2.4pt);
\fill [color=black] (2,-1) circle (2.4pt);

\draw[color=black] (2.4,-1) node {$\alpha_2$};
\draw[color=black] (0,0.4) node {$\alpha_1$};
\draw[color=black] (1,0.4) node {$\alpha_3$};
\draw[color=black] (2,0.4) node {$\alpha_4$};
\draw[color=black] (3,0.4) node {$\alpha_5$};
\draw[color=black] (4,0.4) node {$\alpha_6$};
\draw[color=black] (5,0.4) node {$\alpha_7$};
\draw[color=black] (6,0.4) node {$\alpha_8$};

\end{tikzpicture}
\caption{The $E_8$ lattice (Sterk's convention).}
\label{fig:E10}
\end{figure}

\end{remark}

\begin{theorem}

There exists an injective morphism \[
E_2\rightarrow F_{4,\mathrm{h}}
\] which extends to a morphism of the Baily--Borel compactifications \[
\overline{E}_2^{\mathrm{bb}}\rightarrow \overline{F}_{4,\mathrm{h}}^{\mathrm{bb}}.
\]

\end{theorem}

\begin{proof}

Consider the inclusion of \(U(2)\) into \(U(2)\oplus E_8(2)\) as direct summand. By considering the orthogonal complements in \(\lkt\) we obtain that \[
L_-\subseteq\Lambda_{18}.
\] From this follows from the definitions of \(\Omega_-\) and \(\Omega_{4,\mathrm{h}}\) that we have an inclusion \[
\Omega_-\hookrightarrow\Omega_{4,\mathrm{h}}.
\] Let us show that this descends to a morphism \[
\Omega_-/\Gamma_2\rightarrow\Omega_{4,\mathrm{h}}/\Gamma_{4,\mathrm{h}}.
\] Let \([v],[w]\in\Omega_2\) and assume there exists \(g\in\Gamma\) such that \(g([v])=[w]\). We show that there exists \(h\in\Gamma_{4,\mathrm{h}}\) such that \(h([v])=[w]\). By the definition of \(\Gamma\), \(g=\widetilde{g}|_{L_-}\), there exists \(\widetilde{g}\in O(\lkt)\) such that \(\widetilde{g}\circ I=I\circ\widetilde{g}\) and \(f(e+f,e+f,0)=(e+f,e+f,0)\). Then, by the proof of \cite[Proposition~2.7]{Ste91}, we have that \(\widetilde{g}\) preserves \(e+f\) and \(e-f\) in \(L_+=U(2)\oplus E_8(2)\). In particular, \(\widetilde{g}\) preserves the summand \(U(2)\subseteq U(2)\oplus E_8\). This implies that \(\widetilde{g}\) preserves \(U(2)^\perp=\Lambda_{18}\). In particular, by setting \(h=\widetilde{g}|_{\Lambda_{18}}\) we obtain what we needed.

We now prove that the morphism \(\varphi\colon\Omega_-/\Gamma\rightarrow\Omega_{4,\mathrm{h}}/\Gamma_{4,\mathrm{h}}\) is injective. Let \(x_1,x_2\in\Omega_-/\Gamma\) and assume that \(\varphi(x_1)=\varphi(x_2)\). Let \(S_i\) be the Enriques surface corresponding to \(x_i\). Then \(S_i\) is the quotient of a K3 surface \(T_i\) which is the double cover \(\pi_i\colon T_i\rightarrow{\mathbf{P}}^1\times{\mathbf{P}}^1\) branched along a \((4,4)\) curve \(B_i\) which is invariant with respect to the involution \(\iota\colon(x,y)\mapsto(-x,-y)\). Because of the assumption that \(\varphi(x_1)=\varphi(x_2)\), we must have that

\begin{center}
\begin{tikzpicture}[>=angle 90]
\matrix(a)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.25ex]
{T_1&T_2\\
\PP^1\times\PP^1&\PP^1\times\PP^1\\};
\path[->] (a-1-1) edge node[above]{$\cong$}(a-1-2);
\path[->] (a-1-1) edge node[left]{$\pi_1$}(a-2-1);
\path[->] (a-2-1) edge node[above]{$\cong$}(a-2-2);
\path[->] (a-1-2) edge node[right]{$\pi_2$}(a-2-2);
\end{tikzpicture}
\end{center}

where the bottom isomorphism commutes with \(\iota\) and the top map commutes with \(\widetilde{\iota}\). Let \(\tau_i\) be the deck transformation of the cover \(\pi_i\), so that we have the two Enriques involutions \(\sigma_i=\tau_i\circ\widetilde{\iota}\). Then we have an isomorphism between \(S_1=T_1/\sigma_1\cong T_2/\sigma_2\), which implies that the period points \(x_1,x_2\) are equal.

The morphism \(\Omega_-/\Gamma\rightarrow\Omega_{4,\mathrm{h}}/\Gamma_{4,\mathrm{h}}\) extends to a morphism of the Baily--Borel compactifications by \cite[Theorem~2]{KK72}, and sends boundary components to boundary components.

Next, we describe the cusp correspondence. Recall, \(\overline{E}_2^{\mathrm{bb}}\) has five \(0\)-cusps \(p_1,\ldots,p_5\) corresponding to the following isotropic vectors in \[
L_-=U\oplus U(2)\oplus E_8(2)=\langle e,f\rangle\oplus\langle e',f'\rangle\oplus\langle\alpha_1,\ldots,\alpha_8\rangle.
\]

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(\delta_1=e\);
\item
  \(\delta_2=e'\);
\item
  \(\delta_3=e'+f'+\overline{\alpha}_8\);
\item
  \(\delta_4=2e'+f'+\overline{\alpha}_1\);
\item
  \(\delta_5=2e+2f+\overline{\alpha}_1\).
\end{enumerate}

Note that \(e'\cdot f'=2\) and \(\overline{\alpha}_i\cdot\alpha_j=\delta_{ij}\). We have that \(\delta_1^\perp/\delta_1\cong U(2)\oplus E_8(2)\) and \(\delta_i^\perp/\delta_i\cong U\oplus E_8(2)\) for \(i=2,\ldots,5\).

On the other hand, \(\overline{F}_{4,\mathrm{h}}^{\mathrm{bb}}\) has two \(0\)-cusps \(q_1,q_2\) for which the corresponding isotropic vectors \(\eta_1,\eta_2\in\Lambda_{18}\) satisfy \(\eta_1^\perp/\eta_1\cong U\oplus E_8^{\oplus2}\) and \(\eta_2^\perp/\eta_2\cong U(2)\oplus E_8^{\oplus2}\).

To understand whether \(p_1\mapsto q_1\) or \(p_1\mapsto q_2\), it is enough to compute \[
\delta_1^{\perp\Lambda_{18}}/\delta_1.
\] But this is clear after realizing that \(\Lambda_{18}\cong U\oplus U(2)\oplus E_8^{\oplus2}\), and there is the explicit embedding \begin{align*} L_-=U\oplus U(2)\oplus E_8(2)\subseteq U\oplus U(2)\oplus E_8^{\oplus2}\\ (u,v,w)\mapsto(u,v,w,w). \end{align*} So that it is clear that \[
\delta_1^{\perp\Lambda_{18}}/\delta_1=U(2)\oplus E_8(2).
\]

\end{proof}

\begin{lemma}

\begin{quote}
Cusp correspondence 1
\end{quote}

We have a cusp correspondence from polarized \(\left\{{X_i, p_i}\right\}\) in \(\partial \Mpol\) to hyperelliptic \(\left\{{Z_i, r_i}\right\}\) in \(\partial \Mhe\): \hfill\newline

\input{tikz/cusp_correspondence_two}

\end{lemma}

\section{The Baily-Borel compactification}\label{the-baily-borel-compactification}

\begin{remark}

The Baily-Borel and toroidal compactifications are defined for quotients of Hermitian symmetric spaces by actions of arithmetic subgroups of their automorphism groups, i.e.~those that can be written as \(\dcosetl{\Gamma}{\Omega}\). BB compactifications are generally small, e.g.~\(\dim F_2 = 19\) but \(\operatorname{codim}\partial \bbcpt{F_2} = 18\), and this often precludes having a satisfactory modular interpretation of its boundary points. In particular, given an arc in this compactification with endpoint in the boundary, one can not generally construct a birationally unique limit. Toroidal compactifications \(\torcpt{\dcosetl{\Gamma}{\Omega}}\) are obtained as certain blowups of \(\bbcpt{\dcosetl{\Gamma}{\Omega}}\), and e.g.~for \(F_2\) some boundary components become divisors (codimension 1). However these are highly non-unique and depend on choices of fans. One might hope there are canonical such choices. The semitoric compactifications of Looijenga interpolate between \(\bbcpt{\dcosetl{\Gamma}{\Omega}}\) and \(\torcpt{\dcosetl{\Gamma}{\Omega}}\).

\end{remark}

\begin{definition}

The group \(G\) acts transitively on the set of boundary components \(F \subseteq \partial \cD_L \coloneqq\tilde\cD_L\setminus \cD_L\), and \({\operatorname{Stab}}_G(F)\leq G\) is a maximal parabolic subgroup. Taking stabilizers establishes a bijection \begin{align*} \left\{{\text{Boundary components $F\subseteq \partial \cD_L$}}\right\} &\to \left\{{\text{Maximal parabolic subgroups $P\leq G$}}\right\} \\ F &\mapsto P_F \coloneqq{\operatorname{Stab}}_G(F) \end{align*}

For \(G \coloneqq{\operatorname{SO}}(V)\), parabolic subgroups \(P\) are stabilizers of flags of isotropic subspaces in \(V\), and since \(\operatorname{sig}(V) = (2, n)\), a flag has length at most 3 and a maximal flag is of the form \(p \subseteq I\subseteq J\) where \(p\) is a point, \(I\) is an isotropic line, and \(J\) is an isotropic plane. The only flags that define maximal parabolic subgroups of \({\operatorname{SO}}(V)\) are of length 1, consisting of either a single line or a single plane. Thus we have bijections \begin{align*} \left\{{\text{Rational boundary components of } ({\operatorname{SO}}_{2, n}({\mathbf{R}}), {\operatorname{SO}}_2({\mathbf{R}}) \times {\operatorname{SO}}_n({\mathbf{R}})) }\right\} \\ \reflectbox{\rotatebox[origin=c]{90}{$\rightleftharpoons$}}\hspace{10em} \\ \left\{{\text{Maximal parabolic subgroups of } {\operatorname{SO}}(V) }\right\} \\ \reflectbox{\rotatebox[origin=c]{90}{$\rightleftharpoons$}}\hspace{10em} \\ \left\{{\text{Isotropic lines $I\subset V$ }}\right\} \cup \left\{{\text{Isotropic planes $J \subset V$ }}\right\} \end{align*} where a boundary component \(F\) is rational if \({\operatorname{Stab}}_G(F)\) is defined over \({\mathbf{Q}}\). For an arithmetic subgroup \(\Gamma \leq G\), letting \(\partial(\cD_L)_{\mathbf{Q}}\) be the set of all rational boundary components of \(\cD_L \subset \tilde \cD_L\), we produce a compactification \[
\overline{ \dcosetl{\Gamma}{\cD_L} }
= 
\dcosetl{\Gamma}{\cD_L} 
\bigcup_{F\in \partial(\cD_{L})_{\mathbf{Q}}} 
\dcosetl{(G_F({\mathbf{Q}}) \cap\Gamma)}{F}
\]

\end{definition}

\begin{definition}

Let \(L\) be a lattice of signature \((2, n)\) for \(n\geq 1\), let \(\Omega_L\) be the associated period domain, let \({\operatorname{O}}^+(L) \leq {\operatorname{O}}(L)\) be the subgroup preserving \(\Omega_L\) and let \(\tilde \Omega_L\) be the affine cone over \(\Omega_L\). Let \(n\geq 3\), let \(k\in {\mathbf{Z}}\), and let \(\Gamma \leq {\operatorname{O}}^+(L)\) be a finite-index subgroup with \(\chi: \Gamma\to {\mathbf{C}}^*\) a character. A holomorphic functional \(f: \Omega_L \to {\mathbf{C}}\) is called a \textbf{modular form of weight $k$ and character $\chi$ for $\Gamma$} if

\begin{itemize}
\tightlist
\item
  Factor of automorphy: \(f(\lambda z) = \lambda^{-k} f(z)\) for any \(\lambda \in {\mathbf{C}}^*\).
\item
  Equivariance: \(f(\gamma z) = \chi(\gamma) f(z)\) for all \(\gamma\in \Gamma\).
\end{itemize}

\end{definition}

\begin{definition}

Let \(\Omega_L\) as above and let \(M_k(\Gamma, \chi)\) be the \({\mathbf{C}}{\hbox{-}}\)vector space of such modular forms of weight \(k\) for \(\Gamma\) with character \(\chi\). The \textbf{Baily-Borel compactification} can be defined as \[
\bbcpt{\dcosetl{\Gamma}{\Omega_L}}
\coloneqq
\mathop{\mathrm{Proj}}\bigoplus_{k\geq 1} M_k(\Gamma, \chi_{\mathrm{triv}})
\] where \(\chi_{\mathrm{triv}}\) is the trivial character.

\end{definition}

\begin{remark}

\(\partial \bbcpt{\dcosetl{\Gamma}{\Omega_L}}\) decomposes into points \(p_i\) and curves \(C_j\), which are in bijection with \(\Gamma{\hbox{-}}\)orbits of isotropic lines \(i\) and isotropic planes \(j\) in \(L_{\mathbf{Q}}\). Moreover \(p_i \in \overline{C_j} \iff\) one can choose representatives lines \(i\) and planes \(j\) such that \(i \subseteq j\).

\end{remark}

\begin{remark}

A theorem of Baily-Borel gives the existence of an ample automorphic line bundle \(\cL\) on \(\overline{\cD_L}\) giving it the structure of a normal projective variety isomorphic to a canonical model \(\mathop{\mathrm{Proj}}\bigoplus_{k\geq 0} H^0(L^k)??\).\dzg{I don't quite remember what this graded ring is.} We denote this compactification \(\bbcpt{\dcosetl{\Gamma}{\cD_L}}\).

\end{remark}

\subsection{Toroidal and semitoroidal compactifications}\label{toroidal-and-semitoroidal-compactifications}

\begin{definition}

A \textbf{toroidal compactification} \(\torcpt{\dcosetl{\Gamma }{\cD_L}}\) is a certain blowup of \(\bbcpt{\dcosetl{\Gamma}{\cD_L}}\), so there is a birational map \(\torcpt{\dcosetl{\Gamma}{\cD_L}} \dashrightarrow\bbcpt{\dcosetl{\Gamma}{\cD_L}}\). It is defined by a collection of admissible fans \(\left\{{F_i}\right\}_{i\in I}\) where \(I\) ranges over an index set for all cusps.

\end{definition}

\begin{definition}

A \textbf{semitoroidal compactification} is a generalization due to Looijenga for which the cones of \(F_i\) are not required to be finitely generated.

\end{definition}

\begin{remark}

\cite{AE21} shows that semitoroidal compactifications are characterized as exactly the normal compactifications \(\semitorcpt{M}\) fitting into a tower

\[\begin{tikzcd}
{\torcpt{M}} \\
\\
{\semitorcpt{M} } \\
\\
{\bbcpt{M} }
\arrow[hook, from=5-1, to=3-1]
\arrow[hook, from=3-1, to=1-1]
\end{tikzcd}\]

where \(\torcpt{M}\) is some toroidal compactification of \(M\).

\begin{quote}
Promising stuff here: https://dept.math.lsa.umich.edu/\textasciitilde idolga/EnriquesOne.pdf\#page=567\&zoom=160,-136,765
\end{quote}

\end{remark}

\begin{remark}

On the toroidal compactification associated with \(\dcosetl{\Gamma}{\Omega_L}\): for a cusp \(C_i\) of the BB compactification, let \(F\subset L_{\mathbf{Q}}\) be the corresponding \(\Gamma{\hbox{-}}\)orbit of an isotropic line or plane. Consider its stabilizer \(S(F) \coloneqq{\operatorname{Stab}}_{{\operatorname{O}}^+(L_{\mathbf{R}})}(F)\), and its unipotent radical \(U(F)\). Then \(U(F)\) is a vector space containing a lattice \(U(F) \cap\Gamma\) and an open convex cone \(C(F)\). Let \(\rc{C(F)}\) be the rational closure of the cone, so the union of \(C(F)\) and rational rays in its closure. We then choose a fan \(\Sigma(F)\) with \(\mathop{\mathrm{supp}}(\Sigma(F)) = \rc{C(F)}\) which is invariant under \(S(F) \cap\Gamma\) and produce an associated toric variety \(X_{\Sigma(F)}\). If one does this for every \(F\) to produce a \(\Gamma{\hbox{-}}\)admissible collection of polyhedra \(\Sigma\), their quotients by \(\Gamma\) glue to give a toroidal compactification \(\torcpt{\dcosetl{\Gamma}{\Omega_L}}\), which has the structure of a (complex) algebraic space. There is a surjection \(\torcpt{\dcosetl{\Gamma}{\Omega_L}} \twoheadrightarrow\bbcpt{\dcosetl{\Gamma}{\Omega_L}}\). Why \(e^\perp/e\) shows up: if \(e\) is an isotropic line in \(L\) corresponding to a cusp \(F\), there is an isomorphism of lattices \(U(F) \cap\tilde {\operatorname{O}}^+(L) \cong e^\perp/e\) where \(\tilde {\operatorname{O}}^+ \coloneqq\ker\qty{{\operatorname{O}}^+(L) \to {\operatorname{O}}(A_{L})}\).

\end{remark}

\subsection{Misc}\label{misc}

\begin{definition}[Log CY pairs]

A \textbf{log Calabi-Yau (CY) pair} is a pair \((X, D)\) with \(X\) a proper variety and \(D\) an effective \({\mathbf{Q}}{\hbox{-}}\)Cartier divisor such that the pair is log canonical and \(K_X + D \sim_{\mathbf{Q}}0\).

\end{definition}

\begin{definition}

A degeneration \(\pi: {\mathcal{X}}\to \Delta\) is a \textbf{CY degeneration} if \(\pi\) is proper, \(K_{{\mathcal{X}}}\sim_{\mathbf{Q}}0\), and \(({\mathcal{X}}, {\mathcal{X}}_0)\) is dlt. This implies that \({\mathcal{X}}_t\) is a Calabi-Yau variety for \(t\neq 0\) and \({\mathcal{X}}_0\) is a union of log CY pairs \((V_i, D_i)\). If \({\mathcal{X}}_t\) is a strict CY of dimension \(n\), so \(\pi_1 {\mathcal{X}}_t = 0\) and \(h^i({\mathcal{X}}_t, {\mathcal{O}}_{{\mathcal{X}}_t}) = 0\) for \(1 \leq i \leq n-1\), and \(\dim \Gamma({\mathcal{X}}_0) = n\), we say \({\mathcal{X}}\) is a large complex structure limit or equivalently a maximal unipotent or MUM degeneration.

\end{definition}

\begin{remark}

If \(n = 2\), Kulikov shows that \(\Gamma({\mathcal{X}}_0)\) is always isomorphic to a 2-sphere \(S^2\). Whether \(\Gamma({\mathcal{X}}_0) \cong S^n\) or a quotient thereof for \(n\geq 3\) more generally is an open question, posed by Kontsevich-Soibelman. It has recently been shown by Kollár-Xu that in the case of degenerations of Calabi-Yau or hyperkähler manifolds, the dual complex is always a rational homology sphere.

\end{remark}

\begin{remark}

Why this is useful to us: one formulation of mirror symmetry is the formulation due to Strominger-Yau-Zaslow, aptly called SYZ mirror symmetry. Conjecturally, the general fiber \({\mathcal{X}}_t\) of a punctured family of CYs \({\mathcal{X}}^\circ \to \Delta^\circ\) can be given the structure of a special Lagrangian torus fibration \({\mathcal{X}}_t \to B\), one can ``dualize'' the fibration to obtain a mirror CY \(\widehat{{\mathcal{X}}_t} \to B\) over the same base.\dzg{This might have something to do with the discrete Legendre transform Phil mentions.} The common base \(B\) of these two fibrations is conjecturally of the form \(\Gamma({\mathcal{X}}_0)\), the dual complex of a degeneration \({\mathcal{X}}\to \Delta\) extending \({\mathcal{X}}^\circ\).

\end{remark}

\subsection{Baily-Borel cusps and incidence diagrams}\label{baily-borel-cusps-and-incidence-diagrams}

\subsection{Other compactifications}\label{other-compactifications}

\begin{remark}

\begin{quote}
See https://arxiv.org/pdf/2010.06922.pdf\#page=1\&zoom=auto,-17,32
\end{quote}

Write \(F_{2d} \coloneqq\dcosetl{\Gamma_{2d}}{D_{2d}}\). A cusp \(p_i\) of \(\bbcpt{F_{2d}}\) determines a cone \(C_i\). Toroidal and semitoroidal compactifications are then determined by a collection of \(\Gamma_{2d}{\hbox{-}}\)invariant fans supported on \(C_i\) for \(i\) ranging over an index set for all cusps. If \(d=1\) (or more generally if \(2d\) is squarefree), there is a single 0-cusp \(p_{2d}\) whose cone \(C_{2d}\) has a description as the positive light cone in the rational closure of \(C_{2d}\) with respect to a certain lattice \(M_{2d}\). This can be written \(C^{\mathrm rc }_{2d} \coloneqq\operatorname{Conv}(\overline{C}_{2d} \cap M_{2d}\otimes_{\mathbf{Z}}{\mathbf{R}})\). A semitoroidal compactification of \(F_{2d}\) is then determined by a semitoric fan in \(M_{2d, {\mathbf{R}}}\) supported on \(C_{2d}^{\mathrm rc}\) which is invariant for a particular subgroup \(\Gamma^+_{2d} \leq {\operatorname{O}}(M_{2d})\). \dzg{Todo: can spell out what $M_{2d}, \Gamma, \Gamma^+$ are.} In this case, one can make a canonical choice for such a semitoric fan: the Coxeter fan \(\Sigma_{2d}^{\mathrm{Cox} }\) whose cones are precisely the fundamental domains for a Weyl group action on \(C_{2d}^{\mathrm rc}\), see AET19.

\end{remark}

\section{Scattone's Baily Borel compactifications}\label{scattones-baily-borel-compactifications}

\subsection{\texorpdfstring{Degree \(2d\) compactifications}{Degree 2d compactifications}}\label{degree-2d-compactifications}

\begin{remark}

The main reference for this section is \cite{scattone1987on-the-compactification-of-moduli}, which describes the Baily-Borel compactifications of \(F_{2d}\). The main result of this work is to describe \(\partial\bbcpt{F_{2d}}\) using lattice-theoretic techniques, giving partial cusp diagrams for certain arithmetically constrained values of \(d\). In particular, it shows that the number of 1-cusps is asymptotic to \(d^8\), and the number of 0-cusps is 1 when \(d\) is squarefree, and otherwise is given by the function ???. Complete details are given for the cases \(d=1,2\).

\end{remark}

\begin{remark}

One first notes that by the global Torelli theorem for algebraic K3 surfaces \cite{PSS71}, \(F_{2d}\) admits a coarse space of the form \(D_{L_d}/\Gamma_{L_d}\) for a certain choice of lattice \(L_d\). Note that \(D_{L_d}\) is a 19-dimensional bounded symmetric domain of type IV and \(\Gamma_{L_d}\) is an arithmetic group acting upon it. Recall that the theory of automorphic forms realizes \(\bbcpt{D_{L_d}/\Gamma_{L_d}}\) as a projective variety. If \(L_d\) is the primitive cohomology of a polarized K3 surface, there is a correspondence between \(n\)-dimensional boundary cusps and \(\Gamma_{L_d}\)-orbits of \(n+1\)-dimensional isotropic subspaces in \(L_d\). The boundary of the Baily-Borel compactification is ``small'' in the sense that it has very high codimension, and thus the geometric information it contains is insufficient to reconstruct a birationally unique family from a family over the punctured disc.

\end{remark}

\begin{remark}

A polarization of degree \(2d\) on a K3 surface is a primitive divisor \(H\) with \(H^2 = 2d > 0\) which is pseudoample, i.e.~\(HD\geq 0\) for any effective \(D\in \operatorname{Div}(X)\). A primitively polarized K3 surface is a pair \((X, H)\). Choose a marking \(\phi: H^2(X; {\mathbf{Z}})\to \lkt\) such that \(\phi(H) = h\) where \(h\) is a fixed primitive vector satisfying \(h^2=2d\). So \({\operatorname{O}}(\lkt)\) acts transitively on the set of primitive vectors of a fixed square, such an isometry can always be found and the choice of \(h\) is irrelevant. Note that \((H, \omega_X) = 0\), and thus the period of \((X, H)\) lies in \(\Omega_{2d} \coloneqq\Omega_S\) for \(S \coloneqq h^{\perp \lkt}\). Set \(\Gamma_{2d} \coloneqq{\operatorname{Stab}}_{{\operatorname{O}}(\lkt)}(h)\); then \(\Gamma_{2d}\curvearrowright\Omega_{2d}\) discontinuously and \(\Omega_{2d}/\Gamma_{2d}\) is a normal complex analytic space which serves as a coarse space for \(F_{2d}\) by \cite{PSS71,friedman1984a-new-proof}. Because any two choices of \(h\) are equivalent modulo \({\operatorname{O}}(\lkt)\), the isomorphism class of \(h^{\perp \lkt}\) depends only on \(d\). Making an appropriate choice of \(h\), one can identify \[
L_{2d} \coloneqq h^{\perp \lkt} \cong \left\langle{-2d}\right\rangle \oplus U^{\oplus 2} \oplus E_8^{\oplus 2}
.\] One can now define the period domain as \(\Omega_{2d} \coloneqq\Omega_{L_{2d}}\). This consists of two connected components interchanged by conjugation, so \(\Omega_{2d} = D_{2d} \cup\tilde D_{2d}\), and we fix once and for all a choice of one component which we will denote \(D_{2d}\). It is well known that \[
D_{2d} \cong {{\operatorname{SO}}^0_{2, 19} \over {\operatorname{SO}}_2 \times {\operatorname{SO}}_{19} }
.\] We set \(\tilde {\operatorname{O}}(L_{2d})\) to be the image of \(\Gamma_{2d}\) under the injection \(\Gamma_{2d}\hookrightarrow{\operatorname{O}}(L_{2d})\) induced by restriction -- note that this coincides with the general definition \(\tilde {\operatorname{O}}(L) \coloneqq\ker({\operatorname{O}}(L) \to {\operatorname{O}}(q_L))\). We set \({\operatorname{O}}_-(L_{2d})\) to be the index 2 subgroup that preserves the component \(D_{2d}\), and \(\Gamma_{2d} \coloneqq\tilde {\operatorname{O}}(L_{2d}) \cap{\operatorname{O}}_-(L_{2d})\), we obtain identifications \[
F_{2d} \cong \Omega_{2d}/\tilde{\operatorname{O}}(L_{2d}) = D_{2d}/\Gamma_{2d}
.\] We focus our attention on the latter definition, \(F_{2d} \coloneqq D_{2d}/\Gamma_{2d}\), and more generally on compactifications of general \(D/\Gamma\).

