# 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 $\Kthree$ surface^[A $\Kthree$ surface is a smooth projective surface $X$ with $K_X = 0$ and $h^1(\OO_X) = 0$]. 
For both $\Kthree$ 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 $\Kthree$ 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 $\Kthree$ surfaces (see \cite{ABE22,AE21,AE22nonsympinv,AEH21,AET23}). 
In particular, there are explicit and effective answers to this question for $\Kthree$ 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 $4\dash$dimensional subfamily of the moduli space. 
One of the obstructions to extending similar results to the entire $10\dash$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 **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 $\rho: X\to \PP^1\times\PP^1$ branched over a divisor $D\in -2K_{\PP^1\times\PP^1}$ 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.


# Preliminaries

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

## Lattices

:::{.remark}
By a \emph{lattice} we mean a finitely generated free abelian group $L$ of finite rank equipped with a nondegenerate symmetric bilinear form $b\colon L\times L\rightarrow \ZZ$. 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 $b(v,v)\in2 \ZZ$ for all $v\in L$. Given a lattice $L$ we denote by $L^*$ its dual $\mathrm{Hom}_{\ZZ}(L,\ZZ)$. 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 $q_L\colon A_L\rightarrow \QQ/\ZZ$ by sending $v+L\mapsto b(v,v)~\textrm{mod}~\ZZ$.
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 $A_L\cong \ZZ_2^a$ for some positive integer $a$ are called **$2\dash$elementary**.

:::

## K3 surfaces, and nonsymplectic involutions

:::{.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 $H^2(X,\ZZ)$, endowed with the cup product, is an even, unimodular lattice of signature $(3,19)$. It follows that $H^2(X,\ZZ)$ 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 $\lkt$.

:::

:::{.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 $H^2(X,\ZZ)$ and we denote by $S$ its $(+1)\dash$eigenspace. It turns out that $S$ is a hyperbolic $2\dash$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 $S$, $A_S\cong \ZZ_2^a$, and $\delta$ is the so-called \emph{coparity} of $L$: $\delta=0$ provided $q_L(v)\equiv0~\mathrm{mod}~\ZZ$, and $\delta=1$ otherwise.

:::

# Moduli via period domains

## A general construction
\label{sec:period_domain}

:::{.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 $T \da S^{\perp \Lambda}$ its orthogonal complement in $\Lambda$. Define the **period domain associated to $S$** to be
\[
\Omega_S^{\pm} \da \ts{[v] \in \PP(S \tensor_\ZZ \CC)  \mid v^2=0 ~\text{and}~ v\overline{v} = 0}
.\]
As a matter of notation, we also set
\[
\Omega^S \da \Omega_{S^\perp} \da \Omega_T
.\]
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 $\operatorname{Gal}(\CC/\RR)$. We fix a choice of component $\Omega^+_S$, and let $\Orth(S)^+ \leq \Orth(S)$ be the subgroup fixing this component. We then form a locally symmetric space and a corresponding Baily-Borel compactification
\[
F(S) \da \leftquotient{
\Orth^+(S)
}{
\Omega^+_S
}
\qquad
\bbcpt{F(S)} \da \bbcpt{\leftquotient{
\Orth^+(S)
}{
\Omega^+_S
}}
.\]
More generally, one can let $\Gamma$ be any neat arithmetic group that acts properly discontinuously on $S\Omega^+_S$. One can then similarly form
\[
F(S, \Gamma) \da \leftquotient{\Gamma}{\Omega^+_S}, \qquad \bbcpt{ F(S, \Gamma) } \da \bbcpt{
\leftquotient{\Gamma}{\Omega^+_S}
}
.\]
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
\[
\mathcal{H}_{-2} \da 
\qty{\Union_{\delta\in \Phi_{N}} \delta^\perp}\intersect \Omega^+_N
= \bigcup_{\substack{
\delta\in N,\\\delta^2=-2}}
\ts{ [v]\in\Omega_N^+ \mid v \cdot \delta = 0}
.\]
and define
\[
F(S, \Gamma, \cH_{-2})
\da 
\leftquotient{\Gamma}{
\qty{\Omega^+_S \setminus \cH_{-2}}}, 
\qquad 
\bbcpt{ F(S, \Gamma, \cH_{-2} ) } 
\da 
\bbcpt{
\leftquotient{\Gamma}{
\qty{\Omega^+_S \setminus \cH_{-2} } }
}
\]
:::