\end{remark}

\subsection{General theory}\label{general-theory}

\begin{remark}

Let \(D\) be a symmetric bounded domain and \(\Gamma \leq \mathop{\mathrm{Aut}}(D)\) a discrete arithmetic subgroup of automorphisms. Equivalently, we can write \(D = G({\mathbf{R}})/K\) for \(G\) a connected linear algebraic group defined over \({\mathbf{Q}}\) and \(K\) a maximal compact subgroup of \(G({\mathbf{R}})\). We then require that \(\Gamma \leq G\) is arithmetic, i.e.~\(\Gamma \subseteq G({\mathbf{Q}})\) and is commensurable with \(G({\mathbf{Z}})\). We will generally define \(\overline{D/\Gamma}\) as \(D^*/\Gamma\) where \(D \subseteq D^* \subseteq D {}^{ \vee }\) is a subset of the compact dual via the Borel embedding, comprised of \(D\) and rational boundary components.

\end{remark}

\begin{remark}

Regarding \(D \subseteq D {}^{ \vee }\), we have \(\partial D = {\textstyle\coprod}F_i\) where each \(F_i\) is a boundary component, i.e.~a maximal connected complex analytic set. We set \[
N_F \coloneqq\left\{{g\in G({\mathbf{R}}) {~\mathrel{\Big\vert}~}gF = F}\right\} \coloneqq{\operatorname{Stab}}_{G({\mathbf{R}})}(F)
\] to be the stabilizer of a boundary component and note that the maximal parabolic subgroups of \(G({\mathbf{R}})\) are precisely those of the form \(N_F\). A boundary component is rational when \(N_F({\mathbf{C}})\) is defined over \({\mathbf{Q}}\). Let \(B(D)\) be the set of proper rational boundary components of \(D\). Then there is a bijection \begin{align*} B(D) &\rightleftharpoons\left\{{\text{Proper maximal parabolic ${\mathbf{Q}}$-subgroups of } G({\mathbf{C}})}\right\} \\ F &\rightleftharpoons N_F({\mathbf{C}}) \end{align*}

We can write \[
D^* = D\cup\amalg_{F\in B(D)} F
.\] Then \(\overline{D/\Gamma} \coloneqq D^*/\Gamma\) can be written as \[
D/\Gamma \cup\coprod_{[F]\in B(D)/\Gamma} V_F
\] where \(V_F\) are varieties and we index over orbits of rational boundary components modulo \(\Gamma\).

\end{remark}

\begin{remark}

We can identify the \(V_F\) explicitly: write \(G_F \coloneqq{\operatorname{Stab}}_{G({\mathbf{R}})}(F)/ \mathrm{Fix}_{G({\mathbf{R}})}(F)\) and \(N_{\Gamma, F} \coloneqq{\operatorname{Stab}}_\Gamma(F)/ \mathrm{Fix}_\Gamma(F)\), then \(V_F = F/N_\Gamma(F)\). Note that in applications to K3 surfaces, we have \(G({\mathbf{R}}) = {\operatorname{SO}}^0_{2, 19}\). In this situation, we have a correspondence \begin{align*} \partial D_L &\rightleftharpoons{\operatorname{OGr}}(L_{\mathbf{R}}) \\ F &\rightleftharpoons E \end{align*} where \(E\) corresponds to \(F\) iff \({\operatorname{Stab}}_{{\operatorname{O}}(L_{\mathbf{R}})}(E) = {\operatorname{Stab}}_{G({\mathbf{R}})}(F) \coloneqq N_F\). Restricting to rational boundary components corresponds to \({\operatorname{OGr}}(L_{\mathbf{Q}})\), which are further identified with \({\operatorname{OGr}}(L)\), the primitive isotropic sublattices of \(L\). For any subgroup \(\Gamma_L \leq G({\mathbf{Z}})\) we obtain a bijection \[
B(D_L)/\Gamma_L \rightleftharpoons{\operatorname{OGr}}(L)/\Gamma_L
\] which preserves incidence relations.

\end{remark}

\begin{example}

As an example, one can take the symplectic form on \({\mathbf{Z}}^{2g}\), which yields \(G({\mathbf{R}}) = {\mathbf{P}}{\mathsf{Sp}}_g({\mathbf{R}})\) and \(D\) is the Siegel upper half space \(\cH^g\). Let \(\Gamma = G({\mathbf{Z}}) = {\mathbf{P}}{\mathsf{Sp}}_g({\mathbf{Z}})\) be the full Siegel modular group. Then maximal parabolic \({\mathbf{Q}}\)-subgroups of \(G_{\mathbf{R}}\) correspond to stabilizers of rational isotropic subspaces of \(L_{\mathbf{R}}\). Let \(H^n \coloneqq\cH^n/{\mathsf{Sp}}_n({\mathbf{Z}})\), then \(\overline{D/\Gamma} = H_g \cup H_{g-1}\cup\cdots \cup H_1\cup H_0\) coincides with the Satake compactification.

Consider now the example of \(F_{2d}\). Recall that \(D\) is a component of \(\Omega \coloneqq\left\{{z\in {\mathbf{P}}L_{\mathbf{C}}{~\mathrel{\Big\vert}~}z^2 = 0,\, z\overline{z} > 0}\right\}\) and \(D {}^{ \vee }= \left\{{z\in {\mathbf{P}}L_{\mathbf{C}}{~\mathrel{\Big\vert}~}z^2 = 0}\right\}\) is a quadric. Stratify \(\partial D = \partial_1 D {\textstyle\coprod}\partial_0 D\), noting that \(\partial_0 D = D {}^{ \vee }\cap{\mathbf{P}}L_{\mathbf{R}}\). Then all points of \(\partial_0 D\) are of the form \({\mathbf{P}}(\left\langle{v}\right\rangle_{\mathbf{C}})\) where \(v\in L_{\mathbf{R}}, v^2 = 0\) is isotropic. All components in \(\partial_1 D\) are of the form \({\mathbf{P}}(\left\langle{v, w}\right\rangle_{\mathbf{C}}) \cap\partial_1 D\) where \(\left\langle{v, w}\right\rangle_{\mathbf{R}}\) varies in \({\operatorname{OGr}}_2(L_{\mathbf{R}})\). Restricting to rational components, one considers \({\operatorname{OGr}}_2(L)\) instead, i.e.~subspaces \(E_{\mathbf{R}}\) arising from sublattices \(E \leq L\). We thus obtain bijections \begin{align*} B_0(D) &\rightleftharpoons{\operatorname{OGr}}_1(L) \\ {\mathbf{P}}\left\langle{v}\right\rangle_{\mathbf{C}}&\rightleftharpoons\left\langle{v}\right\rangle_{\mathbf{Z}} \end{align*} and \begin{align*} B_1(D) &\rightleftharpoons{\operatorname{OGr}}_2(L) \\ {\mathbf{P}}\left\langle{v, w}\right\rangle_{\mathbf{C}}\cap\partial_1 D &\rightleftharpoons\left\langle{v, w}\right\rangle_{\mathbf{Z}} \end{align*}

One can then write \begin{align*} \overline{D/\Gamma} =  D/\Gamma \cup \coprod_{[\left\langle{v}\right\rangle_{\mathbf{Z}}] \in {\operatorname{OGr}}_1(L)/\Gamma } p_v \cup \coprod_{[\left\langle{v, w}\right\rangle_{\mathbf{Z}}] \in {\operatorname{OGr}}_2(L)/\Gamma } {{\mathbf{P}}\left\langle{v, w}\right\rangle_{\mathbf{C}}\cap\partial_1 D \over N_\Gamma(\left\langle{v, w}\right\rangle_{\mathbf{Z}})} \end{align*} where \(N_\Gamma(E) = {\operatorname{Stab}}_\Gamma(E) /  \mathrm{Fix}_\Gamma(E)\) can be identified with the image of \({\operatorname{Stab}}_\Gamma(E)\) in \({\operatorname{SL}}(E) \cong {\operatorname{SL}}_2({\mathbf{Z}})\).

\end{example}

\section{\texorpdfstring{Moduli of polarized K3 surfaces of degree \(2d\)}{Moduli of polarized K3 surfaces of degree 2d}}\label{moduli-of-polarized-k3-surfaces-of-degree-2d}

\begin{remark}[K3 surfaces]

A \textbf{K3 surface} is a smooth projective surface with trivial canonical bundle \(\omega_X \cong {\mathcal{O}}_X\) and \(h^1({\mathcal{O}}_X) = 0\). Prototypical examples include double branched covers of sextic curves in \({\mathbf{P}}^2\) and smooth quartic hypersurfaces in \({\mathbf{P}}^3\). All K3 surfaces are diffeomorphic, and thus have the Hodge diamond shown in \cref{fig:hodge-k3-surface}.

\begin{figure}[H]
\centering
\begin{tikzcd}[column sep=small,row sep=small]
    && {h^{2,2}} &&&&&& 1 \\
    & {h^{2,1}} && {h^{1,2}} &&&& 0 && 0 \\
    {h^{2,0}} && {h^{1,1}} && {h^{0,2}} & {=} & 1 && 20 && 1 \\
    & {h^{1,0}} && {h^{0,1}} &&&& 0 && 0 \\
    && {h^{0,0}} &&&&&& 1
\end{tikzcd}
\caption{The Hodge diamond of a K3 surface.}
\label{fig:hodge-k3-surface}
\end{figure}

\end{remark}

\begin{remark}[Lattice theory in moduli]

The cup product endows the singular cohomology \(H^2(X, {\mathbf{Z}})\) with the structure of a lattice, where by a \textbf{lattice} we mean a finitely generated free \({\mathbf{Z}}\)-module with a nondegenerate \({\mathbf{Z}}\)-valued symmetric bilinear form. It is isometric to the K3 lattice: \[
H^2(X; {\mathbf{Z}}) \cong \lkt \coloneqq U\oplus U\oplus U \oplus E_8 \oplus E_8
,\] where \(U\) is the hyperbolic lattice, the unique even unimodular lattice of rank 2 with Gram matrix \(\bigl( \begin{smallmatrix}0 & 1\\ 1 & 0\end{smallmatrix}\bigr)\) and \(E_8\) is the negative-definite lattice associated to the \(E_8\) Dynkin diagram.

By the weak Torelli theorem for K3 surfaces \cite[Cor.\, 8.1.1.2]{BHPV04}, the moduli theory of K3 surfaces is regulated by this lattice structure. Of particular importance is the \textbf{Néron-Severi} lattice \({\operatorname{NS}}(X) \coloneqq H^{1,1}(X) \cap H^2(X, {\mathbf{Z}})\) of integral \((1,1)\) forms, and its orthogonal complement in \(H^2(X, {\mathbf{Z}})\), the \textbf{transcendental lattice}. We refer to these as \(S_X\) and \(T_X\) respectively. By the Lefcshetz \((1,1)\) theorem \cite[Thm.\, 4.2.13]{BHPV04}, the first Chern class \(c_1: \operatorname{Pic}(X) \to {\operatorname{NS}}(X)\) induces an isometry. Because the naive construction of a coarse moduli space of projective K3 surfaces yields a non-Hausdorff space, we restrict our attention to \textbf{polarized K3 surfaces of degree \(2d\)} -- pairs \((X, L)\) where \(X\) is a K3 surface and \(L\) is an ample line bundle on \(X\) satisfying \(L^2 = 2d > 0\).

\end{remark}

\begin{remark}[Lattice polarized K3 surfaces]

Let \(S\) be a non-degenerate lattice of signature \((1, n)\) which admits a primitive embedding into \(\lkt\) (which is of signature \((3, 19)\)) and \(T \coloneqq S^{\perp \lkt}\) which is of signature \((2, 19-n)\). We then define the period domain associated to \(S\) as a connected component \(D_S\) of \[
\Omega_S \coloneqq\left\{{ [\sigma]\in {\mathbf{P}}\qty{T\otimes{\mathbf{C}}} {~\mathrel{\Big\vert}~}\sigma^2 = 0, \sigma\overline{\sigma }> 0}\right\}
,\] yielding a Hermitian symmetric domain of Type IV. Let \(\Gamma_S \coloneqq\tilde {\operatorname{O}}(T)\) where \[
\tilde {\operatorname{O}}(T) \coloneqq\ker({\operatorname{O}}(T)\to {\operatorname{O}}(A_T))
\] and \(A_T \coloneqq T {}^{ \vee }/T\) is the discriminant group of \(T\). It can be shown that \[
F_S \coloneqq\dcosetl{\Gamma_S}{D_S}
\] is a coarse moduli space of \(S\)-polarized K3 surfaces. Taking \(S\coloneqq\left\langle{h}\right\rangle\) the sublattice generated by an ample class \(h\) satisfying \(h^2 = 2d\) recovers \(F_{2d} \coloneqq F_{\left\langle{h}\right\rangle}\), noting that \(\left\langle{h}\right\rangle^{\perp \lkt} \cong \left\langle{-2d}\right\rangle\oplus U^{\oplus 2}\oplus E_8^{\oplus 2}\). By \cite{piateski-shapiro1971torelli}, \(F_{2d}\) constructed in this way is a coarse moduli space of degree \(2d\) primitively polarized K3 surfaces.

To such surfaces, by \cite{BB66} there is a canonically defined quasiprojective compactification \(\dcosetl{\Gamma_S}{D_S} \hookrightarrow\bbcpt{ \dcosetl{\Gamma_S}{D_S} }\) whose boundary consists of 0-cusps (points) and 1-cusps (curves). The 0-cusps are in bijection with \(\Gamma_S\)-orbits of isotropic lines \(I \subseteq T\), and the 1-cusps are in bijection with orbits of isotropic planes \(J \subseteq T\). By \cite{AMRT75}, there exists a class of compactifications \(\dcosetl{\Gamma_S}{D_S} \hookrightarrow\overline{\dcosetl{\Gamma_S}{D_S}}^{{\mathcal{F}}}\) defined by the combinatorial data of a fan \({\mathcal{F}}\coloneqq\left\{{{\mathcal{F}}_I}\right\}\) where \(I\) ranges over \(\Gamma\)-orbits of isotropic lines in \(T\), equivalently as \(I\) ranges over the 0-cusps of \(\partial \bbcpt{\dcosetl{\Gamma_S}{D_S}}\). These are referred to as toroidal compactifications. Noting that \(\operatorname{sig}(I^{\perp T}/I) = (1, 18)\), we obtain a hyperbolic lattice and can construct a model of hyperbolic space \({\mathbb{H}}^{18}\) as a projectivization of the positive cone \[
C^+ \coloneqq\left\{{v\in I^\perp/I\otimes{\mathbf{R}}{~\mathrel{\Big\vert}~}v^2 > 0}\right\}
.\] Letting \(\Gamma_{S, I} \coloneqq{\operatorname{Stab}}_{\Gamma_S}(I)\), the data of \({\mathcal{F}}_I\) is specified by a \(\Gamma_{S, I}\)-invariant rational polyhedral tiling of \({\mathbb{H}}^{18}\). The combinatorics of such a tiling determines a union of toric varieties which are adjoined as the boundary strata of \(\dcosetl{\Gamma_S}{D_S}\) at the 0-cusp corresponding to \(I\). This produces a divisorial boundary with mild singularities. The work of \cite{Loo86} introduces semitoroidal compactifications, allowing for the tiling to be \emph{locally} rationally polyhedral, which simultaneously generalizes the Baily-Borel and toroidal compactifications described above.

\end{remark}

\begin{remark}

A modular alternative to these compactifications was introduced in \cite{kollar1988threefolds-and-deformations}, denoted the space of stable slc pairs, which is proper. A pair \((X, R)\) of a projective variety \(X\) with a \({\mathbf{Q}}\)-divisor \(D\) is stable if \(K_X + R\) is ample and \({\mathbf{Q}}\)-Cartier and \((X, R)\) has slc singularities. These strictly generalize the stable curves that appear in \(\overline{{ \mathcal{M}_{g, n} }}\), and naturally generalize these notions to all dimensions. For an appropriately universal choice of polarizing divisor \(R\) on the generic K3 surface in \(F_{2d}\), one can define a compactification \(\overline{F_{2d}}^{R}\) as the closure of the space of pairs \((X, \varepsilon R)\) in a Zariski open subset of \(F_{2d}\) in the space of stable slc pairs. For example, consider a degree 2 polarized K3 surface \((X, L)\). The linear system \({\left\lvert {L} \right\rvert}\) induces a branched 2-to-1 cover \(X\to {\mathbf{P}}^2\), and one can choose \(R\) to be the ramification divisor of the covering involution. By ???, when \(R\) is a \emph{recognizable divisor}, there is a unique semifan \({\mathcal{F}}_R\) such that the normalization of \(\overline{F_{2d}}^R\) is isomorphic to the semitoroidal compactification \(\overline{F_{2d}}^{{\mathcal{F}}_R}\).

\end{remark}

\begin{remark}[Kulikov models]

Any degeneration of K3 surfaces is birational to a \textbf{Kulikov model}: after a birational modification and a ramified base change, it may be put in the form of a degeneration \(\pi: {\mathcal{X}}\to \mathbf{D}\) over the complex disc such that \(\pi\) is semistable with trivial canonical \(\omega_{{\mathcal{X}}} \cong {\mathcal{O}}_{{\mathcal{X}}}\).

By ?, the central fiber of a type III Kulikov model is encoded in an integral affine 2-sphere, abbreviated \(\mathrm{IAS}^2\).

\end{remark}

\section{Boundary}\label{boundary}

Goal for this section: describe how Coxeter-Vinberg diagrams are used to get models at BB cusps.

\subsection{The case of abelian varieties}\label{the-case-of-abelian-varieties}

Let us consider the setup first for moduli of principally polarized abelian varieties. We have the following:

\begin{theorem}

There is an isomorphism \[
\eta: \torcptf{{\mathcal{A}}_g}  { \, \xrightarrow{\sim}\, }(\ksbacpt{{\mathcal{A}}_g})^\nu
\] for \(F\) the second Voronoi fan. \dzg{Is there always a natural morphism from toric compactifications to KSBA? Not in general! We can talk about it when we meet.}

\end{theorem}

\begin{corollary}

As a result, any punctured 1-parameter family \({\mathcal{X}}^\circ \to \Delta^\circ\) has a unique limit \({\mathcal{X}}_0\) which can combinatorially be described as a tropically polarized abelian variety \((X_{\mathrm{trop}}, \Theta_{\mathrm{trop}})\) with a tropical \(\Theta\) divisor \(\Theta_{\mathrm{trop}}\).

More is true: the fan \(F\) is itself a moduli space for such tropical abelian varieties.

\end{corollary}

\begin{remark}[Motivation from abelian varieties]

To see how this works, consider a 1-parameter family of abelian varieties. These are tori of the form \[
{\mathcal{X}}_t \coloneqq\operatorname{coker}(\phi_t: {\mathbf{Z}}^g\hookrightarrow({\mathbf{C}}^*)^g)
,\] where \(\phi_t\) are embeddings that vary in the family. Write this embedding as a matrix \(M\); this is a matrix of periods. Then for \(t\approx 0\) one exponentiates \(M_{ij}\) to get a symmetric positive-definite \(g\times g\) matrix \(B\). There is a cone \(C \subseteq \left\{{B = B^t > 0}\right\}\) in \(\operatorname{GL}_g({\mathbf{Z}})\) and a Coxeter fan \(F\) supported on its rational closure \(\rc{C}\) which corresponds to affine Dynkin diagram \(\tilde A_2\): \[
\tilde A_2: \quad \dynkin[extended, edge length=1cm]A2
\] This corresponds to a triangular fundamental domain for a reflection group that acts on \(C\). For a cartoon picture, think of a hyperbolic disc \(\mathbf{D}\) and let the fundamental domain be a triangle with ideal vertices:

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/tesselate_disc.jpg}
\caption{Tessellating a hyperbolic disc by triangles}
\label{fig:tesselateone}
\end{figure}

Note that the straight lines forming the edges should ``really'' be curved hyperbolic geodesics. One continues reflecting in order to tessellate the hyperbolic disc, then puts this disc in \({\mathbf{R}}^3\) at height one and cones it to the origin to get an infinite-type fan \(F\):

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/tesselate_disc2.jpg}
\caption{Coning off the hyperbolic disc to form a fan.}
\label{fig:tesselatetwo}
\end{figure}

Note that the entire fan \(F\) admits an \({\operatorname{SL}}_2({\mathbf{Z}})\) action. Now if one rewrites \(B\) as a form \(B(x,y) = ax^2 + by^2 + c(x+y)^2\) with \(a,b,c\in {\mathbf{Z}}_{\geq 0}\), the coordinate vector \(\vec v \coloneqq(a,b,c)\) defines a point in some chamber of \(\rc{C}\). In turn, \(\vec v\) defines a 1-parameter degeneration of abelian varieties, and thus a pair \((X_{\mathrm{trop}}, \Theta_{\mathrm{trop}})\). When \(B\) is integral it defines an embedding \(B:\Lambda\hookrightarrow\Lambda {}^{ \vee }\)\dzg{What is $\Lambda$?} and thus one can construct a torus \(T \coloneqq\Lambda {}^{ \vee }_{\mathbf{R}}/\Lambda \cong {\mathbf{R}}^2/{\mathbf{Z}}^2\) which has finitely many integral points defined by \(\Lambda\).

Recall that for any lattice \(L\) there is an associated Voronoi tessellation by polytopes \(P_i\), one such \(P_i\) centered around each lattice point \(\ell_i\). Let \(\mathrm{Vor}_B\) be the Voronoi tessellation of \(\Lambda\); this can be identified with a hexagonal honeycomb tessellation of \({\mathbf{R}}^2\):

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/Vor_tess.jpg}
\caption{The Voronoi tesselation associated to $\tilde A_2$, the triangular lattice.}
\label{fig:tesselatethree}
\end{figure}

One then defines \(\Theta_{\mathrm{trop}} \coloneqq B(\mathrm{Vor}_B)/\Lambda\), a quotient of the image of the hexagonal tessellation. Although the blue vertices in \(\mathrm{Vor}_B\) generally have vertices with fractional coordinates, the vertices in the image have integral coordinate vertices with respect to \(\Lambda {}^{ \vee }\). The image of a regular hexagon is now a hexagon with side lengths \(a,b,c\), and since we've quotiented by \(\Lambda\), \(\Theta_{\mathrm{trop}}\) is determined by two hexagons with side lengths determined by \(\vec v = (a,b,c)\) which are glued together:

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/honeycomb_sidelengths.jpg}
\caption{A picture of two relevant polytopes in the image of the Voronoi tessellation, which tessellates the entire dual lattice. After quotienting, these will be the only two relevant polygons.}
\label{fig:honeycomb}
\end{figure}

The claim is that this picture describes an entire degeneration \({\mathcal{X}}\) of abelian varieties. To see the central fiber: every vertex \(w_i\) in this new tessellation defines an honest fan via \(\operatorname{Star}(w_i)\); here there are 2 vertices of valence 3 and 3 edges in the quotient, so the central fiber \({\mathcal{X}}_0\) is two copies of \({\mathbf{P}}^2\) corresponding to \(w_1, w_2\) glued together along three curves corresponding to \(a,b,c\). To see the entire family: put this entire picture at height 1, cone to the origin to get a fan, and quotient that fan by a \({\mathbf{Z}}^2\) action to get \({\mathcal{X}}\).

Note that in the K3 case, things are harder because the combinatorics only describes \({\mathcal{X}}_0\) and not the entire family \({\mathcal{X}}\), so one has to appeal to abstract smoothing results to obtain the existence of a family \({\mathcal{X}}\) extending \({\mathcal{X}}_0\).

Moreover, the original fan \(F\) is a moduli of these polyhedral pictures. One can degenerate \(\Theta_{\mathrm{trop}}\) by sending some coordinates \(a,b,c\) to zero. This degenerates the honeycomb 6-gons into 4-gons if just one side goes to zero. For example, if \(a\to 0\), this corresponds to being on a wall in \Cref{fig:tesselatetwo}. If two coordinates degenerate, say \(a,b\to 0\), this corresponds to being on a ray. This can be read off by recalling \(B(x,y) = ax^2 + by^2 + c(x+y)^2\) and labeling the ideal vertices with monomials \(x^2, y^2, (x+y)^2\) as in \Cref{fig:tesselateone}. Thus varying \(\vec v = (a,b,c)\) corresponds to varying the side lengths of hexagons and correspondingly moving through \(\rc{C}\). Staying in the fundamental chamber doesn't change the overall combinatorial type of \Cref{fig:honeycomb}, but passing through a wall will flip the hexagonal tiling in various ways.

\end{remark}

\subsection{To the K3 case}\label{to-the-k3-case}

The claim is that a similar story more or less goes through for K3s: the Coxeter diagram is much more complicated, and the relevant combinatorial device is an \(\mathrm{IAS}^2\) with 24 singularities instead of a tropical variety. We have the following:

\begin{theorem}

There is a morphism \[
\eta: \torcptf{F_2} \to (\ksbacpt{F_2})^\nu
\] where \(F\) is fan of a Coxeter diagram associated to a cusp of \(F_2\), and the Stein factorization of \(\eta\) is through a semitoroidal compactification.

\end{theorem}

\begin{corollary}

Any punctured 1-parameter family \({\mathcal{X}}^\circ\to \Delta^\circ\) has a unique limit \({\mathcal{X}}_0\) which can be combinatorially described as a singular integral-affine sphere with an integral-affine divisor \((\mathrm{IAS}^2, R_{\mathrm{IAS}})\).

\end{corollary}

\begin{remark}

This is a much harder theorem than the \({\mathcal{A}}_g\) case: periods of K3 surfaces are highly transcendental, and the period map is not well-understood. Also note that the relevant Coxeter diagram for \({\mathcal{A}}_g\) was relatively simple, while the diagram for \(F_2\) is the following:

\begin{figure}[H]
\centering
\input{tikz/Coxeter_19_1_1}
\caption{The Coxeter diagram for type $(19, 1, 1)$.}
\label{fig:coxeter1911}
\end{figure}

Nodes in \Cref{fig:coxeter1911} correspond to roots spanning a hyperbolic lattice \[
N\coloneqq U \oplus E_8(-1) { {}^{ \oplus{2} }  } \oplus A_1(-1),\qquad \operatorname{sig}(N) = (1, 18) 
\] which is the Picard lattice of the Dolgachev-Nikulin mirror K3. Decorated nodes \(v_i\) record self-intersection numbers \(v_i^2\), and edges between \(v_i\) and \(v_j\) record the intersection numbers \(v_i.v_j\). Note that the Coxeter diagram also captures the data of all \((-2)\) curves on the mirror K3 surface and their intersections.

This diagram again describes the fundamental chamber of a reflection group, and the cone in this case \(C = \left\{{v^2 > 0}\right\}\). Toroidal compactifications of \(F_2\) correspond to fans whose support is \(\rc{C}\) (i.e.~the interior, plus rational rays on the boundary). There is a natural fundamental chamber defined by \(\left\{{v \mathrel{\Big|}v.r_i \geq 0}\right\}\) where \(\left\{{r_i}\right\}\) are roots, the difference is that now some vertices of the fundamental chamber may be ideal vertices:

\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{figures/F2HyperbolicReflection.jpg}
\caption{A fundamental chamber $F$ for a reflection group. Reflecting over walls of $F$ successively generates a tiling of the hyperbolic disc by copies of $F$. Note that one vertex is an ideal vertex, i.e. it is in $\partial{\overline{{\mathbb{H}}^n}}$.}
\label{fig:hyperbreflection}
\end{figure}

Proceeding similarly to take the cone over this picture and allow rational boundary points yields the cone \(\rc{C}\) and a corresponding infinite-type fan \({\mathcal{F}}\) -- this is a fan since the faces are rationally generated, \(F\) is a fundamental chamber for the reflection group \(W(N)\), the fan is \(W{\hbox{-}}\)invariant by construction and moreover invariant under \({\operatorname{O}}(N)\). Since there is a short exact sequence \[0\to W(N) \to {\operatorname{O}}(N)\to S_3\to 0\] the index of \(W(N)\) is finite and thus \(F\) is finite volume.

Points in this fan can naturally be interpreted as period points, so a choice of a point in the fan yields a degenerating family of K3 surfaces by the Torelli theorem. Let \(v\in F\) be a point in the fundamental chamber, we will next consider how this corresponds to a combinatorial object, the same way \(\vec v = (a,b,c)\) did in the case of \({\mathcal{A}}_g\).

First consider a fan with 18 rays, corresponding to a toric surface \(\Sigma\) with 18 curves. Note that the rays alternative between long and short vectors:

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/toric18.jpg}
\caption{The starting point: a toric surface with 18 rays.}
\label{fig:toric18rrays}
\end{figure}

This corresponds to a polytope \(P_\Sigma\) which is an 18-gon (not necessarily regular) which is the moment polytope for \(X_\Sigma\) where \(\Sigma\) is the fan in \Cref{fig:toric18rrays} and has edge lengths \(\ell_0, \cdots, \ell_{17}\in {\mathbf{R}}\), which determines a polarization \(L\) for \(X_\Sigma\). Although not shown in the picture here, we can call each edge ``long'' if it was dual to a long vector, and similarly ``short'' if dual to a short vector.

Note also that each edge can be written as \(\ell_i v_i\) for \(v_i\) some unit vectors, and it is a nontrivial condition on \(\vec \ell\) that this polygon closes. In particular, one needs \(\sum_{i=0}^{17} \ell_i v_i = 0\).

Now cut triangles out of sides \(0, 6, 12\) and call the resulting polygon non-convex polygon \(P\). Each triangle cut corresponds to a non-toric blowup of \(X_\Sigma\), i.e.~a blowup at a point \(p\) which is not \(T{\hbox{-}}\)invariant. This introduces three new length parameters \(\ell_{18}, \ell_{20}, \ell_{20}\) corresponding to the heights of these three triangles. Each will introduce an \(\rm{I}_1\) singularity to the moment polytope.

\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{figures/toric18polytope.jpg}
\includegraphics[width=0.4\textwidth]{figures/toric18polytope2.jpg}
\caption{The Symington polytope: an 18-gon, before and after a nontoric blowup corresponding to cutting out triangles.}
\label{fig:symblowup}
\end{figure}

Regarding such polytopes as the Symington polytopes, which are bases of Lagrangian torus fibrations, these are in particular elliptic fibrations and these singularities precisely correspond to introducing singular type \(\rm{I}_1\) fibers in Kodaira's classification.

Take two copies of \(P\), say \(P\) and \(P^{\mathrm op}\), and glue them together along the outer edges and call the result \(B\). This is topolologically the gluing of two discs, and thus \(B\) is homeomorphic to \(S^2\). Each gluing along the outer edges introduces a new \(\rm{I}_1\) singularity, yielding \(3+3 = 6\) singularities in the hemispheres and \(18\) singularities along the equator for a total of \(24\) singularities of type \(\rm{I}_1\) and thus an \(\mathrm{IAS}^2\) with charge \(24\).

Note that there are now 24 length parameters: \(\ell_0,\cdots, \ell_{17}\) along the equator, \(\ell_{18}, \ell_{19}, \ell_{20}\) in the northern hemisphere, and \(\ell_{21}, \ell_{22}, \ell_{23}\) in the southern hemisphere. The tuple \(\vec \ell = (\ell_1, \cdots, \ell_{23})\) turns out to correspond to 24 vectors in a 19 dimensional space, and there are enough conditions to ensure the polygons actually close.

This produces the tropical sphere \(\mathrm{IAS}^2\), so one also needs to describe its tropical divisor \(R_{\mathrm{IAS}}\). The above construction works for any K3 with a nonsymplectic involution, e.g.~an elliptic K3, and the \(\mathrm{IAS}^2\) is naturally equipped with an involution \(\iota\) that swaps \(P\) and \(P^\opp\). The ramification divisor of \(\iota\) is the equator, highlighted in blue in the following cartoon picture of \(B\), and one takes \(R_{\mathrm{IAS}}\) to be the sum of the blue edges with coefficient 2 for even (short?) sides and coefficient 1 for odd (long?) sides:

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/IASPuPop.jpg}
\caption{Caption}
\label{fig:iaspupop}
\end{figure}

We now describe how one obtains a degeneration \({\mathcal{X}}\) of K3 surfaces from this combinatorial picture. One must first extend this \(\mathrm{IAS}^2\) to a complete triangulation by basis triangles. This triangulation should be done on \(P\) first, before the doubling construction, so that the vertices and edges in the northern hemisphere are perfectly matched with those in the southern. Here is a cartoon of what this might look like on one copy of \(P\), before gluing:

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/IAScompletetriang.jpg}
\caption{The $\mathrm{IAS}$ on $P$ extended to a complete triangulation by basis triangles.}
\label{fig:IAS_complete_triangulation}
\end{figure}

This is again a cartoon picture, meant to show how vertices and triangles in the hemispheres should match in pairs exchanged by the involution \(\iota\). Here e.g.~the blue triangles are meant to match, as well as \(\operatorname{Star}(\tilde w_1)\) and \(\operatorname{Star}(\tilde w_1^\opp)\):

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/IASdoubletriang.jpg}
\caption{A completely triangulated $\mathrm{IAS}^2$ defined by $B \coloneqq P\cup P^\opp$.}
\label{fig:iasdoubletriang}
\end{figure}

This final picture describes the central fiber \({\mathcal{X}}_0\) of a Kulikov degeneration of K3 surfaces in the following way: there are many non-singular vertices \(p_i\), and exactly 24 singular vertices \(w_i\) and \(\tilde w_i, \tilde w_i^\opp\). For the \(p_i\), there is a fan defined by \(\operatorname{Star}(p_i)\) which defines a toric surface \(V_i\). For the 24 singular vertices, there is a modified recipe to cook up a semi-toric surface -- since the singularity is type \(\rm{I}_1\), this will be a charge 1 surface, and thus realizable as a toric surface with a single non-toric blowup, a \textit{semitoric} surface. How to make this blowup is uniquely determined by an additional omitted decoration called the \textit{monodromy ray} at the singular vertex. Roughly, this is a preferred ray cooked up from the Picard-Lefschetz monodromy operator around the singular vertex. One can think of this as a ``singular fan''.

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{figures/IASmonodromy.jpg}
\caption{$\operatorname{Star}(\tilde w_2)$ in \Cref{fig:IAS_complete_triangulation} with the extra data of a monodromy vector.}
\label{fig:iascompletetriangulation}
\end{figure}

So \[ {\mathcal{X}}_0 = \bigcup_i V_i \cup\bigcup_{j=1}^{24} W_j\] where the \(V_i\) are all toric surfaces and the \(W_j\) are all semitoric surfaces of charge 1, and the triangulation determines how they are all glued together. To see how this gluing is done, consider the following local picture in the triangulation:

\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{figures/IASgluing.jpg}
\caption{Local gluing in the $\mathrm{IAS}^2$ of two toric surfaces $\Sigma_7$ and ${\mathbf{P}}^2$}
\label{fig:iasgluing}
\end{figure}

At the orange vertex, taking the star we see three rays and thus a copy of \({\mathbf{P}}^2\). At the green vertex, we see 7 rays, and thus some toric surface \(\Sigma_7\) which is probably something like a Hirzebruch surface \({ \mathbf{F} }_n\) with 3 toric blowups. Since the orange and green vertices are adjacent by exactly one edge, this means we glue \({\mathbf{P}}^2\) to \(\Sigma_7\) along the the curves determined by rays pointing along that edge. Moreover, whenever there is a triangle, this corresponds to three surfaces glued together along a triple point.

The general case is that the \(\mathrm{IAS}^2\) has 24 copies of \(\rm{I}_1\) singularities; these singularities can collide to produce semitoric surfaces with multiple nontoric blowups.

\end{remark}

\begin{remark}

Note that this \textit{only} describes \({\mathcal{X}}_0\) and not an entire family \({\mathcal{X}}\). Friedman solved this problem: there is a technical condition called \textbf{$d{\hbox{-}}$semistability}, and if this is satisfied then \({\mathcal{X}}_0\) is smoothable. Moreover the smoothing will have the correct period and/or monodromy vector \(\lambda\).\dzg{Haven't discussed $\lambda$ here yet!}

To obtain all degenerations, one considers all of the ways this combinatorial object can degenerate. Sending some \(\ell_i\to 0\) causes the 18-gon to collapse into a small polygon, or causes some hemispherical singularities to descend into the equator. This corresponds to moving an interior point of original fundamental chamber \(F\) onto a wall, and wall-crossing mutates the \(\mathrm{IAS}^2\) in some other ways.

\end{remark}

\begin{remark}

Some miscellaneous remarks:

\begin{itemize}
\tightlist
\item
  The Kulikov models are highly non-unique, differing by flops. Adding the divisor \(R_{\mathrm{IAS}}\) fixes this and pins down \({\mathcal{X}}\) uniquely.
\item
  It seems one can read off the stable model from the \(\mathrm{IAS}^2\). For the honeycombs in the \({\mathcal{A}}_g\) case, everything was contracted down to two \({\mathbf{P}}^2\)s glued along their 3 boundary curves in a \(\Theta{\hbox{-}}\)graph. In the \(F_2\) case, one contracts everything in the \(\mathrm{IAS}^2\) except for the equator, i.e.~the interiors of the hemispheres are contracted. The most general degeneration is 18 copies of \({\mathbf{P}}^2\) glued in a cycle; one can then send some \(\ell_i\to 0\) to collide the vertices and get fewer than 18 surfaces.
\item
  It seems one can also read off Type II degenerations from the \(\mathrm{IAS}^2\). Here there are 4 Type II cusps, 3 correspond to collapsing the 18-gon in the \(\mathrm{IAS}^2\) in the equatorial plane to an interval. The 4th involves collapsing the 18-gon to a point with bits sticking out. Type I degenerations correspond to collapsing everything to a point.
\item
  Why everything works simply here: there is only one relevant cusp in \(F_2\), and the involution propagates to everything including the \(\mathrm{IAS}^2\). The Coxeter diagram is also highly symmetric, hinting at how to make the right toric and \(\mathrm{IAS}\) construction.
\end{itemize}

\end{remark}

\subsection{Notes from Phil's talk}\label{notes-from-phils-talk}

\begin{remark}

For \(\Sigma_g\) a compact complex curve of genus \(g\), choose a symplectic basis \(\left\{{\alpha_i, \beta_i}\right\}_{i\leq g}\) of \(H_1(\Sigma_G; {\mathbf{Z}})\), then there is a unique basis \((\omega_1,\cdots, \omega_g)\) of \(H^0(\Omega_{\Sigma_g})\) such that \(\int_{\alpha_i} \omega_j = \delta_{ij}\). In this basis, form the \textbf{period matrix} \(\tau = (\int_{\beta_i} \omega_j )_{i,j=1}^g\). This satisfies \(\tau^t = \tau\) and \(\Im(\tau) > 0\) is positive-definite, and is thus an element in The \textbf{Siegel upper half-space} \[
\cH_g \coloneqq\left\{{\tau \in \operatorname{Sym}_{g\times g}({\mathbf{C}}) \mathrel{\Big|}\Im(\tau) > 0}\right\}
.\] The \textbf{Jacobian} of \(\Sigma\) is defined as \(\mathrm{Jac}(\Sigma) \coloneqq{\mathbf{C}}^g / ({\mathbf{Z}}^g\oplus \tau {\mathbf{Z}}^g)\). Note that we made a choice of ``marking'' by choosing the symplectic basis \(\left\{{\alpha_i, \beta_i}\right\}\), and any two such choices are related by \({ \begin{bmatrix} {A} & {B}  \\ {C} & {D} \end{bmatrix} } \in {\mathsf{Sp}}_{2g}({\mathbf{Z}})\), the isometry group of \({\mathbf{Z}}^{2g}\) with the standard symplectic form, where the action is \({ \begin{bmatrix} {A} & {B}  \\ {C} & {D} \end{bmatrix} } { \begin{bmatrix} {I_g} \\ {\tau } \end{bmatrix} }= { \begin{bmatrix} {A+B\tau} \\ {C+D\tau} \end{bmatrix} }\), the analogue of a linear fractional transformation. To renormalize the 1-forms, we change basis to get a similar matrix \({ \begin{bmatrix} {I_g} \\ {(A+B\tau)^{-1}(C+D\tau)} \end{bmatrix} }\). Thus to get an invariant, we consider \[
[\tau]\in \dcosetl{{\mathsf{Sp}}_{2g}({\mathbf{Z}})}{\cH_g} \coloneqq{\mathcal{A}}_g
,\] the moduli space of PPAVs. Here one can realize the polarization on \(A\) as a symplectic form on \(H_1(A;{\mathbf{Z}})\) which is represented by a holomorphic line bundle \(L\in \operatorname{Pic}(A)\), i.e.~identifying a symplectic form on \(H_1(A;{\mathbf{Z}})\) as an element of \(H^2(A;{\mathbf{Z}})\) which we want to be a \((1, 1)\) form.

\end{remark}

\begin{remark}

Now \({\mathcal{A}}_g\) is not compact, so we consider degenerations over \({\mathcal{X}}^\circ\to \Delta^\circ\) and let \(\Delta^\circ\to {\mathcal{A}}_g\) be the associated period mapping -- how does the period map degenerate as \(t\to 0\)? The answer is that a certain isotropic subspace \(I \leq ({\mathbf{Z}}^{2g}, \omega_\text{std})\) becomes distinguished by the fact that periods against \(I^\perp\) remain finite.

\end{remark}

\begin{example}

Let \(y^2 = x^3 + x^2 + t\) be a family of elliptic curves over \({\mathbf{A}}^1\setminus{0}\). At \(t=0\) this degenerates to a nodal cubic. There is a vanishing cycle \(\alpha\), and the distinguished isotropic subspace is precisely \(I\coloneqq{\mathbf{Z}}\alpha\). One shows \(I^\perp = {\mathbf{Z}}\alpha\) as well. In this case, to be in \(I^\perp\) means to be a curve that does not pass through the thinning neck of the torus that degenerates; any curve that does pass through should intuitively have a period that blows up. We normalize by picking a \(c_t\) such that \(\int_\alpha c_t {dx\over y} = 1\), then \(\int_\beta \omega_t = \tau_t \in {\mathbf{C}}\). However, this isn't well-defined: one can parallel-transport \(\beta\) around \(t=0\) and the monodromy action will be a Dehn twist, so integrals against \(\beta\) are only well-defined up to \({\mathbf{Z}}p\) where \(p\) are periods against \(\alpha\), here \(p\) is normalized to 1. So \(\int_\beta \omega_t\in \tau_h + {\mathbf{Z}}\in {\mathbf{C}}/{\mathbf{Z}}\). As \(t\to 0\), one was \(\tau_t \to +i\infty\) if \(\alpha, \beta\) are oriented properly. We can fix this ambiguity by exponentiation, getting a well-defined invariant \(\exp(2\pi i \int_\beta \omega_t) \in {\mathbf{C}}^*\).

\end{example}

\begin{remark}

How this works for \(g\geq 1\): assume \(I\) is Lagrangian, so \(I^\perp = I\), corresponding to a maximally unipotent degeneration. If this were a genus \(g\) curve, we could pinch \(\leq g\) disjoint cycles simultaneously, and a maximal degeneration will pinch exactly \(g\). Since \({\mathsf{Sp}}_{2g}({\mathbf{Z}})\) acts transitively on Lagrangian subspaces in \(\mathrm{LGr}(V)\), we can assume \(I = \bigoplus_i {\mathbf{Z}}\alpha_i\) is generated by the \(\alpha\) curves. Generalizing the \({\mathbf{C}}^*\) embedding in the previous case, we obtain a torus embedding \begin{align*} E: \cH_g &\hookrightarrow({\mathbf{C}}^*)^{g\choose 2} \\ \tau &\mapsto  \left[ \begin{matrix}     \exp(2\pi i \tau_{11}) & \cdots & \\     \vdots & \ddots & \vdots \\         & \cdots & \exp(2\pi i \tau_{gg} ) \end{matrix} \right] \end{align*} Since \(\tau\) is symmetric, the image \(E(\tau)\) is again symmetric. Note that the \(\beta_i\) cycles are well-defined up to translation in \(I\), but because the 1-form was normalized so that integrals of \(\alpha_j\) along \(\beta_i\) were 1 or 0, the entries in this matrix are well-defined up to integers. Thus we can exponentiate every entry in the period matrix to get a well-defined symmetric matrix. The unipotent orbit theorem of Schmid gives an asymptotic estimate \[
E(\tau_t) \sim_{t\to 0}
\left[
\begin{matrix}
    c_{11} t^{n_{11}} & c_{12} t^{n_{12}} & \cdots & \\ 
    \vdots & \ddots & & \vdots \\
        & \cdots & &c_{gg} t^{n_{gg}}
\end{matrix}
\right] \in \operatorname{Mat}_{n\times n}({\mathbf{C}}^*)
\] which is a cocharacter of \(({\mathbf{C}}^*)^{g\choose 2}\), i.e.~an inclusion \({\mathbf{C}}^* \hookrightarrow({\mathbf{C}}^*)^{g\choose 2}\) which is a composition of a group morphism and a translation. Here the \(c_{ij}\) are the translation parts, and if \(c_{ij} = 1\) for all \(i,j\) this yields an honest group morphism. Such a cocharacter is called a unipotent orbit. This asymptotic estimate is quantified, so there is a precise speed at which the period matrix approaches the cocharacter.

Setting \(N \coloneqq(n_{ij})\), we have \(N\in \operatorname{Sym}_{g\times g}({\mathbf{Z}})\) and \(N > 0\). These entries capture the relative speeds at which the various cycles are collapsing. Since the \(c_{ij}\) are ultimately just translations, we'll omit them from here onward.

\end{remark}

\begin{remark}

Define a cone \[
P_g \coloneqq\left\{{N\in \operatorname{Sym}_{g\times g}({\mathbf{Z}}) \mathrel{\Big|}N > 0}\right\} \subseteq {\mathbf{Z}}^{g\choose 2}
\] and consider the family \[
({\mathbf{C}}^*)^g \over  \left\langle  
(t^{n_{11}}, t^{n_{12}}, \cdots, t^{n_{1g}}), 
(t^{n_{21}}, t^{n_{22}}, \cdots, t^{n_{2g}}),
\cdots ,
    (t^{n_{g1}}, t^{n_{g2}}, \cdots, t^{n_{gg}})
 \right\rangle 
\] over \(\Delta^\circ\). How does one extend this family over \(t=0\)? If \(N\) has full rank, this entire expression is isomorphic to \(({\mathbf{C}}^g/{\mathbf{Z}}^g)/\tau{\mathbf{Z}}^g\). There are two answers, one by fans and one by polytopes.

\end{remark}

\begin{remark}

The following is the fan construction due to Mumford, which most easily generalizes to K3 surfaces.

Consider the example \[
N = { \begin{bmatrix} {2} & {1}  \\ {1} & {3} \end{bmatrix} } \leadsto 
{({\mathbf{C}}^*)^2 \over 
 \left\langle 
(t^2, t),
(t, t^3)
 \right\rangle }
\]

Note \(N > 0\), since \(\operatorname{det}N > 0, \mathrm{Trace} N > 0\), and \(N({\mathbf{Z}}^g)\) is generated by the vector \((2,1)\) and \((1, 3)\). First quotient \({\mathbf{R}}^2\) by this lattice to get a flat real 2-torus, then take a polyhedral tiling whose vertices are integer points. Here we take a tiling of the fundamental domain and translate it everywhere. This gives a tiling \({\mathcal{F}}_0\) on the universal cover \({\mathbf{R}}^g\). Now put this picture at height 1 in \({\mathbf{R}}^{g+1}\) to get a tiling \(\tilde {\mathcal{F}}_0\) of \({\mathbf{R}}^g\times\left\{{1}\right\}\subseteq {\mathbf{R}}^{g+1}\), and let \(\tilde {\mathcal{F}}\coloneqq{ \mathrm{Cone} }(\tilde {\mathcal{F}}_0) \subset {\mathbf{R}}^{g+1}\) be its cone. Taking the toric variety \(X(\tilde {\mathcal{F}})\), and define \(X({\mathcal{F}}) \coloneqq X(\tilde {\mathcal{F}})/N({\mathbf{Z}}^g)\), where the quotient makes sense precisely because \(N({\mathbf{Z}}^g)\) acts on \({\mathbf{R}}^g\times \left\{{1}\right\}\) by translation, and this extends to a linear action on \({\mathbf{R}}^g\), which moreover preserves \(\tilde {\mathcal{F}}\) and thus acts on the toric variety. There is a morphism \(\phi: X({\mathcal{F}})\to {\mathbf{A}}^1\) induced by the morphism of fans given by the height function: projection in \({\mathbf{R}}^{g+1}\) onto the last coordinate, whose image is in \({\mathbf{R}}_{\geq 0}\). This map descends to the quotient since the linear action preserves the height function.

This produces a degenerating fan of abelian varieties. A fiber \(\phi^{-1}(t)\) of \(X(\tilde F)\) for \(t\neq 0\) yields \(({\mathbf{C}}^*)^g\), and the action \(N({\mathbf{Z}}^g)\) acts by translations of the form \((t^{n_{i1}}, t^{n_{i2}}, \cdots, t^{n_{ig}})\) in the original family. Thus we recover the original family as an infinite quotient of a toric variety. But the toric variety has a toric boundary, encoded in the tiling. The fiber \(\phi^{-1}(0)\) has dual complex \(\Gamma(X_0({\mathcal{F}})) = {\mathcal{F}}_0\) equal to the original tiling, and \(X_0({\mathcal{F}})\) is a union of toric varieties.

In the original lattice, in the quotient there are precisely 3 0-cells, and we interpret the star of each 0-cell as the fan of a toric surface. They are glued according to the tiling.

\end{remark}

\begin{remark}

The polytope construction, which builds the projective coordinate ring instead. One defines \(Q\) to be the hull of certain points, constructs a theta function, and takes Proj of a certain graded algebra generated by such functions with an explicit multiplication rule and structure constants. These define a certain PL function with a ``bending locus'' which gives a polyhedral decomposition of \({\mathbf{R}}^g/{\mathbf{Z}}^g\). For any \(N\in P_g\) one can define the Delaunay decomposition \(\mathrm{Del}(N)\), and the central fiber \({\mathcal{X}}_0\) of the family will have \textbf{intersection} complex \(\mathrm{Del}(N)\) -- the loci where the PL function is linear will be polytopes which are the cells of the Delaunay decomposition. The second Voronoi fan \(F^\mathrm{Vor}\) is a decomposition of \(P_g\) into loci where \(\mathrm{Del}(N)\) is constant. One then takes \(\dcosetl{ \operatorname{Sym}_{g\times g}({\mathbf{Z}}) }{\cH_g} \hookrightarrow({\mathbf{C}}^*)^{g\choose 2} \to X(F^\mathrm{Vor})\). One the quotients by conjugation in \(\operatorname{GL}_g({\mathbf{Z}})\) to get \(X(F^\mathrm{Vor}) /\operatorname{GL}_g({\mathbf{Z}}) \hookrightarrow\overline{{\mathcal{A}}_g}^{\mathrm{Vor}}\). Correspondingly, for any \({\mathcal{X}}^\circ\) in \({\mathcal{A}}_g\), tracing through this construction gives a proper family \({\mathcal{X}}\) in \(\overline{{\mathcal{A}}_g}^{\mathrm{Vor}}\) -- note that we've only described what toric compactification to take for the maximally unipotent degenerations, but one can carry out similar constructions for the other cusps of \(\bbcpt{{\mathcal{A}}_g}\).

\end{remark}

\begin{remark}

One should ask if \(\overline{{\mathcal{A}}_g}^{\mathrm{Vor}}\) actually solves a moduli problem, and the answer is yes (up to normalization) by a theorem of Alexeev. The moduli problem is the moduli of semi-abelic pairs. Define \(\overline{{\mathcal{A}}_g}^\Theta\) to be the closure of pairs \((X, {\varepsilon}R)\) where \(R\) is their theta divisors, then Alexeev shows \[
\overline{{\mathcal{A}}_g}^{\mathrm{Vor}} = (\overline{{\mathcal{A}}_g}^\Theta)^\nu
\]

\end{remark}

\begin{remark}

How do we do something similar for K3 surfaces? Fix \(v\in \lkt\) primitive with \(v^2 = 2d\) and define \begin{align*} \Omega_{L} &\coloneqq\left\{{{\mathbf{C}}x \in {\operatorname{Gr}}_1(\lkt_{\mathbf{C}}) \mathrel{\Big|}(x,x) = 0, (x, \overline{x}) > 0 }\right\} \\ \Omega_{2d} &\coloneqq v^{\perp \Omega_{\lkt} } = \left\{{x\in \lkt_{\mathbf{C}}\mathrel{\Big|}(x, v) = 0}\right\} \\ \Gamma_{2d} &\coloneqq{\operatorname{Stab}}_{{\operatorname{O}}(\lkt)}(v) = \left\{{\gamma\in {\operatorname{O}}(\lkt) \mathrel{\Big|}\gamma(v) = v}\right\} \\ F_{2d} &\coloneqq\dcosetl{ \Gamma_{2d}}{\Omega_{2d} } \end{align*}

Here \(\Omega_{2d}\) plays the role of \(\cH^g\) in the abelian variety case, and is a Hermitian symmetric domain of type IV or \({\operatorname{SO}}_{2, n}\), and \(F_{2d}\) is an arithmetic quotient. Fixing a marking \(\phi: H^2(X;{\mathbf{Z}}) \to \lkt\), the period map for a family \({\mathcal{X}}^\circ \to \Delta^\circ\) is given by taking \(H^{2, 0}(X) = {\mathbf{C}}\omega\) and looking at \([\omega] \coloneqq\phi(\omega) \in F_{2d}\), since \([\omega]\in \Omega_{2d}\) but is ambiguous up to change of marking (elements of \(\Gamma\)). This is a map \(\Delta^\circ\to F_{2d}\).

Given a degenerating family, there is a distinguished isotropic lattice \(I \leq v^\perp\) where \(\operatorname{sig}v^\perp = (2, 19)\). Note \(I\) can only have rank 1 or 2. The rank 1 case (Type III degenerations) is a maximally unipotent degeneration; the central fiber is as singular as possible, and \({\mathcal{X}}_0\) will always have 0-strata. In contrast, in the rank 2 case (Type II degenerations) there are models of the degeneration with no 0-strata.

In the rank 1/Type III case, there is a vanishing cycle \(\delta\) associated to a 0-stratum in \({\mathcal{X}}_0\) which is topologically a 2-torus. It turns out that \(\delta\) is an isotropic vector that spans the isotropic lattice, so we can write \(I = {\mathbf{Z}}\delta\subseteq v^\perp\). In the degeneration, the 2-torus collapses to a point.