## Generally finding cusps

:::{.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 $\partial F(S, \Gamma, \cH_{-2}) = \Union_{i\geq 0} \partial F(S, \Gamma, \cH_{-2})_i$ for a stratification of the boundary by $i\dash$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 $\Gamma\dash$orbits of primitive isotropic lines and 1-cusps to orbits of isotropic planes.
:::

## Moduli of K3 surfaces with nonsymplectic involution

:::{.remark title="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 $H^2(X; \ZZ) \cong\lkt$. Fix a marking $\varphi: H^2(X; \ZZ)\to \lkt$ and a polarization $L$ of degree $2d$, and let $h \da \varphi([L]) \in \lkt$. One can then show that $h^\perp\cong L_{2d}$.
Let 
\[
\Stab_{\Orth(\lkt)}(h) \da \ts{\gamma \in \Orth(\lkt) \mid \gamma(h) = h}
\] 
be the stabilizer of $h$ in $\lkt$ and define 
\[
\Gamma_h \da {\Stab}_{\Orth(\lkt)}(h)^+
\] 
to be the finite index subgroup fixing $\Omega_{\Lambda_{2d}}^+$. Letting $\cF_{2d}^{\mathrm{qp}}$ be the moduli stack of quasi-polarized $\Kthree$ 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, $\cF_{2d}^{\mathrm{qp}}$ is generally not a separated stack. We can instead use the stack $\cF_{2d}^{\mathrm{ADE}}$ of polarized $\Kthree$s with ADE singularities, since there is an isomorphism $F_{2d}^{\mathrm{ ADE}} \cong F_{2d}^{\mathrm{qp}}$ at the level of coarse spaces.
:::

:::{.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 $2\dash$elementary sublattice which is the $(+1)\dash$eigenspace of an involution $\rho$ of $\lkt$. A \emph{$\rho\dash$marking} of $(X,\iota)$ is an isometry $\varphi\colon H^2(X,\ZZ)\rightarrow \lkt$ such that $\iota^*=\varphi^{-1}\circ\rho\circ\varphi$.
Fix such a marking $\rho$. We have a **period domain** $\Omega_S^+$ associated to $S$, and we define the change-of-marking group associated to $\rho$ to be
\[
\Gamma_\rho=\ts{ \gamma\in \Orth(\lkt)\mid\gamma\circ\rho=\rho\circ\gamma}
.\]
One can then show that the coarse moduli space of $\rho\dash$markable K3 surfaces is analytically isomorphic to the locally symmetric space
\[
F_S \da F(S^\perp, \Gamma_\rho, \cH_{-2}) \da
\leftquotient{\Gamma_\rho}{\qty{\Omega_{S^\perp}\setminus\cH_{-2}}}
.\]
In particular, the point corresponding to $(X,\iota)$ is $[\varphi(\mathbb{C}\omega_X)]$. 
:::

## Hodge theoretic compactifications

:::{.remark}
Hodge theory provides different ways to compactify $\Omega_{S^\perp}/\Gamma$ for any finite index subgroup $\Gamma\subseteq \Orth(S^\perp)$. 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 $0\dash$cusps and $1\dash$cusps which correspond to $\Gamma\dash$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 $C_{K,\QQ}$ 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
$\Gamma\dash$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.