In the rank 2/Type II case, there are two linearly independent isotropic vectors \(\delta\) and \(\lambda\) in \(v^\perp\) corresponding to 2-tori collapsing simultaneously not to isolated points as in the previous case, but rather to circles in \({\mathcal{X}}_0\). They are in the singular locus of \({\mathcal{X}}_0\), which is an elliptic curve.

\end{remark}

\begin{remark}

We henceforth assume \(\operatorname{rank}_{\mathbf{Z}}I = 1\) and write \(I = {\mathbf{Z}}\delta\) for \(\delta\) the isotropic vanishing cycle. Normalize \(\omega_t\) so that \(\int_\delta \omega_t = 1\) for \(t\neq 0\). Let \(\left\{{\gamma_i}\right\}_{i=1}^{19}\) be a basis of \(\delta^\perp/\delta\). Since \(\delta \in v^\perp\) was isotropic of signature \((2, 19)\), we have \(\operatorname{sig}(\delta^\perp/\delta) = (1, 18)\) and this gives us a hyperbolic lattice of rank 19. Consider the integral \(\int_{\gamma_i} \omega_t \in {\mathbf{C}}\). For this to make sense, one needs to lift the \(\delta_i\) from \(\delta^\perp/\delta\) to \(\delta^\perp\), and the choice of lift is ambiguous up to a multiple of \(\delta\). By the normalization of the integral, we get a well-defined period \[
\int_{\gamma_i} \omega_t\in {\mathbf{C}}/{\mathbf{Z}}
\] As in the PPAV case, we use the exponential to get rid of the quotient by \({\mathbf{Z}}\). Letting \(U_\delta\leq \Gamma_{2d}\) be the unipotent subgroup stabilizing \(\delta\), we get the following torus embedding \begin{align*} \dcosetl{U_\delta}{\Omega_{2d}} \xrightarrow{\psi} ({\mathbf{C}}^*)^{19} \\ {\mathbf{C}}[\omega_t] &\mapsto \qty{ \exp\qty{2\pi i \int_{\gamma_1}\omega_t}, \cdots, \exp\qty{2\pi i \int_{\gamma_{19}} \omega_t } } \end{align*} and a nilpotent orbit theory yielding an asymptotic estimate \[
\psi_t \sim (c_1 t^{\lambda _1}, \cdots, c_{19} t^{\lambda_{19}} )
\] with \(c_i\in {\mathbf{C}}^*\), so the periods are approximated by a cocharacter where the \(\lambda_i\) measure how fast the periods degenerate.

\end{remark}

\begin{remark}

Degenerations: a theorem of KPP shows that after a finite base change and birational modifications, any degeneration of K3s has a model where

\begin{itemize}
\tightlist
\item
  \(X\) is smooth
\item
  \(X_0\) is RNC
\item
  \(K_X = {\mathcal{O}}_X\)
\end{itemize}

The most famous degeneration of K3s is the Fermat degeneration is a non-example, since smoothness fails: \[
V(x_0 x_1x_2 x_3 = t(x_0 ^4 + x_1^4 + x_2^4 + x_3^4))
\] This threefold has precisely 24 conifold singular points. The central fiber at \(t=0\) is a tetrahedron, 4 planes \({\mathbf{P}}^2\) in \({\mathbf{P}}^3\), and the singular points come from intersecting each edge of the tetrahedron with the residual quartic. One can get a smooth threefold by taking a small resolution of the singular points. There are choices for the resolutions, differing by flops, so here is a heuristic of a symmetric choice where along each edge there are two resolutions extending into each component:

image

The result has four components \(V_i\) which are isomorphic to \(\operatorname{Bl}_6{\mathbf{P}}^2\), 2 points on each of 3 lines in \({\mathbf{P}}^2\).

An observation originally due to GHK: there is an IAS on \(\Gamma(X_0)\), i.e.~there are charts to \({\mathbf{R}}^2\) up to post-composition with \({\operatorname{SL}}_2({\mathbf{Z}}) \rtimes{\mathbf{R}}^2\).

Here is an example of \({\mathcal{X}}_0 = \cup V_i\) for a Kulikov degeneration (written as a decomposition into irreducible components). Each unlabeled edge has an implicit label of \(-1\):

image

This forms a tiling of the sphere. Each tile corresponds to an irreducible component \(V_i\) of \({\mathcal{X}}_0\). The edges correspond to components \(V_i, V_j\) glued along an anticanonical cycle of rational curves \(V_{ij}\). The edge numbers record the self-intersection numbers of the cycles \(V_{ij}\) regarded as a cycle in \(V_i\) and \(V_{ji} = V_{ij}\) regarded as a cycle in \(j\).

A general fact about degenerations of CYs: \({\mathcal{X}}_0\) is generally a union of log CY varieties, i.e.~there are meromorphic 2-forms on components and they are glued along their poles so that the residues agree. The red lines in the image denotes the pole locus of these forms. Each triple point is where 3 surfaces are glued. Since the overall variety is a SNC surface, there are only double curves and triple points. Note that this picture is the \textbf{intersection complex} of \({\mathcal{X}}_0\), and not the dual complex \(\Gamma({\mathcal{X}}_0)\). To obtain the dual complex, take the dual tiling, regard each integral 0-cell in the result as a fan, and glue the fans.

Here are the fans:

image

Here is how this interacts with the original diagram:

image

This works fine at most vertices, but at most 24 components are non-toric. Note that from toric geometry, if \((V, D)\) is a toric pair then \(-D_i^2 v_i = v_{i-1} + v_{i+1}\) and so one can enforce this formula on such components. For example, the following pair has all \(-1\) curves since \(v_2 = v_1 + v_3\):

image

Enforcing this formula locally, non-toric points force some \({\operatorname{SL}}_2({\mathbf{Z}})\) monodromy in the IAS.

\end{remark}

\begin{remark}

This is the analogue of the Mumford fan construction. Note that in the PPAV case, the lattice didn't specify a Kulikov degeneration since it was not a complete triangulation. But completing this to a complete triangulation of the corresponding real 2-torus does yield a Kulikov model. For K3s, instead of a complete triangulation on \(T^2\), we're taking a complete triangulation of an \(\mathrm{IAS}^2\). Note that unlike the PPAV case, a triangulated \(\mathrm{IAS}^2\) only gives \({\mathcal{X}}_0\) (glued from ACPs) and not the entire family \({\mathcal{X}}\). An abstract theorem of Friedman says it smooth to a K3, but one does not get an explicit construction of the smoothing \({\mathcal{X}}_t\).

There is also no polytope construction here whatsoever, only the fan construction for the central fiber. GHK and Siebert have been working on the polytope side. It's hard: it's not clear what the multiplication rule for theta functions should be. We represent an \(\mathrm{IAS}^2\) with the following data:

\dzg{Missing, see video.}

This recovers \({\mathcal{X}}_0\) by taking fans at vertices.

\end{remark}

\begin{remark}

Joint work with Valery: a polarizing divisor is a divisor \(R\) in the generic K3 surface in \(F_{2d}({\mathbf{C}})\). This corresponds to a choice of ample divisor on a Zariski open subset of \(F_{2d}({\mathbf{C}})\). For such a choice, we define \(\overline{F_{2d}}^R\) to be the closure of K3 pairs \((X, {\varepsilon}R)\) in the space of KSBA stable pairs. A generic K3 has Picard rank 1, and it's in the ample class, so any divisor on the generic K3 is automatically ample. Thus \(K_X + {\varepsilon}R > 0\) since \(K_X = 0\). The pair also has slc singularities. Note that we've allowed all K3s to have ADE singularities, these are examples of slc, and taking \({\varepsilon}\) small enough resolves any problems. One needs \(R\) not to pass through log canonical centers, and there are no log canonical centers on an ADE K3. Their theorem gives an explicit description of such a moduli space.

\end{remark}

\begin{remark}

We say \(R\) is recognizable if it extends to a unique divisor \(R_0\) on any Kulikov surface. Idea: for \({\mathcal{X}}_0\) there are many different smoothing families \({\mathcal{X}}_i\) and choices of divisors \(R_i\). For any 1-parameter family, taking the Zariski closure of \(R_i\) yields a flat limit \(R_{i, 0}\) on \({\mathcal{X}}_0\). If \(R\) is recognizable, these flat limits do not vary, so the choice of divisor can be made on \textit{any} K3, even a smooth K3. If \(R\) is a recognizable polarizing divisor, there is a unique semifan \(F_R\) such that \[
\overline{F_{2d}}^{F_R} = (\overline{F_{2d}}^R)^\nu
\] This relates a Hodge-theoretic compactification on the left with a geometric compactification on the right.

\end{remark}

\begin{remark}

A semitoroidal compactification simultaneously generalizes toroidal and BB compactifications.

Recall that assocaited to a degeneration of K3s we had \(\vec \lambda \coloneqq(\lambda_1, \cdots, \lambda_{19}) \in \delta^\perp/\delta\), a signature \((1, 18)\) lattice. Friedman-Scattone show that \(\vec \lambda^2\) is the number of triple points in \({\mathcal{X}}_0\). The semifan \(F_R\) is a locally polyhedral \(\Gamma_\delta \coloneqq{\operatorname{Stab}}_\Gamma(\delta)\) invariant decomposition of the positive cone \(C^+ \subset \delta^\perp/\delta\). This is the future light cone in the corresponding hyperbolic space. Roughly \(\overline{F_{2d}}^{F_R}\) is \(X(F_R)/\Gamma_\delta\).

Why this? We had a torus embedding of the first partial quotient \(\dcosetl{U_\delta}{\mathbf{D}} \to ({\mathbf{C}}^*)^{19}\) and the latter is canonically identified with \(\delta^\perp/\delta\otimes{\mathbf{C}}^*\). The monodromy invariant \(\vec \lambda\) was approximated by the cocharacter \(\lambda \otimes{\mathbf{C}}^*\). We extend that torus by a toric variety whose fan has support in \(\delta^\perp/\delta\).

Note that here semitoroidal corresponds to \textit{locally} polyhedral. A globally polyhedral tiling condition would just yield a usual fan. For instance, the cones here might have infinitely many rational polyhedral walls. On the other hand, the BB compactification corresponds to the trivial compactification of \(C^+\) which is just the entirety of \(C^+\).

\end{remark}

\begin{remark}

AE prove that recognizable divisors \(\rc{R} \coloneqq\sum_{C\in {\left\lvert {L} \right\rvert}, C^\nu \cong {\mathbf{P}}^1} C\) exist. The rational curve divisor is always recognizable for any degree \(2d\), so this exhibits some semitoroidal compactifications with geometric meaning.

AET give some explicit examples for \(F_2\). Degree 2 K3s are generically 2-to-1 covers \(\pi:X\to {\mathbf{P}}^1\) branched over a sextic, take \(L \coloneqq\pi^* {\mathcal{O}}_{{\mathbf{P}}^1}(1)\). One takes the \(R\) to be the ramification divisor \(R\in {\left\lvert {3L} \right\rvert}\); it is a recognizable divisor. They construct a semifan \(F_R\) which is a coarsening of the Coxeter fan for the root system in \(\delta^\perp/\delta\); one takes a subset of the root mirrors.

The construction of the singular K3 surface: start with the heart \(\mathrm{IAS}^2\), triangulate completely, double this construction, replace each vertex with the surface defined by the star. Note the cuts introducing shears along the boundary. The cuts introduce 3 singularities in each hemisphere, and angular defects of the polygonal gluing introduce 18 singularities along the equator. This \(\mathrm{IAS}^2\) has an involution, and this \({\mathcal{X}}_0\) naturally has an involution. The ramification divisor of the \(\mathrm{IAS}^2\) is in blue, it's a tropical ramification divisor. It is the dual complex of the limit of ramification divisors.

Why is this recognizable? \({\mathcal{X}}_0\) admits an involution \(\iota_0\). From this we can determine the limit of \(\mathrm{Fix}(\iota)\). Note that \({\mathcal{X}}_0\) alone determines \(R_0\), the limit of \(R_t\), and \(R_0 =  \mathrm{Fix}(\iota_0)\). This implies recognizability since the choice of divisor \(R\) can be made on any Kulikov surface.

\end{remark}

\begin{remark}

On joint work with ABE for elliptic K3s. Take \(X\to {\mathbf{P}}^1\) an elliptic fibration with fiber \(f\) and section \(s\). This is not of the form \(F_{2d}\), since here one takes \(H = {\mathbf{Z}}s \oplus {\mathbf{Z}}f\) for the polarization. Generically the fibration has 24 singular fibers. They show \(R \coloneqq s + m\sum f_i\) for \(f_i\) the singular fibers is recognizable for any multiple \(m\). The lattice \({\rm II}_{1, 17}\) is reflective, and \(F_R\) here refines the Coxeter chamber into 9 subchambers. This is a fan which is strictly not a semifan. There is a corresponding tropical elliptic K3 given by the following \(\mathrm{IAS}^2\).

image

Here one glues the top too the bottom, identifying the segments by a vertical shear. Note it has an \(S^1\) fibration which tropicalizes the elliptic fibration. The blue vertical lines are limits of singular fibers, the blue horizontal is the limit of the section.

\end{remark}

\section{Coxeter Theory}\label{coxeter-theory}

\begin{quote}
From https://math.ucr.edu/home/baez/twf\_dynkin.pdf\#page=1\&zoom=160,-141,556
\end{quote}

\subsection{Coxeter groups and diagrams}\label{coxeter-groups-and-diagrams}

\begin{remark}

Main ideas:

\begin{itemize}
\tightlist
\item
  Elliptic subdiagrams of rank \(r\) correspond to codimension \(r\) faces of a polytope \(P\)
\item
  Parabolic subdiagrams (of rank \(n-1\)) correspond to cusps of \(P\)
\end{itemize}

\end{remark}

\begin{remark}[A summary of hyperbolic Coxeter diagram conventions]

Regarding this as a group of reflections in hyperplanes, we have the following interpretations:

\input{snippets/coxeter-vinberg-conventions}

Note that generally

\begin{itemize}
\tightlist
\item
  \(\cos(\angle(H_i, H_j)) = -(h_i, h_j)\) when \({\left\lvert { (h_i, h_j) } \right\rvert} < 1\) and
\item
  \(\cosh(\rho(H_i, H_j)) = -(h_i, h_j)\) when \({\left\lvert { (h_i, h_j) } \right\rvert} > 1\).
\end{itemize}

Here \(\rho(H_i, H_j)\) is the length of a common perpendicular to \(H_i\) and \(H_j\).

Moreover,

\begin{itemize}
\tightlist
\item
  \(H_i \pitchfork H_j \iff {\left\lvert {w_{ij}} \right\rvert} = {\left\lvert {(h_i, h_j)} \right\rvert} < 1\),
\item
  \(H_i \parallel H_j \iff {\left\lvert {w_{ij}} \right\rvert} = {\left\lvert {(h_i, h_j)} \right\rvert} = 1\),
\item
  \(H_i \diverge H_j \iff {\left\lvert {w_{ij}} \right\rvert} = {\left\lvert {(h_i, h_j)} \right\rvert} > 1\).
\end{itemize}

For a hyperbolic Coxeter polytope \(P\) bounded by hyperplanes \(\left\{{H_1,\cdots, H_n}\right\}\), one constructs the Gram matrix \(G(P) = (g_{ij}) \in \operatorname{Mat}_{1\leq i,j\leq n}\) defined by \[
g_{ij} = 
\begin{cases}
1 & i=j \\
-\cos(\pi/m_{ij}) & H_i\pitchfork H_j, \angle(H_i, H_j) = \pi/m_{ij} \\
-\cosh(\rho(H_i, H_j)) & H_i \diverge H_j \\
-1 & H_i \parallel H_j
\end{cases}
.\] Note that \(g_{ij} = (h_i, h_j)\) is the Gram matrix of the corresponding intersection form.

\end{remark}

\begin{definition}[Coxeter groups]

A group \(W\) is a \textbf{Coxeter group} if it has a presentation of the following form: \[
W =  \left\langle r_1, \cdots, r_n \mathrel{\Big|}(r_i r_j)^{m_{ij}}  \, \, \forall 1\leq i,j\leq n \right\rangle \qquad m_{ij} \in {\mathbf{Z}}_{\geq 1} \cup\left\{{\infty}\right\}
\] where

\begin{itemize}
\tightlist
\item
  \(m_{ii} = 1\) for all \(i\),
\item
  \(m_{ij} \geq 2\) for \(i\neq j\), and
\item
  \(m_{i,j} = \infty\) means there is no relation imposed.
\end{itemize}

If \(S = \left\{{r_1, \cdots, r_n}\right\}\) is a fixed generating set, we call the pair \((W, S)\) a \textbf{Coxeter system}.

\end{definition}

\begin{definition}[Coxeter diagrams]

Given a Coxeter system \((W, S)\), the \textbf{the pre-Coxeter diagram} of \((W, S)\) is weighted undirected graph with a single vertex \(v_i\) for each \(r_i\in S\), and for each pair \(i\neq j\), an edge \(e_{ij}\) of weight \(w_{ij} \coloneqq m_{ij}\) connecting \(v_i\) to \(v_j\). Note that this yields a complete\footnote{Recall that a graph is \textbf{complete} if every vertex is adjacent to every other vertex.} graph on \({\left\lvert {S} \right\rvert}\) vertices. The \textbf{Coxeter diagram} \(D(W)\) of \((W, S)\) is the partially weighted graph obtained from the pre-Coxeter diagram by the following modifications:

\begin{itemize}
\tightlist
\item
  Edges \(e_{ij}\) of weight \(w_{ij} = 2\) are deleted.
\item
  Edges \(e_{ij}\) of weight \(w_{ij} \geq 7\) are labeled with their weights.
\item
  Edges \(e_{ij}\) of \(w_{ij} = 3,4,5,6\) follow one of two conventions: they are either replaced with an \((w_{ij}-2){\hbox{-}}\)fold multi-edge, or are unmodified and retain their label of \(w_{ij}\).
\item
  Edges \(e_{ij}\) of weight \(w_{ij} = \infty\) are replaced by bold/thick edges.
\end{itemize}

\end{definition}

\begin{remark}[Facts about Coxeter diagrams]

We summarize several facts about the full Coxeter diagram:

\begin{itemize}
\tightlist
\item
  Vertices \(v_i\) and \(v_j\) are non-adjacent if and only if \(w_{ij} = 2\),
\item
  Vertices \(v_i\) and \(v_j\) are adjacent if and only if \(w_{ij} \geq 3\),
\item
  Edge weights are suppressed for small weights \(w_{ij} \leq 6\), and explicitly included for every \(w_{ij} \geq 7\).
\end{itemize}

\end{remark}

\begin{remark}[How to read a group presentation from a Coxeter diagram]

One can recover the presentation of a Coxeter group from any Coxeter diagram. Explicitly, given a diagram \(D\), one constructs a group \(W\) such that \(D = D(W)\) in the following way: first one transforms the Coxeter diagram into a pre-Coxeter diagram by adding weight 2 edges between every pair of non-adjacent vertices, forming a complete graph. One then replaces double/triple/quadruple edges with weight 4/5/6 edges respectively. Finally, reads the group presentation off of the weighted adjacency matrix of the resulting graph. Explicitly, the group \(W\) will have a generator for every vertex and a relation \((r_ir_j)^{ w_{ij} }\) for each edge \(e_{ij}\) of weight \(w_{ij}\).d

\end{remark}

\begin{example}[Passing between Coxeter diagrams and Coxeter groups]

Every Coxeter diagram is naturally associated with a weighted graph whose edge weights are all integers \(m_{ij} \geq 2\), and from this presentation, one can immediately read off the group presentation. For example, consider the following diagram and the associated weighted graph: \[
\quad 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=black node] (0) at (0, 2) {};
        \node [style=black node] (1) at (2, 0) {};
        \node [style=black node] (3) at (6, 0) {};
        \node [style=black node] (2) at (0, -2) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) to (1);
        \draw (1) to (2) ;
        \draw [style=double edge] (2) to (0);
        \draw (1)  edge["7"] (3);
    \end{pgfonlayer}
\end{tikzpicture}
\leadsto 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=black node, label=left:$r_1$] (0) at (0, 2) {};
        \node [style=black node, label=$r_2$] (1) at (2, 0) {};
        \node [style=black node, label=$r_3$] (3) at (6, 0) {};
        \node [style=black node, label=left:$r_4$] (2) at (0, -2) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw (0) edge["3"] (1);
        \draw (1) edge["3"] (2);
        \draw (2) edge["4"] (0);
        \draw (1) edge["7"] (3);
        \draw[bend left] (0) edge["2"] (3);
        \draw[bend right] (2) edge["2"] (3);
    \end{pgfonlayer}
\end{tikzpicture} 
\]

Reading generators and relations off of this graph, we obtain a group freely generated by \(r_1,r_2,r_3,r_4\) subject to the following relations: \[
W \coloneqq \left\langle r_1, r_2, r_3, r_4 \,
\middle\vert
\begin{array}{ll}
    r_1^2=r_2^2 =r_3^2=r_4^2=1 \\
    (r_1 r_2)^3=(r_2 r_3)^7 =(r_1 r_4)^4=(r_2 r_4)^3=1 \\
    (r_1 r_3)^2 =(r_3 r_4)^2=1
\end{array}
 \right\rangle 
.\] Letting \(A\) be the weighted adjacency matrix of this weighted graph, we can read this group presentation directly off of the following symmetric matrix: \[
A  =
\left[
\begin{matrix}
1 & 3 & 2 & 4 \\
3 & 1 & 7 & 3 \\
2 & 7 & 1 & 2 \\
4 & 3 & 2 & 1
\end{matrix}
\right]
\] This matrix defines an exact sequence of \({\mathbf{Z}}{\hbox{-}}\)modules \[
0 \to {\mathbf{Z}}^4 \xrightarrow{A} {\mathbf{Z}}^4 \to W \to 0
,\] realizing \(W \cong \operatorname{coker}A\) as a presentation of \(W\) by generators and relations.

\end{example}

\subsection{Coxeter polytopes}\label{coxeter-polytopes}

\begin{remark}

Recall the cosine formula for Euclidean inner product spaces: in \({\mathbb{E}}^n\), the norm is \({\left\lVert {x} \right\rVert} \coloneqq\sqrt{x^2} \coloneqq\sqrt{x.x}\), and we have \[
vw = {\left\lVert {v} \right\rVert}{\left\lVert {w} \right\rVert} \cos(\angle(v, w)) 
= \sqrt{v^2}\sqrt{w^2} \cos(\angle(v, w)) 
= \sqrt{v^2 w^2 } \cos(\angle(v, w))
\] For a general bilinear form, we can define \[
\angle(v, w) \coloneqq\cos^{-1}\qty{vw \over \sqrt{ v^2 w^2 } }
.\] We can thus interpret the pairing as measuring angles in the following way: \[
vw = {\cos(\angle(v, w) \over \sqrt{v^2 w^2}}
,\] which moreover allows one to compute intersections \(vw\) from knowledge of \(v^2, w^2\), and angles \(\angle(v, w)\), which is precisely the data that is encoded in a Coxeter diagram.

\end{remark}

\begin{definition}[Dihedral angles between hyperplanes]

If \(H_i, H_j\) are intersecting hyperplanes in \({\mathbb{E}}^n\), we write \(H_i \pitchfork H_j\). We write \(h_i \coloneqq H_i^\perp\) and \(h_j\coloneqq H_j^\perp\) for unit normal vectors spanning their orthogonal complements, and define the \textbf{dihedral angle} between \(H_i\) and \(H_j\) as \[
\angle(H_i, H_j) \coloneqq\angle(h_i, h_j)
.\] If \(H_i\) is parallel to \(H_j\), we write \(H_i \parallel H_j\) and define \(\angle(H_i, H_j) = 0\). We similarly write \(H_i \perp H_j\) if \(\angle(H_i, H_j) = \pi/2\).

\end{definition}

\begin{remark}

Note that there is a common trick to get rid of the square root in these formulas: one writes \[
(vw)^2 = v^2 w^2 \cos^2(\angle(v, w))
\] For \(\angle(v, w) = \pi/m_{ij}\), this gives a way to recover \(m_{ij}\) from the bilinear form.

\end{remark}

\begin{definition}[Coxeter polytopes]

Let \(X \coloneqq{\mathbb{E}}^n, {\mathbb{S}}^n, {\mathbb{H}}^n\) be a Euclidean, spherical, or hyperbolic geometry. A polytope \(P\subseteq X\) is \textbf{Coxeter polytope} if all dihedral angles between pairs of intersecting facets \(H_i\) and \(H_j\) are of the form \(\pi/m_{ij}\) for \(m_{ij}\in {\mathbf{Z}}_{\geq 2}\), and any two non-intersecting facets are parallel.

\end{definition}

\begin{remark}[Coxeter group $G_P$ of a Coxeter polytope $P$]

Every Coxeter polytope \(P\) defines a Coxeter group \(G_P \leq \mathrm{Isom}(\bX)\) generated by reflections through the supporting hyperplanes \(H_i\) of facets of \(P\) and a corresponding Coxeter diagram \(D_P\). For \(\bX = {\mathbb{E}}^n\), one constructs \(G_P\) in the following way:

\begin{itemize}
\tightlist
\item
  A generator \(r_i\) for each facet \(H_i\) of \(P\) with relation \(r_i^2 = 1\), representing reflection through the hyperplane \(H_i\),
\item
  For any facets \(H_i, H_j\) where \(H_i \pitchfork H_j\), there is a relation \((r_i r_j)^{m_{ij}} = 1\) where \(m_{ij}\) is defined by \(\angle(H_i, H_j) = \pi/{m_{ij}}\).
\item
  For non-intersecting facets \(H_i \parallel H_j\), we set \(m_{ij} = \infty\) and take a relation \((r_i r_j)^\infty = 1\), i.e.~no relation is imposed at all.
\end{itemize}

\end{remark}

\begin{remark}

Note that \(P\) is a fundamental domain for the action of \(G_P\) on \(\bX\). Moreover, if \(G\leq \mathop{\mathrm{Isom}}(\bX)\) is any discrete finitely generated reflection group, then its fundamental domain is always a Coxeter polytope. If \(\bX = {\mathbb{S}}^n\) or \({\mathbb{E}}^n\), Coxeter polytopes are classified and are either simplices or products of simplices respectively, and full lists can be found. For \(\bX = {\mathbb{H}}^n\), the general classification is an open problem. Poincaré classified them in \({\mathbb{H}}^2\). Vinberg showed that no compact Coxeter polytopes exist in \({\mathbb{H}}^n\) for \(n\geq 30\), and no non-compact but finite volume polytopes exist for \(n\geq 996\). These bounds are not sharp. Finding explicit examples of high-dimensional compact Coxeter polytopes is interesting because these can be used to explicitly construct high-dimensional hyperbolic manifolds.

\end{remark}

\begin{definition}[Volumes and covolumes of Coxeter groups/polytopes/diagrams]

We define the \textbf{covolume} of \(G_P\) as the volume of \(P\cong \bX/G_P\), where the metric on the quotient is induced from the metric defining the geometry on \(\bX\).

\end{definition}

\begin{remark}

We collect some facts about the corresponding Coxeter diagram \(D(P)\):

\begin{itemize}
\tightlist
\item
  \(D(P)\) has vertices \(v_i\) corresponding to \(H_i\), where \(v_i, v_j\) are non-adjacent if and only if \(H_i \perp H_j\)\footnote{Recalling that edges with $m_{ij} = 2$ are deleted by convention.},
\item
  Edges \(e_{ij}\) are plain if \(m_{ij} < \infty\) and \(m_{ij} \neq 0\), so \(H_i \pitchfork H_j\),
\item
  Edges \(e_{ij}\) are bold if \(m_{ij} = \infty\), so \(H_i \parallel H_j\) and \(\angle(H_i, H_j) = \pi/\infty = 0\).
\end{itemize}

\end{remark}

\begin{example}[Euclidean Coxeter polytopes]

Consider the following Coxeter diagram: \[
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node] (0) at (0, 0) {};
        \node [style=white node] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) edge["$m$"] (1);
    \end{pgfonlayer}
    \end{tikzpicture}   
\]

This corresponds to a non-compact polytope in \({\mathbb{E}}^2\) bounded by two hyperplanes \(H_1, H_2\) through the origin (i.e.~lines), one corresponding to each node, intersecting at an angle of \(\pi/m\). Without loss of generality, we can take \(H_1\) to be the \(x{\hbox{-}}\)axis and \(H_2\) to be a line of slope \(\pi/m\):

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/chamber_pi_m.png}
    \caption{Caption}
    \label{fig:chamberpim}
\end{figure}

One can note that if \(m=2\), then one deletes the edge by convention to get the Coxeter diagram \[
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node] (0) at (0, 0) {};
        \node [style=white node] (1) at (3, 0) {};
    \end{pgfonlayer}
    \end{tikzpicture}   
\] This is the Dynkin diagram of \(A^1\times A^1\), which indeed has fundamental chamber the first quadrant. Similarly, if one takes \(m=3\) on recovers the standard Dynkin diagram for \(A_2\): \[
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node] (0) at (0, 0) {};
        \node [style=white node] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) to (1);
    \end{pgfonlayer}
    \end{tikzpicture}   
\] We get a fundamental chamber with two walls at a dihedral angle of \(\pi/3\), corresponding to the dual hyperplanes of the two standard short roots\(\alpha\) and \(\beta\) with \(\angle(\alpha, \beta) = 2\pi/3\) in Lie theory:

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/A2RootSystem.png}
    \caption{Caption}
    \label{fig:a2root}
\end{figure}

\end{example}

\begin{example}[Affine examples]

\dzg{Todo: weighted $\tilde A_2$ as a simplex.}

\end{example}

\begin{remark}

Note that taking reflections of the fundamental domain \(C\) by the Weyl group generates a \textbf{tiling} of the hyperbolic disc in these cases.

\end{remark}

\begin{remark}[Importance of tilings]

Why this is important: given \textit{any} tiling of \({\mathbb{E}}^2\) or \({\mathbb{H}}\) the hyperbolic disc, we can place it at height one and take a cone to get an infinite-type toric variety. Alternatively, given any tiling we can construct a surface that is a union of toric pairs by interpreting every vertex of the tiling as a fan and the edges of tiles as gluing instructions.

Finally, we can interpret an \(\mathrm{IAS}^2\) has an irregular spherical tiling, i.e.~a tiling \({\mathbb{S}}^2\) which is not necessarily generated by reflections, but one which has finitely many tiles. We then regard the tiling as a union of toric surfaces as described above.

\end{remark}

\begin{definition}[The Gram matrix of a Euclidean Coxeter polytope]

Let \(P\subset {\mathbb{E}}^n\) be a Euclidean Coxeter polytope, not necessarily compact. One defines the \textbf{Gram matrix} \(G(P)\) of \(P\) as \[
G(P)_{ij} = 
\begin{cases}
1 & i = j \\
-\cos\qty{\pi \over m_{ij}} & 
H_i \pitchfork H_j,
\quad \angle(H_i, H_j) = \pi/m_{ij} \\
-1 &  
H_i \parallel H_j,
\quad \angle(H_i, H_j) = \pi/\infty = 0
\end{cases}
.\]

\end{definition}

\subsection{Hyperbolic Coxeter polytopes}\label{hyperbolic-coxeter-polytopes}

\%https://hal.science/hal-03345221/file/Survey\_Discrete\_Cox\_Gp\_V3\_arxiv.pdf\#page=5\&zoom=auto,-147,687

\begin{remark}

See the section on hyperbolic geometry for a description of \({\mathbb{H}}^n\coloneqq\left\{{x\in {\mathbb{E}}^{n, 1} \mathrel{\Big|}x^2 = -1, x_0 > 0}\right\}\) and terminology (space/time/light-like vectors). As a convention, \({\mathbb{H}}^n\) means the interior of \(\overline{{\mathbb{H}}^n} \coloneqq{\mathbb{H}}^n\cup\partial {\mathbb{H}}^n\) where \(\partial {\mathbb{H}}^n\) is the boundary at infinity consisting of ideal points.

Some unsorted notes:

\begin{itemize}
\item
  The distance \(\rho\) on \({\mathbb{H}}^n\) is defined such that \(\rho(v, w) \coloneqq\mathrm{arccosh}(vw)\).
\item
  In \({\mathbb{H}}^n\) Vinberg defines the dihedral angle as \(\angle(f_i, f_j) \coloneqq\pi - \angle(f_i^\perp, f_i^\perp)\).
\item
  The diagram \(E_{10}\) describes a polytope in \({\mathbb{H}}^9\).
\end{itemize}

\end{remark}

\begin{remark}[Hyperplane incidence relations in hyperbolic spaces]

In hyperbolic geometry (\({\mathbb{H}}^2\) to simplify), there are two types of parallelism: asymptotically parallel (converging) lines, or ultraparallel (diverging) lines. Both are characterized by sharing a common orthogonal line, however, asymptotically parallel lines have a common perpendicular in \(\partial \overline{{\mathbb{H}}^2}\) going through their ideal point of intersection, while ultraparallel lines share a common perpendicular at a point in the interior \({\mathbb{H}}^2\). By the ultraparallel theorem, \(H_i, H_j\) are ultraparallel if and only if \(H_i\cap H_j = \emptyset\) in \(\overline{{\mathbb{H}}^2}\).

\begin{figure}[H]
    \centering
\includegraphics[width=0.9\textwidth]{figures/ultraparallel.jpg}
    \caption{The two types of parallelism in hyperbolic space, visualized in the ball model and half-plane model respectively.}
    \label{fig:ultraparallel}
\end{figure}

Thus given a pair of hyperplanes \(H_i\) and \(H_j\), there are thus three possibilities for their incidence relations:

\begin{itemize}
\tightlist
\item
  \(H_i, H_j\) are not parallel and thus intersect in \({\mathbb{H}}^n\). We write \(H_i \pitchfork H_j\) and define \(\angle(H_i, H_j)\) as the usual dihedral angle.
\item
  \(H_i, H_j\) are asymptotically parallel/converging and thus intersect in an ideal point in \(\partial {\mathbb{H}}^n\). We write \(H_i \parallel H_j\) and define \(\angle(H_i, H_j) = {\pi \over \infty} = 0\).
\item
  \(H_i, H_j\) are ultraparallel/diverging and do not intersect in \(\overline{{\mathbb{H}}^n}\). We write \(H_i \diverge H_j\).
\end{itemize}

\end{remark}

\begin{remark}[Hyperbolic distance between hyperplanes]

Note that in the last case above, \(\angle(H_i, H_j)\) is undefined but there is a minimal distance \(\rho(H_i, H_j)\) between the two hyperplanes. By geometric axioms, if \(H_i \cap H_j = \emptyset\) then there is a unique geodesic \(L_{ij}\) that is simultaneously orthogonal to both \(H_i\) and \(H_j\), intersecting them at points \(p_i\) and \(p_j\). One then defines \(\rho(H_i, H_j)\) as the length of a geodesic segment along \(L_{ij}\) with endpoints at \(p_i\) and \(p_j\).

\end{remark}

\begin{remark}[Extending Coxeter diagrams for hyperbolic polytopes]

Following Vinberg, one can extend the notion of a Coxeter diagram to a weighted graph with positive weights \(w_{ij} > 0\) where all \(w_{ij}\in (0, 1)\) can be written in the form \(w_{ij} = \cos\qty{\pi \over m_{ij}}\) for some \(m_{ij} \in {\mathbf{Z}}_{\geq 2}\) and \(w_{ij} \in [1, \infty]\) can be arbitrary real (possibly infinite) numbers. In this convention,

\begin{itemize}
\tightlist
\item
  \(w_{ij} = \cos\qty{\pi \over {m_{ij}}} \in (0, 1)\) get simple edges of labeled weight \(m_{ij}\) (or multiedges) corresponding to \(H_i \pitchfork H_j\) and \(\angle(H_i, H_j) =\qty{\pi\over m_{ij}}\)
\item
  \(w_{ij} = 1\) get \textbf{bold} unlabeled edges of weight 1 corresponding to \(H_i \parallel H_j\) and \(\angle(H_i, H_j) = {\pi \over \infty} = 0\).
\item
  \(w_{ij}\in (1,\infty)\) get \textbf{dotted} labeled edges of weight \(w_{ij}\) (or unlabeled) corresponding to \(H_i \diverge H_j\) and \(w_{ij}\) corresponds to \(\rho(H_i, H_j)\)
\end{itemize}

More generally, given a Coxeter-Vinberg diagram set \[
g_{ij} =  {h_i h_j \over \sqrt{h_i^2 h_j^2 }}
,\] then one interprets

\begin{itemize}
\tightlist
\item
  \(g_{ij} < 1 \implies g_{ij} = \cos\qty{\angle(h_i, h_j)}\) and \(H_i \pitchfork H_j\) with \(\angle(h_i, h_j) = \pi/m_{ij}\),
\item
  \(g_{ij} = 1 \implies H_i \parallel H_j\) with \(\angle(h_i, h_j) = 0\),
\item
  \(g_{ij} > 1 \implies H_i \diverge H_j\).
\end{itemize}

\end{remark}

\begin{example}[Hyperbolic Coxeter polytopes]

\dzg{Todo: $(\infty,\infty, \infty)$.}

\end{example}

\begin{remark}

As in the Euclidean case that taking reflections of the fundamental domain \(C\) by the corresponding Weyl group naturally constructs a \textbf{tiling} of \({\mathbb{E}}^2\) in all of these cases:

\begin{itemize}
\tightlist
\item
  \(A_1\times A_1\) tiles \({\mathbb{E}}^2\) with 4 non-compact quadrants,
\item
  \(A_2\) tiles \({\mathbb{E}}^2\) with 6 non-compact sectors of angle \(\pi/3\),
\item
  In general, taking \(\circ \to^{m} \circ\) with \(m\in {\mathbf{Z}}_{\geq 1}\) tiles \({\mathbb{E}}^2\) with \(2m\) non-compact sectors of angle \(\pi/m\),
\item
  \(\tilde A_2\) tiles \({\mathbb{E}}^2\) with infinitely many compact equilateral triangles of with internal angles \(\pi/3\).
\end{itemize}

\end{remark}

\begin{definition}[The Gram matrix of a hyperbolic polytope]

Let \(P\subseteq \overline{{\mathbb{H}}^n}\) be a Coxeter polytope, possibly with ideal points. The \textbf{Gram matrix} of \(P\) is the matrix \[
G(P)_{ij} = 
\begin{cases}
    1 & i=j \\
    -\cos\qty{\pi \over m_{ij}} & H_i \pitchfork H_j, \quad \angle(H_i, H_j) = \pi/m_{ij}, \\
    -1 & H_i \parallel H_j, 
    \quad \angle(H_i, H_j) = \pi/\infty = 0, \\
    -\cosh(\rho(H_i, H_j)) & H_i \diverge H_j,
    \quad \angle(H_i, H_j) = \pi/0 = \infty, \\
\end{cases}
.\]

\end{definition}

\begin{remark}

When labeling the Coxeter graph, one often puts \(m_{ij}\) or \(\cosh(\rho(H_i, H_j))\) as the labels, mixing conventions slightly. Edges of weight 2 are deleted, edges of weight 3 are unlabeled simple edges.

\end{remark}

\begin{remark}

If \(P\subseteq{\mathbb{H}}^n\) is a compact hyperbolic Coxeter polytope, the quotients \({\mathbb{H}}^n/G_P\) are hyperbolic orbifolds. The simplest examples of such polytopes are the hyperbolic \(n{\hbox{-}}\)gons defined by integers \(p_1,\cdots, p_k \geq 2\) satisfying \(\sum p_i^{-1}< k-2\).

\end{remark}

\begin{definition}[Simple systems]

We say \(\Delta = \left\{{r_i}\right\}\) is a \textbf{simple system} of generators for a polytope \(P\) if \(r_i r_j \geq 0\) for all \(i\) and \(j\), and \(P\) has a facet presentation by the mirrors \(H_{r_i}\). This allows one to write \[
P = \left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = 0,\, r_i v \geq 0}\right\}
.\] We call \(P\) a \textbf{Weyl chamber}\footnote{This is also sometimes notated $C$.}. The closure \(\overline P\) is a fundamental domain for the action of \(W(L)\) and the Weyl group acts simply transitively on the set of chambers.

\end{definition}

\begin{remark}[Decomposing the future orthogonal group into a Weyl and symmetry group]

Let \(W(L)\) be the reflections in all negative norm vectors. There is an identification \[ 
{\operatorname{O}}^+(L) \cong W(L) \rtimes S(C),\qquad S(C) \coloneqq{\operatorname{Stab}}_{{\operatorname{O}}^+(L)}(C)
\] \dzg{The semidirect might be in the wrong direction here, which one is normal?} where \(C \subset \BB^n\) be a fundamental chamber of \(W(L)\) with respect to some choice of a simple set of generators.

\end{remark}

\begin{definition}[Reflective lattices]

We say \(L\) is \textbf{reflective} if \(W(L)\leq {\operatorname{O}}^+(L)\) is finite-index. More generally, if we define \({\operatorname{O}}^+(L)_k\) as the subgroup generated by all \(k{\hbox{-}}\)reflections, i.e.~reflections in roots \(v\) with \(v^2 = k\), we say \(L\) is \(k{\hbox{-}}\)reflective if \(W(L)\) is finite index in \({\operatorname{O}}^+(L)\).

\end{definition}

\begin{remark}

If \(L\) as above is reflective, it is well-known \(C\) is a hyperbolic Coxeter polytope of finite volume.

\end{remark}

\begin{definition}[Vinberg-Coxeter diagrams]

A \textbf{Vinberg-Coxeter diagram} is an extension of a Coxeter diagram with adds the following decorations:

\begin{itemize}
\tightlist
\item
  Black edges
\item
  Double-circled edges
\item
  Dotted edges
\item
  Thick edges
\end{itemize}

It is a weighted graph with positive edge weights \(w_{ij} > 0\) where we require that any \(w_{ij}\in (0, 1)\) is of form \(w_{ij} = \cos\qty{\pi \over m_{ij}}\) for some \(m_{ij} \in {\mathbf{Z}}_{\geq 2}\), but we explicitly allow some \(w_{ij} \in [1, \infty]\) to be real (possibly infinite) numbers. We additionally specify vertex weights \(r_{i}\) for each vertex \(v_i\). In this convention,

\begin{itemize}
\tightlist
\item
  \(w_{ij} = \cos\qty{\pi \over {m_{ij}}} \in (0, 1)\) get \textbf{simple edges} of labeled weight \(m_{ij}\) (or unlabeled multi-edges of multiplicity \(m_{ij} - 2\) for \(m_{ij} = 3,4,5,6\)),
\item
  \(w_{ij} = 1\) get \textbf{bold unlabeled edges} of weight 1
\item
  \(w_{ij}\in (1,\infty)\) get \textbf{dotted labeled edges} of weight \(w_{ij}\).
\end{itemize}

\end{definition}

\subsection{Elliptic and Parabolic subdiagrams}\label{elliptic-and-parabolic-subdiagrams}

\begin{remark}

Given these weights, one can construct the weighted adjacency matrix \(A\) with \(a_{ij} = w_{ij}\) if \(v_i, v_j\) are adjacent and zero otherwise.

A matrix \(A\) is a \textbf{direct sum of matrices} \(A_i\) if \(A\) is similar via permutations of rows and columns to the block diagonal matrix whose blocks are the \(A_i\). If \(A\) can not be written as a direct sum of two matrices, we say \(A\) is \textbf{indecomposable}. Every matrix has a unique representation as a sum of indecomposable components. We say a Coxeter polytope is indecomposable if its Gram matrix \(G_P\) is indecomposable. Any matrix \(G_P\) arising from an irreducible Coxeter polytope is either positive-definite, positive-semidefinite, or indefinite. We say a diagram \(D_P\) is elliptic if \(G_P\) is PD, parabolic if every subdiagram is elliptic and it has at least one degenerate irreducible component.

Connected components of the diagram correspond to indecomposable sub-block matrices of \(A\). A diagram is elliptic of \(A\) is positive-definite, and is parabolic if any indecomposable component of \(A\) is degenerate and positive-semidefinite. There are finitely many indecomposable elliptic and parabolic diagrams. If a Coxeter diagram describes a Coxeter polytope \(P\), elliptic subdiagrams of codimension 1 correspond to facets of \(P\). Moreover, \(P\) has finite volume iff every such elliptic subdiagram can be extended in exactly 2 ways to either an elliptic subdiagram of rank \(n\) or a parabolic subdiagram of rank \(n-1\), corresponding to every facet of the polytope meeting each of its adjacent facets at either an interior point or an ideal point of \({\mathbb{H}}^n\) respectively.

\end{remark}

\begin{remark}

Idea: a subdiagram is elliptic if the Gram matrix is negative definite of full rank, and parabolic if negative semidefinite of corank equal to the number of components of the diagram. Elliptic diagrams of rank \(r\) biject with codimension \(r\) faces of \(C\). Parabolic diagrams of corank 1 correspond to ideal points of \(C\). Vinberg's algorithm produces a simple system \(\Delta\) of generators for \(W(L)\) which determines a hyperbolic polytope \(C\) via the corresponding Weyl chamber. If the algorithm terminates, \(C\) is of finite volume.

\end{remark}

\begin{definition}[Ranks of subdiagrams]

The \textbf{rank} of a subdiagram is its number of vertices minus its number of connected components.

\end{definition}

\begin{definition}[Elliptic and parabolic Coxeter subdiagrams]

A Coxeter diagram \(G\) is called \textbf{elliptic} (resp. \textbf{parabolic}) if every connected component of \(G\) is a Coxeter diagram underlying classical (resp. affine) Dynkin diagram. This is summarized in the following table; note that the classical diagrams \(B_n\) and \(C_n\) become identified when the arrow is omitted:

\begin{quote}
A possible reference that mostly agrees with this: http://webdoc.sub.gwdg.de/ebook/serien/e/mpi\_mathematik/2005/8.pdf\#page=3\&zoom=180,-91,721
\end{quote}

\begin{quote}
Another reference, although they seem to include a mysterious G\_2\^{}m\ldots{} \% https://www.maths.dur.ac.uk/users/anna.felikson/talks/HypCoxPoly17.pdf\#page=8\&zoom=80,48,172
\end{quote}

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Elliptic &
  Parabolic \\ \midrule
$A_n\,$ \dynkin[label, labels={1,2,n-1, n},arrows=false, edge length=.75cm]A{} &
  $\tilde A_1 = I_\infty\,$ \dynkin [Coxeter,gonality=\infty, edge length=.75cm, o/.style=black,label,labels={0,1}]I{} $= 
\begin{tikzpicture}[baseline=-0.5ex, scale=0.75, transform shape]
    \begin{pgfonlayer}{nodelayer}
        \node [style=black node, label=below:$0$] (0) at (0, 0) {};
        \node [style=black node, label=below:$1$] (1) at (1, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
$ \\
 &
  $\tilde A_n\,$ \dynkin[extended, arrows=false, edge length=.75cm, o/.style=black,label,labels={0,1,2,n-1, n}]A{} \\
$B_n = C_n\,$ \dynkin[label, labels={1,2,,n-1, n},arrows=false, edge length=.75cm]B{} &
  $\tilde B_n\,$ \dynkin[extended, arrows=false, edge length=.75cm,o/.style=black,label, labels={0,1,2,3,n-2, n-1, n}]B{} \\
 &
  $\tilde C_n\,$ \dynkin[extended, arrows=false, edge length=.75cm,o/.style=black, label, labels={0,1,,,n-1,n}]C{} \\
$D_n\,$ \dynkin[label, labels={1,2,,,n-1,n},arrows=false, edge length=.75cm]D{} &
  $\tilde D_n\,$ \dynkin[label, labels={0,1,2,3,,,n-1,n},extended, arrows=false, edge length=.75cm,o/.style=black]D{} \\
$E_6\,$ \dynkin[label, arrows=false, edge length=.75cm]E6 &
  $\tilde E_6\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E6 \\
$E_7\,$ \dynkin[label, arrows=false, edge length=.75cm]E7 &
  $\tilde E_7\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E7 \\
$E_8\,$ \dynkin[label, arrows=false, edge length=.75cm]E8 &
  $\tilde E_8\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E8 \\
$F_4\,$ \dynkin[label, arrows=false, edge length=.75cm]F4 $=$ \dynkin[label, Coxeter]{F}{4} &
  $\tilde F_4\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]F4 $=$ \dynkin[label, Coxeter, extended, o/.style=black]{F}{4} \\
$G_2\,$ \coxeterGtwo $=$ \dynkin[label, Coxeter,gonality=6]{G}{2} &
  $\tilde G_2\,$ \coxeterGtwoAffine $=$ \dynkin[label, Coxeter, extended, o/.style=black]{G}{2} \\
$H_3\,$ \coxeterHthree $=$ \dynkin[label, Coxeter]{H}{3} &
   \\
$H_4\,$ \coxeterHfour $=$ \dynkin[label, Coxeter]{H}{4} &
   \\ \bottomrule
\end{tabular}%
}
\caption{Classification of elliptic and parabolic subdiagrams of a Coxeter diagram}
\label{tab:elliptic-parabolic-subdiagrams}
\end{table}

\end{definition}

\subsection{Some discrepancies}\label{some-discrepancies}

Note the following discrepancies when comparing the classification of diagrams of Coxeter diagrams to the usual notions of Dynkin diagrams:

\begin{itemize}
\tightlist
\item
  These are not Dynkin diagrams: we forget the arrows on double, triple, etc edges.
\item
  \textbf{Warning}: in a \textbf{Coxeter} diagram, an edge of label \(m\) always corresponds to an \((m-2){\hbox{-}}\)fold edge. \textbf{In a Dynkin diagram, a 3-fold edge corresponds to $m=6$. We do not use this convention in the table above!} Compare \(G_2, \tilde G_2\) in the table, which have 4-fold edges corresponding to \(m=6\) to the following classical diagrams for \(G_2\) and \(\tilde G_2\) which still correspond to \(m=6\): \[
  G_2: \dynkin[label, edge length=0.75cm]G2 \qquad \tilde G_2: \dynkin[label,extended, edge length=0.75cm]G2
  \] The reason for this discrepancy: in a \textbf{Dynkin} diagram, the edge labels \(m\) must satisfy a crystallographic condition and thus \(m=2,3,4,6\). Since \(m=5\) is not possible, this makes the interpretation in that special case unambiguous.
\item
  This discrepancy also occurs for \(H_i\); here a triple edge truly corresponds to \(m=5\).
\item
  In the affine case, we do not distinguish the ``new'' node, usually denoted by a white dot labeled \(0\). Compare to the usual diagram e.g.~for \(\tilde A_n\): \[
  \dynkin[label, edge length=0.75cm, extended]A{}
  \]
\end{itemize}

\begin{remark}

Elliptic subdiagrams are a disjoint union of classical Dynkin diagrams, while parabolic subdiagrams are a disjoint union of \textit{affine} Dynkin diagrams.

Why these matter: we are working with Coxeter polytopes \(P\) in a hyperbolic space, i.e.~hyperbolic Coxeter polytopes. Vinberg has a general theory which says the Coxeter diagram \(D\) records the combinatorics of \(P\):

\begin{itemize}
\tightlist
\item
  Facets of \(P \rightleftharpoons\) nodes of \(D\),
\item
  Dihedral angles between two facets of \(P \rightleftharpoons\) edges of \(D\),
\item
  \(k{\hbox{-}}\)faces of \(P \rightleftharpoons\) elliptic subdiagrams of \(P\) of co-rank \(k\),
\item
  Ideal vertices of \(P \rightleftharpoons\) parabolic subdiagrams of rank \(k\).
\end{itemize}

Idea: in \(F_2\), Type II strata are classified by maximal parabolic subdiagrams of the single Coxeter diagram, and Type III strata by elliptic subdiagrams. Dimensions of strata correspond to number of vertices in these subdiagrams, and inclusion of diagrams corresponds to degenerations (smaller diagrams correspond to ``more degenerate'').

\end{remark}

\subsection{Edge conventions for Coxeter diagrams}\label{edge-conventions-for-coxeter-diagrams}

The interpretation of these Coxeter diagrams in terms of root systems:

Needs some notation from \cite{AN06}:

\[\begin{array}{ll}
V(M) &\text {the light cone } V(M)=\left\{x \in M \otimes \mathbb{R} \mathrel{\Big|}x^2>0\right\} \text { of a hyperbolic lattice } M \\
V^{+}(X) &\text {the half containing polarization of the light cone } V\left(S_X\right) \\
\mathcal{L}(S)=V^{+}\left(S_X\right) / \mathbb{R}^{+} & \text {the hyperbolic space of a surface } S \\
W^{(2)}(M) & \text{the group generated by reflections in all $f \in M$ with $f^2=-2$} \\
W^{(4)}(M) & \text{the group generated by reflections in all $(-4)$ roots of $M$} \\
W^{(2,4)}(M) & \text{the group generated by reflections in all $(-2)$ and $(-4)$ roots of $M$} \\
\mathcal{M}^{(2)} & \text {a fundamental chamber of } W^{(2)}(S) \text { in } \mathcal{L}(S) \\
\mathcal{M}^{(2,4)} & \text {a fundamental chamber of } W^{(2,4)}(S) \text { in } \mathcal{L}(S) \\
P^{(2)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-2) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
P^{(4)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-4) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
(X, \theta) &\text{a $\mathrm{K}3$ with involution $\theta$} \\
X^\theta &\text{the fixed locus of an involution} \\
P(X)_{+I} & \text {the subset of exceptional classes of }(X, \theta)  \text{ of type } \rm{I}
\end{array}\]

\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Description                                                                          & Symbol                                                                    \\ \midrule
Black vertices: $f \in P^{(4)}\left(\mathcal{M}^{(2,4)}\right)$, i.e. $f^2 = -4$     & \tikz[baseline] { \node [style=black node, label=$f$] (0) at (0, 0) {}; } \\
White vertices: $f \in P^{(2)}\left(\mathcal{M}^{(2,4)}\right)$, i.e. $f^2 = -2$     & \tikz[baseline] { \node [style=white node, label=$f$] (0) at (0, 0) {}; } \\
Double-circled vertices: $f \in P(X)_{+I}$, i.e. the class of a rational component of $X^\theta$. &
  \tikz[baseline] { \node [style=doubled node, label=$f$] (0) at (0, 0) {}; } \\
No edge: $f_1\neq f_2 \in P\left(\mathcal{M}^{(2,4)}\right)$ with $f_1 \cdot f_2=0$, so $\angle(f_1 f_2) = \pi/2$ & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
\end{tikzpicture}
\\
Simple edges of weight $m$, or $m-2$ simple edges when $m$ is small: ${ 2 f_1 f_2 \over  \sqrt{f_1^2 f_2^2} } = 2 \cos \frac{\pi}{m}$, so $\angle(f_1 f_2) = \pi/m$ &
  
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) edge["$m$"] (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
Thick edges: $\frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2$                      & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
Broken edges of weight $t$: ?                                                          & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=dashed edge] (0) edge["$t$"] (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\ \bottomrule
\end{tabular}%
}
\caption{Edge conventions for Coxeter diagrams}
\label{tab:edge-notation}
\end{table}

Edge conventions for Coxeter polytopes: nodes correspond to facets \(f_i, f_j\) of \(P\) and edges record relations in \(G_P\).

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Description                                         & Diagram            \\ \midrule
$\angle(f_i f_j) = \pi/2$                           & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
\end{tikzpicture}
\\
$\angle(f_i f_j) = \pi/m$                           & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) edge["$m$"] (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
$\angle(f_i f_j) = \pi/3$                           & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0)to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
$\angle(f_i f_j) = \pi/4$                           & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=double edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
$\angle(f_i f_j) = \pi/5$                           & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=triple edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
$f_i, f_j$ do not intersect                         & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=dashed edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\
$f_i, f_j$ intersect in $\partial \overline{{\mathbb{H}}^n}$ & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
\\ \bottomrule
\end{tabular}%
}
\caption{}
\label{tab:vinberg-stuff}
\end{table}

\subsection{Surfaces associated with Coxeter diagrams}\label{surfaces-associated-with-coxeter-diagrams}

\begin{remark}

As described in \cite[Prop. 4.6]{AE22nonsympinv}, \% https://arxiv.org/pdf/2208.10383.pdf\#page=18\&zoom=180,-45,169 for the Halphen case \(S \coloneqq\HalphenInvariants\), there exists a \(\mathrm{K}3\) surface with \(S_X^+ = S\) with \(\pi: X\to Y\coloneqq X/\iota\) where \(\operatorname{Nef}(Y)\) can be identified with the Coxeter chamber for the full reflection group \(W_r\).

Moreover, \cite[Cor. 4.8]{AE22} \% https://arxiv.org/pdf/2208.10383.pdf\#page=18\&zoom=160,134,179 shows that the Coxeter diagram of \(S\) can be used to write the dual graph of exceptional curves on \(Y\) under the following modifications:

\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lcl@{}}
\toprule
Description                               & Symbol               & Description                               \\ \midrule
A single-circled white vertex                    & \tikz[baseline]{ \node [style=white node, label=$F$] at (0, 0) {}; } & $F \cong {\mathbf{P}}^1$ with $F^2 = -1$  \\
\addlinespace[2em] A double-circled white vertex & \tikz[baseline]{\node [style=doubled node, label=$F$] at (0, 0) {};} & $F \cong {\mathbf{P}}^1$ with $F^2 = -4$  \\
\addlinespace[2em] A black vertex                & \tikz[baseline]{\node [style=black node, label=$F$] at (0, 0) {}; }  & $F \cong {\mathbf{P}}^1$ with $F^2 = -2$. \\
\addlinespace[2em] Any single, plain edge & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$F_i$] (0) at (0, 0) {};
        \node [style=white node, label=$F_j$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
& $F_i, F_j \cong {\mathbf{P}}^1$ with $F_i F_j = 1$ \\
\addlinespace[2em] Any bold edge          & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$F_i$] (0) at (0, 0) {};
        \node [style=white node, label=$F_j$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw [style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
& $F_i, F_j \cong {\mathbf{P}}^1$ with $F_i F_j = 2$ \\
\addlinespace[2em] Any double edge        & 
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$F_i$] (0) at (0, 0) {};
        \node [style=white node, label=$F_j$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw [style=double edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
& ??                                        \\ \bottomrule
\end{tabular}%
}
\caption{How to read a surface off of a Coxeter diagram}
\label{tab:surface_coxeter}
\end{table}

\dzg{Unclear what double edges are, need to read further.}

\end{remark}

\subsection{Incidence diagrams/dual complexes}\label{incidence-diagramsdual-complexes}

\subsection{Dual complexes}\label{dual-complexes}

\begin{definition}[Dual complex]

Let \(D\) be an RSNC divisor. The \textbf{dual complex} \(\Gamma(D)\) of \(D\) is the PL-homeomorphism type of the simplicial complex whose \(d{\hbox{-}}\)cells correspond with codimension \(d\) strata in \(D\), i.e.~irreducible components of \(d{\hbox{-}}\)fold intersections \(V_{i_0} \cap\cdots \cap V_{i_d}\).

\end{definition}

\begin{example}

Let \({\mathcal{X}}\to \Delta\) be a semistable degeneration and let \({\mathcal{X}}_0 = V_1 \cup\cdots \cup V_n\) be the smooth surfaces forming the irreducible components of the central fiber. Writing \(C_{ij} \coloneqq V_i \cap V_j\) and \(p_{ijk} \coloneqq V_i \cap V_j \cap V_k\) for their intersections along curves and points, we call each irreducible component of \(C_{ij}\) a \textbf{double curve} and the points \(p_{ijk}\) \textbf{triple points}.

Semistability ensures that the dual complex has dimension at most 3, i.e.~there are at worst triple points. Thus concretely the dual complex has

\begin{itemize}
\item
  a vertex for each component \(V_i\),
\item
  an edge from \(V_i\) to \(V_j\) for each double curve \(C_{ij}\), and
\item
  a 2-simplex spanning \(V_i, V_j, V_k\) for each triple point \(p_{ijk}\).
\end{itemize}

\end{example}

\begin{remark}

For a double curve \(C = C_{ij} = C_{ji}\) regarded as a curve in \(V_i\) and \(V_j\) respectively, Persson's triple point formula holds: \[
C_{ij}^2 + C_{ji}^2 = - T_C
\] where \(T_C\) is the number of triple points on \(C\).

\end{remark}

\begin{example}

Let \(H_i \subset {\mathbf{P}}^3\) for \(0\leq i \leq 3\) be the four standard coordinate hyperplanes, i.e.~\(H_i = \left\{{[z_1:z_2:z_3:z_4] \mathrel{\Big|}z_i =0}\right\}\) and let \(D = \sum H_i\). Any 2 planes intersect in a line and any 3 planes intersect in a point, so there are \({4\choose 2} = 6\) double curves \(C_{ij} \coloneqq H_i \cap H_j\) and \({4\choose 3} = 4\) triple points \(p_{ijk} \coloneqq H_i \cap H_j \cap H_k\). The dual complex is the standard tetrahedron:

\includegraphics[width=0.9\textwidth]{figures/dual_complex.jpg}

\end{example}

\begin{definition}[Incidence complex]

Let \((X, D)\) be a RNC compactification. The \textbf{incidence complex $I(D)$ of $D$} is the simplicial complex built in the following way: let \(D = \sum_i D_i\) be a decomposition into prime divisors, and take a complex \(I(D)\) whose \(k{\hbox{-}}\)dimensional cells are in bijection with irreducible components of \(k{\hbox{-}}\)fold intersections of the \(D_i\). The \textbf{colored incidence complex} is \(I(D)\) with an integer weight (or a coloring) attached to each 0-cell indicating the dimension of the corresponding stratum.

\end{definition}

\begin{remark}

In our case of interest, \((X, D)\) will be a Baily-Borel compactification of a moduli space where \(D \coloneqq\partial \overline{X}\) is a union of boundary strata of various dimensions. Because we primarily work with hyperbolic lattices, \(D\) will only contain strata of dimensions 0 and 1, i.e.~points and curves. Thus \(I(D)\) will reduce to a graph whose vertices are in bijection with points and curves in \(D\) and whose edges record when a point \(p_j\) is contained in the closure of a curve \(C_i\). We can thus form the colored incidence complex \(I(D)\) with two colors, taking points to be black and curves to be white.

\end{remark}

\begin{definition}[Cusp incidence diagrams]

Let \(\Omega_N\) be the period domain associated with a lattice \(N\) and let \(\Gamma \subseteq {\operatorname{O}}(N)\) be a finite-index subgroup. The Baily-Borel compactification \(\bbcpt{ \Omega_S/\Gamma}\) is a projective variety with a boundary stratification \[
\bbcpt{ \Omega_S/\Gamma} = \qty{\Omega_S/\Gamma} \cup \mathcal{I} \cup \mathcal{J},\qquad
\partial \bbcpt{ \Omega_S/\Gamma} = \mathcal{I} \cup \mathcal{J}
\] where

\begin{itemize}
\tightlist
\item
  \(\mathcal{I}\) is a set of points referred to as \emph{$0$-cusps}, which are in bijective correspondence with \(\Gamma\)-orbits of primitive isotropic 2-dimensional sublattices of \(N\), and
\item
  \(\mathcal{J}\) is a set of modular curves referred to as \emph{$1$-cusps}, which are in bijective correspondence with \(\Gamma\)-orbits of primitive isotropic 1-dimensional sublattices of \(N\).
\end{itemize}

We summarize below what information the colored incidence complex \(I(\partial \bbcpt{ \Omega_S/\Gamma})\) captures:

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}l|l|l@{}}
\toprule
Cusp Type    & Type II, $\cJ$                 & Type III, $\cI$                             \\ \midrule
Boundary Strata     & 1-cusps/curves $C_i$    & 0-cusps/points $p_j$                  \\
Vertex type & \tikz[baseline=-0.5ex]{\node[style=white node, label=left:{$C_i$}] at (0, 0) {};} & \tikz[baseline=-0.5ex]{\node[style=black node, label=left:{$p_j$}] at (0, 0) {};} \\
Sublattice Type & Isotropic lines $[{\mathbf{Z}}e] \in \isoGr_1(L)/\Gamma$ & Isotropic planes $[{\mathbf{Z}}e \oplus {\mathbf{Z}}f] \in \isoGr_2(L)/\Gamma$ \\ 
Subdiagram Type & Maximal parabolic & Elliptic \\
\bottomrule
\end{tabular}%
}
\caption{Cusp types}
\label{tab:cusptypes}
\end{table}

Moreover, we draw an edge between a black and white node to denote a point \(p_i\) contained in the closure of a curve \(C_j\): \[ 
\tikz[baseline=-0.5ex]{
\node[style=white node, label=$C_j$] (0) at (0, 0) {};
\node[style=black node, label=$p_i$] (1) at (3, 0) {};
\draw[style=plain edge] (0) to (1);
}
\]

\end{definition}

\begin{example}

Consider the following colored incidence diagram:

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=above:{$C_{12}$}] (1) at (0, 3) {};
        \node [style=black node, label=below:{$p_2$}] (2) at (3, 0) {};
        \node [style=white node, label=above:{$C_2$}] (3) at (3, 3) {};
        \node [style=black node, label=below:{$p_1$}] (4) at (0, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw (4) to (1);
        \draw (1) to (2);
        \draw (2) to (3);
    \end{pgfonlayer}
    \end{tikzpicture}
    \caption{A colored incidence diagram $I(D)$ for $D = \cI \cup\cJ$.}
    \label{fig:incidence_diag_N}
\end{figure}

This represents the boundary stratification of a Baily-Borel compactification for which \(\cI = \left\{{p_1, p_2}\right\}\) consists of two points, \(\cJ = \left\{{C_{12}, C_2}\right\}\) is two curves, where \(p_1, p_2\in \overline{C_{12}}\), \(p_2\in \overline{C_2}\), and \(p_1\not\in \overline{C_2}\). This can be represented by the following configuration of curves and points:

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=none] (0) at (0, 0) {};
        \node [style=none] (1) at (4, 3) {};
        \node [style=none] (2) at (0, 4) {};
        \node [style=none] (3) at (4, 0) {};
        \node [style=black node, label=above:{$p_{2}$}] (4) at (1, 2) {};
        \node [style=black node, label=above:{$p_{1}$}] (6) at (2.75, 3) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw [bend left=45] (0) edge["$C_{12}$" very near start] (1);
        \draw [bend left=60, looseness=0.50] (3) edge["$C_{2}$" near start] (2);
    \end{pgfonlayer}
    \end{tikzpicture}
    \caption{A configuration of curves and points representing $I(D)$ in \Cref{fig:incidence_diag_N}.}
    \label{fig:incidence_diag_N}
\end{figure}

\end{example}

\begin{remark}

Each 0-cusp \(p_i\) has an associated Vinberg diagram \(\mathcal{D}(p_i)\) \todo{Spell out which root system this is attached to? Yes, that would be a good idea.} whose maximal parabolic subdiagrams enumerate the 1-cusps \(C_{ij}\) adjacent to \(p_i\) in the incidence diagram.

\end{remark}

\begin{example}

The following figure shows the Vinberg diagram for the 0-cusp ???? associated to the lattice \(N \coloneqq(18, 0, 0)\):

\begin{figure}
    \centering
    \input{tikz/Vinberg1}
    \caption{Coxeter-Vinberg diagram}
    \label{fig:vin1}
\end{figure}

This has the following two maximal parabolic subdiagrams:

\begin{figure}
    \centering
    \input{tikz/Vinberg1Parabolic1}
    \caption{Caption}
    \label{fig:enter-label}
\end{figure}

\[ 
\tikzfig{Vinberg1Parabolic1}
\qquad 
\tikzfig{Vinberg1Parabolic2} 
\]

That there are exactly 2 such subdiagrams is reflected in the fact that the vertex ??? in the incidence diagram has valence 2.

\end{example}

\textbackslash end\{document\}

\[
\dynkin[Coxeter]{F}{4} = \dynkin[label, arrows=false, edge length=.75cm,o/.style=black]F4
\]

\begin{quote}
Conventions on Coxeter diagrams: https://file.notion.so/f/s/962e72a3-fa3a-45ec-84a3-2bb0e7699a7b/Alexeev-Nikulin\_Del\_Pezzo\_and\_K3\_Surfaces.pdf?id=7e4996b4-08e6-4bd2-ac49-8d23080f7577\&table=block\&spaceId=7cb2f7c7-7373-4d11-91ab-284625335dc8\&expirationTimestamp=1683905929597\&signature=fMZC5iiiI0MbbvXkvnI4JyGwzN6JjCSDIU7Z6itmcUE\&downloadName=AN06.pdf\#page=64\&zoom=auto,-155,146
\end{quote}

The interpretation of these Coxeter diagrams in terms of root systems:

Needs some notation from \cite{AN06}:

\[
\begin{array}{ll}
V(M) &\text {the light cone } V(M)=\left\{x \in M \otimes \mathbb{R} \mathrel{\Big|}x^2>0\right\} \text { of a hyperbolic lattice } M \\
V^{+}(X) &\text {the half containing polarization of the light cone } V\left(S_X\right) \\
\mathcal{L}(S)=V^{+}\left(S_X\right) / \mathbb{R}^{+} & \text {the hyperbolic space of a surface } S \\
W^{(2)}(M) & \text{the group generated by reflections in all $f \in M$ with $f^2=-2$} \\
W^{(4)}(M) & \text{the group generated by reflections in all $(-4)$ roots of $M$} \\
W^{(2,4)}(M) & \text{the group generated by reflections in all $(-2)$ and $(-4)$ roots of $M$} \\
\mathcal{M}^{(2)} & \text {a fundamental chamber of } W^{(2)}(S) \text { in } \mathcal{L}(S) \\
\mathcal{M}^{(2,4)} & \text {a fundamental chamber of } W^{(2,4)}(S) \text { in } \mathcal{L}(S) \\
P^{(2)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-2) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
P^{(4)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-4) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
(X, \theta) &\text{a $\mathrm{K}3$ with involution $\theta$} \\
X^\theta &\text{the fixed locus of an involution} \\
P(X)_{+I} & \text {the subset of exceptional classes of }(X, \theta)  \text{ of type } \rm{I}
\end{array}
\]

\subsection{Edge notation}\label{edge-notation}

\begin{itemize}
\item
  Vertices corresponding to different elements \(f_1, f_2 \in P\left(\mathcal{M}^{(2,4)}\right)\) are not connected by any edge if \(f_1 \cdot f_2=0\).
\item
  Simple edges of weight \(m\) (equivalently, by \(m-2\) simple edges if \(m>2\) is small): \[
  \tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \draw (0) to (1);
    \node [style=none] at (1.5, 0.4) {$m$};
  }
  \qquad \implies \qquad
  \frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2 \cos \frac{\pi}{m}, \quad m \in \mathbb{N}
    \]
\item
  Thick edges: \[
  \tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \draw [line width=2pt] (0) to (1);
  }
  \qquad \implies \qquad
  \frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2
    \]
\item
  Broken edges of weight \(t\): \[
  \tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=white node] (1) at (3, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f_1$};
    \node [style=none] (5) at (3, 0.4) {$f_2$};
    \node [style=none] at (1.5, 0.4) {$t$};
    \draw [dashed] (0) to (1);
  } 
  \qquad \implies \qquad
  \frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=t>2
    \]
\item
  A vertex corresponding to \(f \in P^{(4)}\left(\mathcal{M}^{(2,4)}\right)\) is black: \[
  \tikz{
    \node [style=black node] (0) at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
  } 
  \qquad \implies \qquad
  f^2 = -4?
    \]
\item
  It is white if \(f \in P^{(2)}\left(\mathcal{M}^{(2,4)}\right)\): \[
  \tikz{
    \node [style=white node] (0) at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
  } 
  \qquad \implies \qquad
  f^2 = -2?
    \]
\item
  It is double-circled white if \(f \in P(X)_{+I}\) (i.e.~it corresponds to the class of a rational component of \(\left.X^\theta\right)\). \[
  \tikz{
    \node [style=white node, double] at (0, 0) {};
    \node [style=none] (2) at (0, 0.4) {$f$};
  } 
  \qquad \implies \qquad
  ??
    \]
\end{itemize}

Interpreting this geometrically: consider the cycle of \(2\overline{k}\) white vertices cycling between plain and double-circled:

\begin{quote}
Second to last paragraph here: https://arxiv.org/pdf/2208.10383.pdf\#page=41\&zoom=140,99,262
\end{quote}

\begin{itemize}
\item
  See \cite{AE22}
\item
  Each edge on the outer cycle corresponds to \(\mathbb{P}^2\)
\item
  Single circle vertices (with odd \(i\)) corresponds to a line in \(\mathbb{P}^2\)
\item
  Double-circled vertices (with even \(i\)) correspond to conics on the \(\mathbb{P}^2\)
\item
  Explicit example worked out in \cite[\S 5]{AET23}.
\end{itemize}

\begin{quote}
\% https://arxiv.org/pdf/1903.09742.pdf\#page=28\&zoom=170,-70,442
\end{quote}

\begin{itemize}
\tightlist
\item
  It seems like that from the Coxeter diagram, you draw the fan of a toric surface, you compute the charge, and then, if this is not 24, you fix it by blowing up some non-torus fixed points along the toric boundary.
\end{itemize}

\section{Hermitian Symmetric Domains}\label{hermitian-symmetric-domains}

\subsection{Cusp correspondence for Hermitian symmetric domains}\label{cusp-correspondence-for-hermitian-symmetric-domains}

\begin{definition}[Symmetric spaces]

A \textbf{locally symmetric space} is a connected Riemannian manifold such that every \(x\in M\) is the fixed point of some involution \(\gamma_x\in \mathop{\mathrm{Isom}}(U_x)\), the real algebraic Lie group of holomorphic automorphisms of an open subset \(U_x\subseteq M\), which acts by \(-1\) on \(T_x M\). Equivalently, the covariant derivative of the curvature tensor vanishes, which is analogous to a constant curvature condition. It is a \textbf{symmetric space} if \(\gamma_x\) extends from \(\mathop{\mathrm{Isom}}(U)\) to \(\mathop{\mathrm{Isom}}(M)\).

If \(M\) is a symmetric space, then \(M \cong G/K\) where \(G = \mathop{\mathrm{Isom}}(M)\) is its group of isometries and \(K \coloneqq{\operatorname{Stab}}_G(x)\) is the stabilizer of any point \(x\in M\). We say a manifold \(M\) is \textbf{homogeneous} if \(M\cong G/K\) for some \(G\) and \(K\).

\end{definition}

\begin{remark}

Idea: Hermitian symmetric manifolds are manifolds that are homogeneous spaces such that every point has an involution preserving the Hermitian structure. These were first studied by Cartan in the context of Riemannian symmetric manifolds. They show up often as orbifold covers of moduli spaces, e.g.~polarized abelian varieties (with or without level structure), polarized K3 surfaces, polarized irreducible holomorphic symplectic manifolds, etc. There is a structure theorem: any Hermitian symmetric manifold \(M\) decomposes as a product \(M\cong {\mathbf{C}}^n/\Lambda \times M_c \times M_{nc}\) where \(\Lambda\) is some lattice, \(M_c\) is an HSM of compact type, and \(M_{nc}\) is an HSM of non-compact type. Every HSM of compact type is a flag manifold \(G/P\) for \(G\) a semisimple complex Lie group and \(P\) is a parabolic subgroup. Every HSM of non-compact type admits a canonical so-called Harish-Chandra embedding whose image is a bounded symmetric domain \(D\subseteq {\mathbf{C}}^N\) for some \(N\). Moreover, every HSM of non-compact type admits an associated Borel embedding into an associated HSM of compact type called its compact dual. Moreover, there is a Lie-theoretic classification of HSMs of compact and non-compact type -- they are all of the form \(G/K\) for \(G\) a simple compact (resp. non-compact) Lie group and \(K\leq G\) is a maximal compact subgroup with center isomorphic to \(S^1\cong {\operatorname{U}}_1({\mathbf{C}})\).

By the Harish-Chandra embedding, non-compact HSMs can be realized as bounded domains \(D\subseteq {\mathbf{C}}^N\) and admit a compactification by taking the closure \(\overline{D} \supseteq D\) in \({\mathbf{C}}^N\). There is a partition of \(\overline{D}\) by an equivalence relation related to being connected through chains of holomorphic discs, and each equivalence class is called a boundary component of \(D\). Boundary components are in bijection with their normalizer subgroups, which are precisely maximal parabolic subgroups of \(G\coloneqq\mathop{\mathrm{Aut}}(D)\).

A \textbf{Hermitian symmetric domain} is a Hermitian symmetric space of non-compact type.

\end{remark}

\begin{example}

Some very basic examples of Hermitian symmetric manifolds:

\begin{itemize}
\tightlist
\item
  Tori \({\mathbf{C}}/\Lambda\) with Hermitian structure \(g = dx dx + dy dy\) induced from \({\mathbf{R}}^2\) (constant zero curvature).
\item
  The upper half space \(\cH^1\) with Hermitian structure the hyperbolic metric \(g = y^{-2}dx dy\) (constant negative curvature)
\item
  \({\mathbf{P}}^1({\mathbf{C}})\) with the Fubini-Study metric (constant positive curvature).
\end{itemize}

More advanced examples of symmetric spaces:

\begin{itemize}
\tightlist
\item
  \({\mathbb{E}}^n\), Euclidean space \({\mathbf{R}}^n\).
\item
  \({\mathbb{S}}^n\), the spherical geometry,
\item
  \({\mathbb{H}}^n\), hyperbolic space,
\item
  \(\operatorname{Sym}_{n\times n}^{> 0}({\mathbf{R}}) \leq {\operatorname{SL}}_n({\mathbf{R}})\) the Riemannian manifold of positive-definite symmetric matrices with real entries
\item
  \(X\) defined in the following way: let \(V\) be a Hermitian \({\mathbf{C}}{\hbox{-}}\)module with Hermitian form \(h\) of signature \((p, q)\) and let \(X \subseteq {\operatorname{Gr}}_p(V)\) be the Grassmannian of \(p{\hbox{-}}\)dimensional subspaces \(W\) such that \({ \left.{{h}} \right|_{{W}} }\) is positive definite.
\end{itemize}

We first record their isometry groups:

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{Isom}}({\mathbb{E}}^n) = {\mathbf{R}}^n\rtimes{\operatorname{O}}_n({\mathbf{R}})\).
\item
  \(\mathop{\mathrm{Isom}}({\mathbb{S}}^n) = {\operatorname{O}}_{n+1}({\mathbf{R}})\)
\item
  \(\mathop{\mathrm{Isom}}({\mathbb{H}}^n) = {\operatorname{O}}_{n+1}^+({\mathbf{R}})\) the index 2 subgroup of \({\operatorname{O}}_{n+1}({\mathbf{R}})\) which preserves the upper sheet \(({\mathbb{H}}^n)^+\).\footnote{Note that for $n=1$, we can take the upper half-plane model which has isometry group ${\operatorname{PSL}}_2({\mathbf{R}})$ or the disc model which has isometry group ${\operatorname{PSU}}_{1, 1}({\mathbf{C}})$. These are actually isomorphic as Lie groups.}
\item
  \(\mathop{\mathrm{Isom}}(\operatorname{Sym}_{n\times n}^{> 0}({\mathbf{R}})) = {\operatorname{SL}}_n({\mathbf{R}})\).
\item
  \(\mathop{\mathrm{Isom}}(X) = {\operatorname{SU}}_{p, q}({\mathbf{C}})\)?
\end{itemize}

Computing stabilizers of points, one can show \begin{align*}     {\mathbf{R}}^n &\cong {{\mathbf{R}}^n\rtimes{\operatorname{O}}_n({\mathbf{R}}) \over  {\operatorname{O}}_n({\mathbf{R}})} \\     {\mathbb{S}}^n &\cong {{\operatorname{O}}_{n+1}({\mathbf{R}}) \over {\operatorname{O}}_n({\mathbf{R}})} \\     {\mathbb{H}}^n &\cong {{\operatorname{O}}_{n+1}^+({\mathbf{R}}) \over {\operatorname{O}}_n({\mathbf{R}}) } \\     \operatorname{Sym}_{n\times n}^{> 0}({\mathbf{R}}) &\cong {{\operatorname{SL}}_n({\mathbf{R}}) \over {\operatorname{SO}}_n({\mathbf{R}})}\\      X &\cong {{\operatorname{SU}}_{p, q}({\mathbf{C}}) \over {\operatorname{SU}}_p({\mathbf{C}}) \times {\operatorname{SU}}_q({\mathbf{C}})} \end{align*}

Note that taking \((p, q) = (1,1)\) yields \({\mathbb{H}}^2\).

\end{example}

\begin{definition}

\begin{quote}
\% Probably need to cite:https://dept.math.lsa.umich.edu/\textasciitilde idolga/EnriquesOne.pdf\#page=556\&zoom=160,-136,697
\end{quote}

A \textbf{Hermitian symmetric space} is a locally symmetric space \(M\) which is additionally equipped with an integrable almost-complex structure whose Riemannian metric is Hermitian.

We say \(M\) is \textbf{irreducible} if it is not the cartesian product of two symmetric Hermitian spaces; every irreducible such space is either \({\mathbf{R}}^n\) for some \(n\) or a homogeneous space \(G/K\) for \(G\) a real Lie group and \(K\) a maximal subgroup. We say \(M\) is of \textbf{compact type }if \(G\) is compact and \(K\) is a maximal proper subgroup, and of \textbf{non-compact type} if \(G\) is non-compact. If \(M\) is an irreducible Hermitian symmetric domain of non-compact type, there is an open embedding \(M\hookrightarrow\cD_L \subseteq {\mathbf{C}}^n\) onto a bounded subset \(\cD\) of complex \(n{\hbox{-}}\)space, in which case we call \(\cD\) a \textbf{bounded Hermitian symmetric domain}.

The simplest example is the upper half plane \({\mathbb{H}}\coloneqq G/K\) for \(G= {\operatorname{SL}}_2({\mathbf{R}})\) and \(K = {\operatorname{SO}}_2({\mathbf{R}})\), which is biholomorphic to the bounded domain \(\Delta\) via the Cayley transformation, which is a homogeneous space for \((G, K) = ({\operatorname{SU}}_{1, 1}, B)\) where \(B\) is the subgroup of diagonal matrices. Note that \({\mathbb{H}}\cong \cH_1\) is the Siegel upper half space of genus 1.

If \((V, q)\) is a real quadratic space where \(V \coloneqq L_{\mathbf{R}}\) for \(L\) a lattice, we can define a corresponding domain \[
\cD^\pm_L \coloneqq\left\{{{\mathbf{C}}z \in {\mathbf{P}}(V_{\mathbf{C}}) \mathrel{\Big|}z^2 = 0,\, {\left\lvert {z} \right\rvert} > 0}\right\}
,\] the set of lines spanned by isotropic vectors of positive Hermitian norm \({\left\lvert {z} \right\rvert} \coloneqq z\overline{z}\) in \(V_{\mathbf{C}}\). If \(\operatorname{sig}(L) = (2, n)\), so \(L\) is hyperbolic, this has an irreducible component decomposition into two parts \(\cD^\pm_L = \cD^+_L \coprod \cD^-_L\) interchanged by conjugation \(z\mapsto \overline{z}\). Each component is an irreducible Hermitian symmetric domain of type \[ 
(G, K) = ({\operatorname{SO}}(V) \coloneqq{\operatorname{SO}}_{2, n}({\mathbf{R}}), {\operatorname{SO}}_2({\mathbf{R}}) \times {\operatorname{SO}}_n({\mathbf{R}})).
,\] i.e.~a Type IV domain for \({\operatorname{SO}}_{2, n}\). We let \(\cD_L \coloneqq\cD_L^+\) be a choice of one component and write \({\operatorname{O}}^+(L) \leq {\operatorname{O}}(L)\) for the subgroup which preserves \(\cD_L\) setwise. There is a distinguished divisor attached to \(\cD_L\), the \textbf{discriminant divisor}: \[
\cH_L \coloneqq\bigcup_{v \in R_2(L)} H_v \cap\cD_L
,\] the hyperplane configuration defined by mirrors of roots. When Global Torelli is satisfied, there is a period map \(\phi\) whose image is typically the complement of some hyperplane arrangement \(\cH\). In good cases, the relevant arrangement is precisely \(\cH_L\).

Note that \(\cD_L\) is isomorphic to a flag variety \(G_{\mathbf{C}}/P\) for \(P\) some parabolic subgroup, and thus the compact form \(\tilde\cD_L\) is a projective algebraic variety containing \(\cD_L\).

We say \(\cD_L\) as above is a \textbf{Hermitian symmetric domain of orthogonal type} or a \textbf{type IV Hermitian symmetric domain} in Cartan's classification. The period domains of K3 and Enriques surfaces are examples of such Type IV domains for \(1\leq n \leq 19\).

\end{definition}

\begin{definition}

Let \(G\) be a simple linear algebraic group defined over \({\mathbf{Q}}\). We define \[
G({\mathbf{Z}}) \coloneqq\operatorname{GL}_n({\mathbf{Z}}) \cap G({\mathbf{Q}})
\] where we use the natural embedding of algebraic groups \(G\hookrightarrow\operatorname{GL}_n\) over \({\mathbf{Q}}\). A subgroup \(\Gamma \leq G({\mathbf{Q}})\) is \textbf{arithmetic} if \(\Gamma \cap G({\mathbf{Z}})\) has finite index in both \(\Gamma\) and \(G({\mathbf{Z}})\).

\end{definition}

\begin{definition}[Parabolic subgroups]

Let \(G\) be a linear algebraic group over \({\mathbf{Q}}\). We say \(P\leq G\) is a parabolic subgroup if \(G/P\) is a projective variety.

\end{definition}

\begin{remark}

As the notation suggests, there are other types of irreducible Hermitian symmetric domains. The following are some typical examples of the form \(\Gamma\backslash \Omega\) for various definitions of \(\Omega\):

\begin{itemize}
\tightlist
\item
  Type \(\rm III\): Siegel modular varieties corresponding to \(\Gamma \leq {\mathsf{Sp}}(\Lambda)\), the isometry group of a symplectic lattice, of rank \(n\geq 3\).
\item
  Type \(\rm IV\): Orthogonal modular varieties corresponding to \(\Gamma\leq {\operatorname{O}}^+(\Lambda)\), a connected component of the isometry group of a lattice of signature \((2, n)\) for \(n \geq 3\),
\item
  Type \({\rm I}_{n, n}\): Hermitian modular varieties/Hermitian upper half spaces. These are attached to \(\Gamma\leq {\operatorname{U}}(\Lambda)\) for \(\Lambda\) a Hermitian form \(q\) of signature \((n, n)\) with \(n\geq 2\). The compact dual is the Grassmannian \({\operatorname{Gr}}_{n, 2n}\).
\item
  Type \({\rm II}_{2n}\): Quaternionic modular varieties/quaternionic upper half spaces. These are attached to \(\Gamma\leq {\mathsf{Sp}}_{2n}(H)\) for \(H\) Hamilton's quaternions, attached to a skew-Hermitian space of dimension \(2n\) with \(n\geq 2\). The compact dual is the orthogonal Grassmannian \({\operatorname{O}}{\operatorname{Gr}}_{2n, 4n}\)
\end{itemize}

\dzg{Where do $\cH_g$ and $\Gamma\backslash {\mathbb{H}}^n$ fit in?}

\end{remark}

\begin{remark}

For \(\Lambda\) a lattice of signature \((2, n)\), the Hermitian symmetric domain attached to \(\Lambda\) is the following: define \(Q \subseteq {\mathbf{P}}(\Lambda_{\mathbf{C}})\) be the quadric cut out by \((\omega, \omega) = 0\), then \(\Omega_\Lambda\) is a choice of one of the two connected components of the open set \(Q\) defined by \((\omega, \overline{\omega}) > 0\). Letting \({\operatorname{O}}^+(\Lambda) \leq {\operatorname{O}}(\Lambda)\) be the subgroup preserving the component \(\Omega_\Lambda\) and \(\Gamma \leq {\operatorname{O}}^+(\Lambda)\) be any finite index subgroup, we obtain \[
X_\Lambda(\Gamma) \coloneqq\Gamma \backslash \Omega_\Lambda
.\] Embedding \(\Omega_\Lambda\) in its compact dual, it has 0 and 1-dimensional boundary strata, corresponding to 1 and 2-dimensional isotropic subspaces of \(\Lambda_{\mathbf{Q}}\). The BB compactification \(\bbcpt{X_\Lambda(\Gamma)}\) is the union of \(\Omega_\Lambda\) and these rational boundary components, quotiented by the action of \(\Gamma\), equipped with the Satake topology.

A toroidal compactification \(\torcpt{X_\Lambda(\Gamma)}\) is specificed by a finite collection of suitable fans \(\left\{{F_I}\right\}\), one for each 0-cusp (i.e.~each \(\Gamma{\hbox{-}}\)orbit of isotropic lines \(I\) in \(\Lambda_{\mathbf{Q}}\)).

For each \(I\) there is a tube domain realization given by taking the linear projection from the boundary point, which defines an isomorphism \[
\Omega_\Lambda/U(I)_{\mathbf{Z}}\cong U \subseteq T_I \coloneqq U(I)_{\mathbf{C}}/ U(I)_{\mathbf{Z}}
\]

\end{remark}

The partial compactifications for the 1-cusps are completely canonical, so the overall compactification is defined by gluing onto the boundary of \(X_\Lambda(\Gamma)\) certain natural quotients of all of these partial compactifications to obtain \(\torcpt{X_\Lambda(\Gamma)}\). This yields a compact algebraic space which is proper over \(\operatorname{Spec}{\mathbf{C}}\), and there is a natural morphism \(\torcpt{X_\Lambda(\Gamma)}\to \bbcpt{X_\Lambda(\Gamma)}\). :::

\begin{example}

\begin{verbatim}
Let $G \da \SL_2$ defined over $\QQ$ and let $\Gamma \leq \SL_2(\QQ)$ be an arithmetic subgroup. The (noncompact) modular curve attached to $\Gamma$ is
\[
Y(\Gamma) \da \Gamma\backslash \HH^1
.\]
In this case, rational boundary components are given by $\PP^1(\QQ) = \QQ\union \ts{\infty}\subseteq \PP^1(\CC)$, and a cusp of $Y(\Gamma)$ is a $\Gamma\dash$orbit in $\Gamma \backslash \PP^1(\QQ)$, of which there are finitely many. Adding them yields a compactification 
\[
X(\Gamma) \da \overline{Y(\Gamma)} \da Y(\Gamma) \union \ts{\text{cusps}}
\]
topologized appropriately, where e.g. $\ts{\infty}$ is one such cusp.

Note that one typically takes the following groups for moduli of elliptic curves with level structure:
\end{verbatim}

\begin{itemize}
\item
  \(Y(N) \coloneqq Y(\Gamma(N))\) where \[\Gamma(N) \coloneqq\ker\qty{\phi_N: {\operatorname{SL}}_2({\mathbf{Z}}) \to {\operatorname{SL}}_2({\mathbf{Z}}/N{\mathbf{Z}})}.\] The level structure is a basis for \(E[n]\).
\item
  \(Y_0(N) \coloneqq Y(\Gamma_0(N))\) where \(\Gamma_0(N) \supseteq \Gamma(N)\) is the pullback \(\phi_N^{-1}\qty{{ \begin{bmatrix} {a} & {b}  \\ {0} & {d} \end{bmatrix} }}\). The level structure is an identification \(\mu_N \hookrightarrow E_{\operatorname{tors}}\).
\item
  \(Y_1(N) \coloneqq Y(\Gamma_1(N))\) where \(\Gamma_1(N)\) is the pullback \(\phi_N^{-1}\qty{1b01}\). The level structure is a point \(p\in E\) of order \(N\) in the group structure.

  How parabolic subgroups appear here: for \(G \coloneqq{\operatorname{SL}}_2\), parabolic subgroups are all conjugate to the subgroup \(P\) of upper-triangular matrices, and \(G({\mathbf{Q}})/P({\mathbf{Q}}) \cong {\mathbf{P}}^1({\mathbf{Q}})\) parameterizes all such parabolic subgroups.

  Why automorphic forms matter: consider \(\Gamma \coloneqq{\operatorname{SL}}_2({\mathbf{Z}})\). The graded ring of modular forms \(\bigoplus_k M_k\) is graded-isomorphic to \({\mathbf{C}}[x,y]\) where \({\left\lvert {x} \right\rvert} = 4, {\left\lvert {y} \right\rvert} = 6\), and \(\mathop{\mathrm{Proj}}{\mathbf{C}}[x,y]\cong {\mathbf{P}}^1({\mathbf{C}})\). Letting \(\left\{{f_0, \cdots, f_N}\right\}\) be a basis of \(M_k\), we can write down a map \begin{align*}  \phi_k: Y({\operatorname{SL}}_2({\mathbf{Z}})) &\to {\mathbf{P}}^n({\mathbf{C}}) \\   z &\mapsto [f_0(z): \cdots : f_N(z)]    \end{align*} For \(k=12\) this separates points and tangent directions, giving a projective embedding. Explicitly, the morphism is \[
    \phi_{12}(z) = [E_4(z): E_4(z)^3 - E_6(z)^2] \approx j(z)
    \] modulo some missing constants. In general, finding enough automorphic forms yields a projective embedding. \dzg{Would like to spell this out in terms of line bundles and linear systems too, in this easy case.}
\end{itemize}

\end{example}

\subsection{Misc}\label{misc-1}

\begin{remark}

Let \(L\) be a lattice of signature \((2, n)\) and the associated period domain \(\Omega_L^\pm = \Omega_L^+ {\textstyle\coprod}\Omega_L^-\). Let \({\operatorname{O}}(L)^+\leq {\operatorname{O}}(L)\) be the finite index subgroup fixing \(\Omega_L^+\), equivalently the subgroup of elements of spinor norm one. A modular variety of orthogonal type is a homogeneous space of the form \(F_L(\Gamma) \coloneqq\Gamma\backslash \Omega_L^+\) for an arithmetic subgroup \(\Gamma \leq {\operatorname{O}}(L_{\mathbf{Q}})^+\).

By general theory, such spaces admit BB compactifications \(\bbcpt{F_L(\Gamma)}\) where rational maximal parabolic subgroups correspond to stabilizers of isotropic subspaces of \(L_{\mathbf{Q}}\); since \(\operatorname{sig}(L) = (2, n)\) these are always isotropic lines or planes.

For period spaces of K3 surfaces, one takes \(\Gamma  \coloneqq{\operatorname{O}}(L_{2d})\cap\ker\qty{{\operatorname{O}}(L) \to {\operatorname{O}}(A_{L})}\). Boundary strata correspond to central fibers of KPP models of Type II and Type III.

\end{remark}

\begin{remark}

Let \(L\) be a symplectic lattice of rank \(2g\), ie.e. a free \({\mathbf{Z}}{\hbox{-}}\)module with a nondegenerate alternating form \(({-}, {-})\). Define the associated period space \[
D_L \coloneqq\left\{{V\in {\operatorname{Gr}}_g(L_{\mathbf{C}}) \mathrel{\Big|}(V, V) = 0, \, i(V, \overline{V}) > 0}\right\} \cong {\mathsf{Sp}}_{2g}({\mathbf{R}})/ {\operatorname{U}}_g({\mathbf{C}}) \cong \cH^g
\] which is a Hermitian symmetric domain of type III that can be identified with the Siegel upper half-space of dimension \(g\). We can form the moduli space of PPAV as \[
{\mathcal{A}}_g \coloneqq{\mathsf{Sp}}_{2g}({\mathbf{Z}}) \backslash \cH^g \cong {\mathsf{Sp}}_{2g}({\mathbf{Z}}) \backslash {\mathsf{Sp}}_{2g}({\mathbf{R}}) / {\operatorname{U}}_g({\mathbf{C}})
\] Rational boundary components of \(\bbcpt{{\mathcal{A}}_g}\) correspond to \(\Gamma \coloneqq{\mathsf{Sp}}_{2g}({\mathbf{Z}})\) orbits of totally isotropic subspaces in \(L_{\mathbf{Q}}\). Since \(\Gamma\) acts transitively, such spaces are indexed by their dimension \(i = 0,1,\cdots, g\) and there is a stratification \[
\bbcpt{{\mathcal{A}}_g} = {\textstyle\coprod}_{k=0}^g {\mathcal{A}}_{k} \implies \partial \bbcpt{{\mathcal{A}}_g} = {\textstyle\coprod}_{k=0}^{g-1} {\mathcal{A}}_k
\]

\end{remark}

\begin{remark}

The BB compactification of a locally symmetric domain \(D\): write \(D = H/K\) as a homogeneous space where \(H \coloneqq\mathrm{Hol}(D)^+\) and \(K\leq H\) is a maximal compact subgroup. Then cusps in \(\partial \bbcpt{\Gamma\backslash D}\) correspond to rational maximal parabolic subgroups of \(H\). To get boundary components: apply the Harish-Chandra embedding to \(D\) to embed \(HC: D\hookrightarrow D^{cd}\) and let \(F_P \in \overline{HC(D)}\) be a boundary component. Its normalizer \(N(F_P)\coloneqq\left\{{g\in H \mathrel{\Big|}g(F_P) = F_P}\right\} \leq H\) is a maximal parabolic in \(H\). We say \(F_P\) is \textbf{rational} if \(N(F_P)\) can be defined over \({\mathbf{Q}}\). Since \(\Gamma\) preserves such rational \(F_P\), we can set \({{\partial}}D \coloneqq\) the disjoint union of all rational \(F_P\) and set \(\bbcpt{\Gamma\backslash D} = {D {\textstyle\coprod}{{\partial}}D \over \Gamma}\).

\end{remark}

\subsubsection{Explicit realizations of symmetric spaces}\label{explicit-realizations-of-symmetric-spaces}

\begin{remark}

The symmetric space associated with a Lie group \(G\) is in some sense the most natural space \(G\) acts on. For \(G={\operatorname{O}}_{p, q}({\mathbf{R}})\), the symmetric space is \({\operatorname{Gr}}^+({\mathbf{R}}^{p, q})\), the Grassmannian of maximal positive-definite subspaces of \({\mathbf{R}}^{p, q}\). The right choice of maximal compact subgroup here is \(K\coloneqq{\operatorname{O}}_p({\mathbf{R}}) \times {\operatorname{O}}_q({\mathbf{R}})\), the subgroup fixing \({\mathbf{R}}^{m, 0}\).\dzg{Typo maybe.} When \((p, q) = (2, n)\), these symmetric spaces admit special descriptions. Note that \({\operatorname{O}}_{n+1}({\mathbf{R}})\) is the group of isometries of \(S^n\), so its projectivization \({\mathbf{P}}{\operatorname{O}}_{n+1}({\mathbf{R}})\) is the isometry group of an elliptic geometry. One can similarly obtain isometries of hyperbolic geometry:

\begin{itemize}
\tightlist
\item
  Start with \({\mathbf{R}}^{1, n}\)
\item
  Take the norm 1 vectors \(H^\pm \coloneqq\left\{{v\in {\mathbf{R}}^{1, n} \mathrel{\Big|}v^2 = 1}\right\} = H^+ {\textstyle\coprod}H^-\) to get a 2-sheeted hyperboloid; the pseudo-Riemannian metric on \({\mathbf{R}}^{1, n}\) restricts to a Riemannian metric on \(H\).
\item
  Take one sheet \(H^+\); this is a model of \({\mathbb{H}}^{n}\)\dzg{Indexing might be off here}
\end{itemize}

The group of isometries of \(H^+\) is now \({\mathbf{P}}{\operatorname{O}}_{1, n}({\mathbf{R}})\). Note that in \({\operatorname{O}}_{1, n}({\mathbf{R}})\) there is an index 2 subgroup\footnote{ Apparently, these are elements whose spinor norm equals their determinant. } \[
{\operatorname{O}}_{1, n}({\mathbf{R}})^\pm = \left\{{\gamma \in {\operatorname{O}}_{1, n}({\mathbf{R}}) \mathrel{\Big|}\gamma(H^+) = H^+, \gamma(H^-) = H^-}\right\}
.\]

\end{remark}

\begin{remark}

Forming the symmetric spaces for \({\operatorname{O}}_{2, n}({\mathbf{R}})\): the maximal compact is \(K = {\operatorname{O}}_2({\mathbf{R}}) \times {\operatorname{O}}_n({\mathbf{R}})\) and \({\operatorname{O}}_2({\mathbf{R}})\) is similar enough to \({\operatorname{U}}_1({\mathbf{C}})\) that we should expect the associated symmetric space to be Hermitian. It will be an open subset of a certain quadric:

\begin{itemize}
\tightlist
\item
  Start with \({\mathbf{P}}({\mathbf{C}}^{2, n})\).
\item
  Take the quadric of isotropic vectors \(Y = \left\{{z\in {\mathbf{P}}({\mathbf{C}}^{2, n}) \mathrel{\Big|}z^2 = 0}\right\}\).
\item
  Take the open subset \(U \coloneqq\left\{{z\in Y \mathrel{\Big|}(z, \overline{z}) > 0}\right\}\).
\end{itemize}

Why this matches the previous description: write \(z = x+iy\), then \(x^2 = y^2 > 0\) and \((x, y) = 0\), so \(V\coloneqq{\mathbf{R}}x\oplus {\mathbf{R}}y\) are an orthogonal basis for a positive definite subspace of \({\mathbf{R}}^{2, n}\). Multiplying by a scalar only changes basis, so we essentially get a map \({\mathbf{P}}(U) \to {\operatorname{Gr}}^+({\mathbf{R}}^{2, n})\) naturally. This symmetric space can also be identified with points \(z\in {\mathbf{C}}^{1, n-1}\) with \(\Im(z) \in C^+\), one of two cones of \({\mathbf{R}}^{1, n-1}\), realizing this as a tube domain generalizing \({\mathbb{H}}\).

\end{remark}

\section{Hyperbolic Geometry}\label{hyperbolic-geometry}

\subsection{Hyperbolic lattices}\label{hyperbolic-lattices}

\dzg{Note: some of this mixes conventions, need to fix later.}

\begin{warning}

There is a significant gap in the AG literature vs the physics literature for the terminology for hyperbolic spaces, and the traditional AG terminology can be ``wrong'' in some senses. For example, the AG literature will typically call \(\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 > 0}\right\}\) a ``light cone'', but this is not quite correct: the actual \textit{light cone} in general relativity is \(\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = 0}\right\}\). The following picture is the usual mnemonic:

\begin{figure}[H]
    \centering
    \includegraphics[width=0.7\textwidth]{figures/timelike_lightlike.jpg}
    \caption{$\left\{{v^2=0}\right\}$ is the light cone, its interior is timelike and exterior spacelike.}
    \label{fig:enter-label}
\end{figure}

\end{warning}

\begin{definition}[Hyperbolic lattices]

An indefinite lattice \(L\) is a \textbf{hyperbolic lattice} \footnote{Also called a **Lorentzian lattice**.} if \(\operatorname{sig}(L) = (1, n_-)\) or \((n_+, 1)\) for some \(n_-, n_+ \geq 1\). By convention, by twisting \(L\) to \(L(-1)\) if necessary, we assume hyperbolic lattices have signature \((1, n)\). In this convention, the single positive-definite direction is referred to as \textbf{timelike}, and the remaining directions are \textbf{spacelike}.

\end{definition}

\begin{definition}[Time/light/spacelike vectors]

Let \(L\) be a hyperbolic lattice of signature \((1, n)\). We say a vector \(v\in L_{\mathbf{R}}\) is

\begin{itemize}
\item
  \textbf{timelike} if \(v^2 < 0\),
\item
  \textbf{lightlike} or \textbf{isotropic} if \(v^2=0\).
\item
  \textbf{spacelike} if \(v^2 > 0\), More generally, a subspace \(W \subseteq {\mathbb{E}}^{1, n}\) with the restricted form \(({-}, {-})_W\) is
\item
  \textbf{timelike} if \(({-}, {-})_W\) is negative-definite, or is indefinite and non-degenerate,
\item
  \textbf{lightlike} or \textbf{isotropic} if \(({-}, {-})_W\) is degenerate, or
\item
  \textbf{spacelike} if \(({-}, {-})_W\) is positive-definite.
\end{itemize}

Define \begin{align*}     L^{< 0} \coloneqq\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 < 0}\right\} &\quad \text{The timelike regime} \\     L^{= 0} \coloneqq\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = 0}\right\} &\quad \text{The lightlike regime} \\     L^{> 0} \coloneqq\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 > 0}\right\} &\quad \text{The spacelike regime} \end{align*}

\end{definition}

\begin{remark}

\cite{AE22nonsympinv} refers to the non-spacelike regime \(L^{\geq 0} \coloneqq\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 \geq 0}\right\}\) as the \textbf{round cone}; this is used for a model over \(\overline{{\mathbb{H}}^n}\) with ideal points included, and is often used as the support of a semifan for a semitoroidal compactification.

\end{remark}

\begin{definition}[Past and future light cones]

\label{def:past-future-lightcones} Let \(L\) be a hyperbolic lattice of signature \((1, n)\). The spacelike regime \(L^{> 0}\) of \(L\) has an irreducible component decomposition \[
L^{> 0 } \coloneqq\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 > 0}\right\} = C_L^+ \amalg C_L^-
, \] whose components we refer to as the \textbf{future light cone} and \textbf{past light cone} of \(L\) respectively, and can be distinguished by the sign of the coordinate in the negative-definite direction: \[
C^+_L \coloneqq\left\{{v\in L^{ >0} \mathrel{\Big|}v_{0}  > 0 }\right\},\qquad 
C^-_L \coloneqq\left\{{v\in L^{ >0} \mathrel{\Big|}v_{0}  < 0 }\right\}
.\] We write their closures in \(L_{\mathbf{R}}\) as \(\overline{C_L^+}\) and \(\overline{C_L^-}\) respectively, and write \(C_L \coloneqq C_L^{+}\) for a fixed choice of a \textbf{future} light cone and \(\overline{C_L}\) for its closure.

\end{definition}

\subsection{Models of hyperbolic space}\label{models-of-hyperbolic-space}

\begin{definition}[Euclidean upper-half space]

The upper-half space in \({\mathbb{E}}^n\) is \[
{\mathbb{E}}^n_+ \coloneqq\left\{{(x_1,\cdots, x_{n}) \in {\mathbb{E}}^{n} \mathrel{\Big|}x_1 > 0}\right\}
.\]

\end{definition}

\begin{definition}[Minkowski space]

\label{def:minkowski} The \textbf{\(n{\hbox{-}}\)dimensional Minkowski space} \({\mathbb{E}}^{1, n}\) is the real vector space \({\mathbf{R}}^{n+1}\) equipped with a bilinear form of signature \((1, n)\) which can be explicitly written as \[
vw \coloneqq-v_0 w_0 + \sum_{i=1}^n v_i w_i
\] with the associated quadratic form \[
v^2 \coloneqq Q(v) \coloneqq-v_0^2 + \sum_{i=1}^n v_i^2
.\] This induces a metric \[
\rho(v, w) \coloneqq\mathrm{arccosh}(-vw)
.\]

\end{definition}

\begin{remark}

If \(L\) is hyperbolic of signature \((1, n)\) then \(L_{\mathbf{R}}\cong {\mathbb{E}}^{1, n}\) is a Minkowski space of dimension \(n+1 = \operatorname{rank}_{\mathbf{Z}}L\).

\end{remark}

\subsubsection{Half-plane models}\label{half-plane-models}

\begin{quote}
https://arxiv.org/pdf/1908.01710.pdf\#page=40\&zoom=auto,-95,626
\end{quote}

\begin{definition}[de Sitter space and light cone of a lattice]

Let \(L\) be a hyperbolic lattice and consider the squaring functional \begin{align*} f_L: L_{\mathbf{R}}&\to {\mathbf{R}}\\ v & \mapsto v^2 .\end{align*} One can show that \(\pm 1\) are regular values of \(f_L\) and thus define two canonical ``hyperbolic unit spheres'' which are regular surfaces. We define the \textbf{de Sitter space of \(L\)} as \[
\mathrm{dS}_L \coloneqq f_L^{-1}(1) = \left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = 1 }\right\} \subseteq L^{ > 0}
\] in the spacelike regime and the \textbf{unit hyperboloid of \(L\)} as the two-sheeted hyperboloid \[
H_L \coloneqq f_L^{-1}(-1) = \left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = -1}\right\} \subseteq L^{< 0}
\] in the timelike regime.

\end{definition}

\begin{example}

\Cref{fig:deSitter} shows the de Sitter space and unit hyperboloid for a lattice \(L\) of signature \((2, 1)\) in \({\mathbb{E}}^{2, 1}\), visualized in \({\mathbf{R}}^3\).

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/deSitterHyperboloid.png}
    \caption{The hyperbolic unit spheres: the de Sitter space and light cone for ${\mathbb{E}}^{2, 1}$.}
    \label{fig:deSitter}
\end{figure}

\end{example}

\begin{definition}[Half-plane model/Lobachevsky space of  a lattice]

Let \(L\) be a hyperbolic lattice. The \textbf{half-plane model of \({\mathbb{H}}^n\) associated to \(L\)} or \textbf{Lobachevsky space of \(L\)} is the unit hyperboloid of \(L\) intersected with its future light cone, \[
\bL^n_L \coloneqq H_L \cap C_L \coloneqq\left\{{v \in L_{\mathbf{R}}\mathrel{\Big|}v^2 = -1, v_{0} > 0}\right\} 
,\] given the metric restricted from \(L_{\mathbf{R}}\cong {\mathbb{E}}^{n,1}\). This more simply be described as the future sheet of the unit hyperboloid \(H_L\), using the irreducible component decomposition \[
H_L = H^+_L \amalg H^-_L = \left\{{v\in H_L \mathrel{\Big|}v_0 > 0}\right\} \amalg \left\{{v\in H_L \mathrel{\Big|}v_0 < 0}\right\}
\] and setting \(\bL^n_L \coloneqq H_L^+\).

\end{definition}

\begin{remark}

Note that \(H_L^+\) is in the timelike regime. This gives a model of the hyperbolic space \({\mathbb{H}}^n\) which we often denote \({\mathbb{H}}_L\) when we do not fix a specific choice of model, or simply by \({\mathbb{H}}^n\) when the dependence on \(L\) is not important.

\end{remark}

\begin{remark}[The isometry group of hyperbolic spaces]

It can be shown that the isometries of the timelike regime \(L^{< 0}\) are restrictions of isometries of the ambient Minkowski space \({\mathbb{E}}^{1, n}\), and thus \[
\mathop{\mathrm{Isom}}(L^{ < 0}) \cong \mathop{\mathrm{Isom}}({\mathbb{E}}^{1, n}) \cong {\operatorname{O}}_{1, n}({\mathbf{R}})
.\] Using the half-plane model, we can thus naturally identify \[
\mathop{\mathrm{Isom}}(\bL^n) \cong {\operatorname{O}}^+_{1, n}({\mathbf{R}}) \coloneqq{\operatorname{Stab}}_{{\operatorname{O}}_{1, n}({\mathbf{R}})}(C_L)
,\] the index 2 subgroup which stabilizes the future light cone \(C^+_L\) of \(L\). These are precisely the isometries of \({\mathbb{E}}^{1, n}\) of positive spinor norm.

\end{remark}

\subsubsection{Ball models}\label{ball-models}

\begin{definition}[The Poincar'e ball model]

Let \(L\) be a hyperbolic lattice. The \textbf{Poincar\textquotesingle e ball model of \({\mathbb{H}}^n\) associated to \(L\)} is defined as \[
\bB^n_L \coloneqq{\mathbf{P}}(L^{ < 0})
,\] the projectivization of the timelike regime of \(L\), where \begin{align*} {\mathbf{P}}({-}): {\mathbb{E}}^{1, n}\setminus\left\{{x_{n}\neq 0}\right\} &\to {\mathbb{E}}^{n} \\ (x_0,\cdots, x_{n-1}, x_n) &\mapsto \qty{ {x_0\over x_n}, \cdots, { x_{n-1} \over x_n} } .\end{align*} In this model, there is a natural compactification \(\overline{{\mathbb{H}}^n}\) in \({\mathbf{P}}(S^n)\) such that the interior is given by \(\bB^n_L\) as above and the boundary by \(\partial \overline{{\mathbb{H}}^n} = {\mathbf{P}}(L^{=0})\), i.e.~ideal points correspond to (the projectivization of) the lightlike regime.

\end{definition}

\begin{remark}[An alternative construction]

It can be explicitly constructed by considering the future light cone \(C_L\) described in \autoref{def:past-future-lightcones}. Letting \({\mathbf{R}}_{> 0}\) act on \(L_{\mathbf{R}}\cong {\mathbb{E}}^{1, n}\) by scaling along the timelike direction (i.e.~in the coordinate \(v_0\)), the ball model can be formed as the quotient \[
\BB^n_L \cong C_L/{\mathbf{R}}_{> 0} \subset {\mathbf{P}}({\mathbb{S}}^n)
.\]

\end{remark}

\begin{remark}

The advantage of \(\bB^n_L\) over \(\bL^n_L\) is that the former provides a natural compactification in \({\mathbf{P}}({\mathbb{S}}^n)\). Moreover, it can be easier to work with hyperplanes in the ball model: let \(\pi: {\mathbb{E}}^{1, n}\ \to {\mathbf{P}}({\mathbb{S}}^n)\) be the natural projection, then every hyperplane \(H_v \coloneqq v^\perp\) for \(v\in \bB^n_L\) is of the form \[
H_v = \left\{{\pi(x) \mathrel{\Big|}x\in C_L,\, xv = 0}\right\}
\] One can also concretely interpret the bilinear form geometrically in the ball model in the following way: \begin{align*}     H_v \pitchfork H_w \implies {\left\lvert {vw} \right\rvert} < 1 &     \implies -vw = \cos(\angle(H_v, H_w)) \\     H_v \parallel H_w \implies {\left\lvert {vw} \right\rvert} = 1      &\implies -vw = \cos(\angle(H_v, H_w)) \\     H_v \diverge H_w \implies {\left\lvert {vw} \right\rvert} > 1      &\implies -vw = \cosh(\rho(H_v, H_w)), \end{align*} where \(\rho\) is the hyperbolic metric described in \Cref{def:minkowski}.

\end{remark}

\begin{remark}

\(\mathop{\mathrm{Isom}}(\bB^n) = {\mathbf{P}}{\operatorname{O}}_{1, n}({\mathbf{R}})\).

\end{remark}

\subsubsection{Ideal points}\label{ideal-points}

\begin{remark}

Let \(H_L \cong {\mathbb{H}}^n\) be a model of hyperbolic space associated to a hyperbolic lattice \(L\) of signature \((1, n)\). Boundary points \(\partial\overline{H_L}\) correspond to ideal points in \({\mathbb{H}}^n\), i.e.~points ``at infinity'', which in turn correspond to 1-dimensional isotropic subspaces of \(L\).

\includegraphics{figures/hyperboloid.png}

In this model, points in \({\mathbb{H}}^n\) are points in the interior of the cone and on the hyperboloid. Moreover points on \(\partial{\overline{{\mathbb{H}}^n}}\) correspond to points on the surface of the cone: \[
\partial\overline{{\mathbb{H}}^n} \cong \left\{{v = (v_0,\cdots, v_{n+1}) \in L_{{\mathbf{R}}} \mathrel{\Big|}v_0 > 0}\right\} \cap\left\{{v\in L_{\mathbf{R}}\mathrel{\Big|}v^2 = 0}\right\}
.\] We interpret \(uv = -\cos(\angle(H_u H_v))\), so \(uv = -1\) means \(H_u \cap H_v \in \partial{\overline {\mathbb{H}}^n}\), i.e.~they are ``parallel'' planes. Hyperplanes in \({\mathbb{H}}^n\) correspond to branches of hyperbolas obtained by slicing the hyperboloid by a plane in \(L_{\mathbf{R}}\).

\end{remark}

\begin{remark}

Define Minkowski space as \({\mathbb{E}}^{1, n}\), which is \({\mathbf{R}}^n\) with the form \(vw = v_0w_0 -\sum v_i w_i\). Define Lobachevsky space \(\bL^n\) as the hyperboloid model of hyperbolic space, a certain ``hyperbolic unit sphere'': \[
\bL^n \coloneqq\left\{{v\in {\mathbb{E}}^{1, n} \mathrel{\Big|}v^2 = 1, v_0 > 1}\right\}
.\] The geodesic curves are precisely intersections of the form \(H_2 \cap\bL^n\) where \(H_2\in {\operatorname{Gr}}_2({\mathbf{R}}^{n+1})\) is a standard 2-plane passing through the origin in the ambient space. The hyperbolic metric on \(\bL^n\) is gotten by computing the length in the standard metric in \({\mathbf{R}}^{n+1}\) of any geodesic curve between two points. The associated Poincare ball model is contained in the standard Euclidean ball \(\bB^n \subset {\mathbf{R}}^{n+1}\) and is the projection of \(\bL^n\) onto the hyperplane \(\left\{{x_0 = 0}\right\} \subset {\mathbf{R}}^{n+1}\) using rays passing through \((-1, 0, 0,\cdots, 0)\). Explicitly, the projection is \begin{align*} \phi: \bL^n &\to \bB^n \\ (v_0, \cdots, v_n) &\mapsto {1\over 1 + v_0}(v_1,\cdots, v_n) \end{align*} Geodesics are now straight lines through the origin or arcs of Euclidean circles intersecting \(\partial \bB^n\) orthogonally. Define the hyperbolic upper-half-space as \[
{\mathbb{H}}^n \coloneqq\left\{{x = (x_1,\cdots, x_n) \in {\mathbf{R}}^n \mathrel{\Big|}x_1 > 0 }\right\}
\] which is obtained by taking inversions through certain spheres centered on \(\partial \bB^n\). Geodesics are now straight lines orthogonal to \(\partial {\mathbb{H}}^n\) or half-circles centered on \(\partial {\mathbb{H}}^n\).

\end{remark}

\subsection{Root Systems}\label{root-systems}

\begin{definition}[Primitive vectors]

\begin{verbatim}
Let $L$ be any lattice. A finite set $S \da \ts{s_1,\cdots, s_n}\subseteq L$ of elements in $L$ is **primitive** if $S$ is $\RR\dash$linearly independent and $L\intersect \RR S = \bigoplus_{i=1}^n L s_i$, i.e. no $s_i$ can be replaced with a small vector in the same 1-dimensional subspace which is also in $L$. A primitive set of size one is called a **primitive element**, and we write $L_{\prim}$ for the set of such.
\end{verbatim}

\end{definition}

\begin{definition}[Roots and $k$-roots in lattices]

Let \(L\) be any lattice. For \(k \in {\mathbf{Z}}_{> 0}\), define the set of \textbf{\(k{\hbox{-}}\)roots in \(L\)} as \[
\Phi_k{L} \coloneqq\left\{{v\in L_{{\operatorname{prim}}} \mathrel{\Big|}v^2 = k,\, 2(v, L) \subseteq k{\mathbf{Z}}}\right\}
\] A \textbf{root} is by definition a \(2{\hbox{-}}\)root. We write the set of roots in \(L\) as \(\Phi(L)\), and the complete set of roots as \[
\Phi_\infty{L} \coloneqq\bigcup_{k\geq 1} \Phi_k{L}
.\]

\end{definition}

\begin{remark}

In the theory of 2-elementary lattices, the roots consist of all \((-2){\hbox{-}}\)vectors along with any \((-4){\hbox{-}}\)vector \(v\) with \(\operatorname{div}(v) = 2\).

\end{remark}

\begin{definition}[Reflections]

Let \(L\) be any lattice and \(L_{\mathbf{R}}\) its associated \({\mathbf{R}}{\hbox{-}}\)module. An element \(s\in \operatorname{GL}(L_{\mathbf{R}})\) is a \textbf{reflection} if there exists a vector \(v\in L_{\mathbf{R}}\) and an \({\mathbf{R}}{\hbox{-}}\)linear functional \(f\in { \operatorname{Hom} }_{\mathbf{R}}(L_{\mathbf{R}}, {\mathbf{R}})\), both depending on \(s\), \[
s(x) = x - f(x)v \quad \forall x\in L_{\mathbf{R}},\qquad f(v) = 2
.\] Concretely, \(s\) is an isometry of \(L_{\mathbf{R}}\) which pointwise fixes a hyperplane and is an involution satisfying \(\operatorname{det}(s) = -1\). Every reflection can be written in the form \[
s(u) = s_v(u) = u - {uv \over v^2/2 }v
\] for some \(v^2\neq 0\) in \(L_{\mathbf{R}}\) determined up to scaling. The reflection in \(v\) is only well-defined when \(2 \operatorname{div}(v) \in v^2 {\mathbf{Z}}\) where \(\operatorname{div}(v)\) is the divisibility of \(v\) defined in \autoref{def:divisibility}. The \textbf{reflection hyperplane} associated to \(s\) is the fixed subspace \[
H_v \coloneqq\ker(f) = \ker(\operatorname{id}- s) \cong v^\perp
.\]

\end{definition}

:::\{.remark\} Alternatively: \(s\in \operatorname{GL}_n({\mathbf{C}})\) is a quasi-reflection if it 1 eigenvalue \(\lambda \neq 1\) with an eigenspace of dimension 1 and the remaining eigenvalues all 1. It is a reflection if \(\lambda = -1\). \textbackslash end\{remark\}

\begin{definition}[Mirrors in hyperbolic lattices]

Any root \(v\in \Phi(L)\) defines a reflection \(s_v\) through the mirror \[
H_v \coloneqq v^\perp \coloneqq\left\{{x\in C^+_L \mathrel{\Big|}xv = 0}\right\}
.\] If \(H_v\) is the reflection hyperplane of a root, we say it is a \textbf{mirror} in \(L\). Note that \(H_v\) is nonempty if and only if \(v^2 < 0\).

\end{definition}

\begin{definition}[Weyl group]

Let \(L\) be any lattice. The \textbf{Weyl group of \(L\)} is defined as the group generated by reflections in \(2{\hbox{-}}\)roots, \[
\weylgroup{L} \coloneqq\left\{{s_v \mathrel{\Big|}v\in \Phi(L) }\right\} \leq {\operatorname{O}}_L({\mathbf{R}})
\]

\end{definition}

\begin{definition}[The discriminant locus]

For \(L\) a hyperbolic lattice, define the \textbf{discriminant locus of \(L\)} as the union of all mirrors of \(2{\hbox{-}}\)roots, \[
\Delta(L)
\coloneqq\bigcup_{v\in \Phi(L)} v^\perp
\coloneqq\bigcup_{v\in \Phi(L)} H_v
.\]

\end{definition}

\begin{definition}[Weyl chambers]

The \textbf{chamber decomposition} of \(C_L\)\dzg{Forgot to write down what is $C_L$.} is defined as \[
C_L^\circ \coloneqq C_L \setminus \Delta(L) = C_L \setminus \qty{ \bigcup_{\delta \in \Phi(L)} \delta^\perp}
,\] the complement of all mirrors. This further decomposes into connected components called \textbf{Weyl chambers}: fixing a chamber \(P\), there is a decomposition into orbits \[
C_L^\circ = \amalg_{s_v \in \weylgroup{L} } s_v(P)
.\]

\end{definition}

\begin{remark}

Any Weyl chamber \(P\) is a simplicial cone, so the orbit decomposition yields a decomposition of \(C_L^\circ\) into simplicial cones. Since \(W\) acts on the set of Weyl chambers \(\pi_0 C_L^\circ\) simply transitively and the closure \(\overline P\) of any chamber is a fundamental domain for this action, there is a homeomorphism \(\overline P \cong C_L^\circ/W\).

\end{remark}

\begin{quote}
https://www.ms.u-tokyo.ac.jp/preprint/pdf/2007-12.pdf\#page=7\&zoom=100,88,601
\end{quote}

\begin{definition}[Fundamental chamber]

Let \(P\) be a Weyl chamber of \(L\), define \begin{align*} \Phi(L)^+ &\coloneqq\left\{{v \in \Phi(L) \mathrel{\Big|}(v, P) > 0 }\right\} \\ \Phi(L)^- &\coloneqq\left\{{v\in \Phi(L) \mathrel{\Big|}(v, P) < 0 }\right\} = -\Phi(L)^+ \end{align*} which induces a decomposition \[
\Phi(L) = \Phi(L)^+ \amalg  \Phi(L)^-
.\]

\end{definition}

\begin{remark}

Thus \(P\) can be written as \[
P = \left\{{v\in C_L \mathrel{\Big|}(v,\Phi(L)^+ ) > 0 }\right\} = \left\{{}\right\}
.\] This realizes \(P\) as an intersection of positive half-spaces and thus as a polytope.

\end{remark}

\begin{definition}[Walls]

Let \(\overline P\) be the closure in \(L_{\mathbf{R}}\) of \(P\). We say a mirror \(H_v \subseteq L_{\mathbf{R}}\) for \(v\in \Phi(L)^+\) is a \textbf{wall of \(P\)} if \(\operatorname{codim}_{L_{\mathbf{R}}}(H_v\cap\overline{P}) = 1\).

\end{definition}

\begin{definition}[Simple systems]

Let \(P\) be a Weyl chamber of \(P\) and let \[ 
\Pi(L, P) \coloneqq\left\{{v\in \Phi(L) \mathrel{\Big|}H_v \text{ is a wall of } P}\right\}
\] be the set of walls of \(P\). We can more economically define \(P\) by \[
P = \left\{{v\in C_L \mathrel{\Big|}\qty{ v, \Pi(L, P) } > 0}\right\}
,\] \dzg{Todo, messed up notation here a bit.} where no inequality is redundant. Moreover, \((P, \Pi(L, P))\) forms a Coxeter system, and \(\overline{P}\) is a fundamental domain for \(W(L) \curvearrowright L_{\mathbf{R}}\).

\end{definition}

\begin{definition}[Chambers and $\Orth^+(L)$]

The connected components of \[
V^+_L \coloneqq\left\{{x\in L^\pm \mathrel{\Big|}(\Phi(L), x)\neq 0}\right\}
\] are called \textbf{chambers} of \(L\). Any positive isometry preserves \(L^+\) and \(L^-\) set-wise, motivating the definition of the \textbf{group of positive isometries of \(L\)} \[
{\operatorname{O}}^+(L) \coloneqq\left\{{\gamma\in {\operatorname{O}}(L) \mathrel{\Big|}\gamma(L^+) = L^+, \gamma(L^-) = L^-}\right\}
\]

\end{definition}

\begin{definition}[Positive isometries]

\begin{verbatim}
We say an isometry $\gamma \in \Orth(L)$ is **positive** if it preserves a chamber (i.e. a connected component of $V^+_L$)
\end{verbatim}

\end{definition}

\begin{definition}[Roots, root systems, root lattices]

A vector \(v\in L\) is a \textbf{root} if \(v^2 = - 2\)\footnote{ One occasionally calls any time-like vector $v^2 < 0$ a "root", in which case distinguishes between e.g. $(-2){\hbox{-}}$roots $\Phi_2(L)$ and $(-4){\hbox{-}}$roots $\Phi_4(L)$.}, and we write \(\Phi(L)\) for the set of roots in \(L\). If \(L\) is negative definite and \(L = {\mathbf{Z}}\Phi(L)\)\footnote{i.e. if the roots form a ${\mathbf{Z}}{\hbox{-}}$generating set for $L$}, we say \(L\) is a \textbf{root lattice}. Any root lattice decomposes as a direct sum of root lattices of ADE type.

\end{definition}

\begin{definition}[Weyl group]

The \textbf{Weyl group} of \(L\) is the maximal subgroup of the orthogonal group of \(L\) generated by hyperplane reflections in roots, \[
\weylgroup{L}^2 \coloneqq \left\langle s_v \mathrel{\Big|}v\in \Phi(L)  \right\rangle _{\mathbf{Z}}\leq {\operatorname{O}}(L)
.\]

One can similarly define the group of reflections in \textit{all} vectors, \[
\weylgroup{L}\coloneqq \left\langle s_v \mathrel{\Big|}v\in L \right\rangle _{\mathbf{Z}}\unlhd {\operatorname{O}}(L)
.\] Since conjugating a reflection by any automorphism is again a reflection, this is a normal subgroup. If \(L\) is a hyperbolic lattice, we replace \({\operatorname{O}}(L)\) in the above definition by \({\operatorname{O}}^+(L)\), the isometries that preserve the future light cone.

\end{definition}

\begin{definition}[Mirrors/walls and chambers]

The \textbf{mirror} or \textbf{wall} associated with a root \(v\in \Phi(L)\) is the hyperplane \(H_v \coloneqq v^\perp\). As \(v\) ranges over \(\Phi(L)\), these partition \(L_{\mathbf{R}}\) into subsets called \textbf{chambers}. The Weyl group acts on \(L_{\mathbf{R}}\) by isometries and acts simply transitively on chambers, and we often distinguish a fundamental domain for this action called the \textbf{fundamental chamber}. We write \(D_L\) for the closure in \(L_{\mathbf{R}}\) of a fundamental chamber. A \textbf{cusp} of \(L\) is a primitive isotropic lattice vector \(e\in   D_L \cap L\).

\end{definition}

\subsection{General Period Domains}\label{general-period-domains}

\begin{remark}

Define \(G^{L} \coloneqq G_{L^\perp}\) for \(G\) any algebraic group determined by \(L\).\dzg{Very useful convention: $\Omega^S$ involves $S^\perp$, while $\Omega_S$ involves just $S$.}

Let \(L\leq \lkt\) be a sublattice of signature \((1, r-1)\) so \(\operatorname{sig}(L^\perp) = (2, 19-r+1)\). One can always form the period domain corresponding to \(L{\hbox{-}}\)polarized K3 surfaces as \[
\Omega^{L} \coloneqq\Omega_{L^\perp} \coloneqq\left\{{x\in (L^\perp)_{\mathbf{C}}\mathrel{\Big|}x^2 = 0, x\overline{x} > 0}\right\},
\] The period domain can be described as a Hermitian symmetric space: \[
\Omega^{L} \cong 
{{\operatorname{SO}}^{L}({\mathbf{R}}) \over {\operatorname{SO}}_2({\mathbf{R}}) \times {\operatorname{SO}}_{20-r}({\mathbf{R}}) }
\] \textgreater{} http://content.algebraicgeometry.nl/2020-5/2020-5-021.pdf\#page=10\&zoom=auto,-85,607

For any arithmetic subgroup \(\Gamma\leq {\operatorname{O}}^{L}({\mathbf{R}})\) there is a complex-analytic isomorphism \[
\Gamma\backslash \Omega^L \cong 
\qty{\Gamma \cap{\operatorname{SO}}^{L}({\mathbf{R}})} \backslash \Omega^L
.\] In particular, for \(L\) a primitive sublattice of \(\lkt\), letting \(F_{L}\) be the stack of \(L{\hbox{-}}\)polarized K3 surfaces, the period map \(\tau_L\) yields an open immersion \[
\tau_L: F_{L}({\mathbf{C}}) \hookrightarrow\widetilde{{\operatorname{SO}}^L}({\mathbf{Z}}) \backslash \Omega^L
\] where \(\widetilde{{\operatorname{SO}}^L}\) are isometries of \(L\) which extend to an isometry of \(\lkt\) which fixes \(L\).

For \(\operatorname{sig}L = (2, n)\) let \(G_{L} \coloneqq{\operatorname{SO}}_{L_{\mathbf{Q}}}\) be its associated rational isometry group and let \(\bX\coloneqq\Omega_L\) the associated Hermitian symmetric space as above, forming a Shimura datum \((\bX, G) \coloneqq(\Omega^L, {\operatorname{SO}}_{L_{\mathbf{Q}}})\). We can then realize \[
\mathrm{Sh}_L({\mathbf{C}}) \coloneqq\mathrm{Sh}_{\bK_L}[G_L, \bX_L]({\mathbf{C}}) \cong \widetilde{{\operatorname{SO}}_L}({\mathbf{Z}}) \backslash \bX_L
\] where \(\bK_L \coloneqq\ker\qty{G_L \to \mathop{\mathrm{Aut}}(A_{L}) }(\widehat{{\mathbf{Z}}})\), the admissible morphism in \(G_L(\bA_f)\); the stack \(\mathrm{Sh}_{\bK}[G, \bX]\) is a certain well-known quotient stack attached to a Shimura datum \((G, \bX)\) and a choice of a compact open subgroup \(\bK \leq G(\bA_f)\) of the finite adeles.

Defining the compact dual: \[
\Omega^{L, \mathrm{cd}} \coloneqq\left\{{x\in (L^\perp)_{\mathbf{C}}\mathrel{\Big|}x^2 = 0}\right\}
.\]

\end{remark}

\section{Appendix}\label{appendix}

\subsection{Dynkin Diagram}\label{dynkin-diagram}

\begin{remark}

The following is a table of the classical and affine Dynkin diagrams:

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}llll@{}}
\toprule
\multicolumn{2}{l}{Classical Type}          & \multicolumn{2}{l}{Affine Type}                              \\ \midrule
$A_n$ & \dynkin[label, labels={1,2,n-1, n}, edge length=.75cm]A{}   & $\tilde A_n$ & \dynkin[extended, label, labels={0,1,2,n-1, n}, edge length=.75cm]A{}        \\
$B_n$ & \dynkin[label, labels={1,2,,n-1, n}, edge length=.75cm]B{}  & $\tilde B_n$ & \dynkin[extended, label, labels={0,1,2,3,n-2, n-1, n}, edge length=.75cm]B{} \\
$C_n$ & \dynkin[label, labels={1,2,, n-1, n}, edge length=.75cm]C{} & $\tilde C_n$ & \dynkin[extended, label, labels={0,1,2,n-2, n-1, n}, edge length=.75cm]C{}   \\
$D_n$ & \dynkin[label, labels={1,2,,,n-1,n}, edge length=.75cm]D{}  & $\tilde D_n$ & \dynkin[extended, label, labels={0,1,2,3,,,n-1,n}, edge length=.75cm]D{}     \\
$E_6$ & \dynkin[label, edge length=.75cm]E6 & $\tilde E_6$ & \dynkin[extended, label, edge length=.75cm]E6 \\
$E_7$ & \dynkin[label, edge length=.75cm]E7 & $\tilde E_7$ & \dynkin[extended, label, edge length=.75cm]E7 \\
$E_8$ & \dynkin[label, edge length=.75cm]E8 & $\tilde E_8$ & \dynkin[extended, label, edge length=.75cm]E8 \\
$F_4$ & \dynkin[label, edge length=.75cm]F4 & $\tilde F_4$ & \dynkin[extended, label, edge length=.75cm]F4 \\
$G_2$ & \dynkin[label, edge length=.75cm]G2 & $\tilde G_2$ & \dynkin[extended, label, edge length=.75cm]G2 \\ \bottomrule
\end{tabular}%
}
\caption{Classical and Affine Dynkin Diagrams}
\label{tab:dynkin-diagrams-table}
\end{table}

\end{remark}

\subsection{Images}\label{images}

\subsubsection{Tikz}\label{tikz}

\input{tikz/coble-coxeter-diagram-maximal-parabolic.tikz}

\input{tikz/coble-enriques-cusp-correspondence.tikz}

\input{tikz/e10_coxeter_diagram.tikz}

\input{tikz/e8_coxeter_diagram.tikz}

\input{tikz/enriques-cusp-coxeter-diagrams-maximal-parabolics.tikz}

\input{tikz/enriques-cusp-coxeter-diagrams.tikz}

\input{tikz/enriques-cusp-diagram-detailed.tikz}

\input{tikz/mirror-moves-coble.tikz}

\input{tikz/mirror-moves-enriques-simplified.tikz}

\begin{figure}[H]
\centering
\input{tikz/nikulin_table.tikz}
\caption{Nikulin's table of 2-elementary lattices.}
\label{fig:nikulin_table}
\end{figure}

\input{tikz/sterk-2-ias2-template.tikz}

\subsubsection{SVG}\label{svg}

\includesvg{inkscape/Hyperbolic Cone.svg}

\includesvg{inkscape/ias-sterk2-kulikov-model.svg}

\includesvg{inkscape/triangulated-sphere.svg}

\includesvg{inkscape/triangulated-sphere-fan.svg}

\includesvg{inkscape/type-ii-kulikov-degeneration.svg}

\printbibliography %Prints bibliography

\end{document}