:::

# Hyperelliptic $\Kthree$s

:::{.definition}
Let $X$ be a $\Kthree$ surface and let $L\in \Pic(X)$ be a line bundle with $L^2 > 0$ where the linear system $\abs{L}$ has no fixed components. We say that $\abs{L}$ is a **hyperelliptic linear system on $X$** and $X$ is a **hyperelliptic $\Kthree$ surface** if $\abs{L}$ contains a hyperelliptic curve.
:::

:::{.remark}
The induced morphism $\varphi_{\abs L}: X\to \PP^g$ where $L^2 = 2g-2$ in this case is a generally 2-to-1 morphism onto a surface $F$ of degree $g-1$ in $\PP^g$. By the classification of surfaces, either $F \cong \PP^2$ or $\FF_n$^[A Hirzebruch surface $\FF_n \da \Proj_{\PP^1} (\OO_{\PP^1}(-n) \oplus \OO_{\PP^1})$.] with $n\in \ts{0,1,2,3,4}$ ramified over a curve $C \in \abs{-2 K_F}$.

:::

:::{.remark}
The open locus of $\Mhe$ can be realized using the standard construction of $L\dash$polarized $\Kthree$ surfaces, taking $L = U\sumpower{2} \oplus D_{16}$. More generally, degree $n$ hyperelliptic $\Kthree$ surfaces can be constructed by taking $L = U\sumpower{2} \oplus D_{n-2}$.
:::

## Hyperelliptic quartic K3s

:::{.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 $2\dash$elementary even lattice $S$ is given by $U(2)$, which corresponds to the invariants $(r,a,\delta)=(2,2,0)$.

The Baily--Borel compactification $\overline{\Omega_{S^\perp}/\Gamma}^{\mathrm{bb}}$ ... 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.
:::

:::{.remark}
Following \cite{LO21}, consider the period domain construction described in \autoref{sec:period_domain} using the lattice $\Lambda_N \da U\sumpower{2}\oplus D_{N-2}$ and $\Gamma = \Orth(\Lambda_N)^+$.^[For moduli-theoretic purposes, if $N\equiv 6\mod 8$, one then instead passes to a finite index subgroup as detailed in \cite{LO21}.]
We then obtain a sequence of locally symmetric spaces
\[
\mathcal{F}(N) \da F(\Lambda_N, \Orth(\Lambda_N)^+) \da \leftquotient{\Orth(\Lambda_{N})^+ }{\Omega_{\Lambda_{N}}^+ }
.\]
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 $\Lambda_{18} \injects \Lambda_{19}$ induced by $D_{16} \injects D_{17}$ 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 
\[
\Mhe \cong \operatorname{Chow}_{2,4}\gitquot \SL_4
,\] 
a GIT quotient of the Chow variety of $(2, 4)$ curves in $\PP^3$.

> See section 2.1 here https://arxiv.org/pdf/1801.04845.pdf#page=6&zoom=auto,-87,319

:::

:::{.theorem title="{\cite[Theorem~2.3]{LO21}}"}
The Baily--Borel compactification
\[
\Mhe \cong
\bbcpt{ 
\leftquotient{ \Orth(\Lambda_{18})^+ }{ \Omega_{\Lambda_{18} }^+ }
}
\]
has two $0\dash$cusps (type III boundary components) and eight $1\dash$cusps (type II boundary components). The incidences between $0\dash$cusps and $1\dash$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 $\Kthree$ surfaces $\Mhe$ .}
\label{fig:cusp-diagram-bb-deg-4-hyper-K3}
\end{figure}

:::

:::{.remark}
> See https://arxiv.org/pdf/2006.06816.pdf#page=1&zoom=100,-274,431

If $C \subseteq \PP^1\times\PP^1$ is a smooth curve of bidegree $(4, 4)$ and $\pi: X_C: \to \PP^1\times\PP^1$ is the double cover branched along $C$, then $X_C$ is a smooth hyperelliptic polarized $\Kthree$ surface of degree 4 and thus $X_C\in \Mhe$. Letting $M \da \abs{\OO_{\PP^1\times\PP^1}(4, 4)} \gitquot \Aut(\PP^1\times\PP^1)$ be the GIT quotient, LO21 describes a birational period map $M\rational \Mhe$.
:::

:::{.remark}
The K3 surfaces parameterized by $\overline{\mathbf{K}}$ are double covers of $\PP^1\times\PP^1$ 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 $\PP^1\times\PP^1$ 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 $\PP^{24}$ be the space of coefficients, up to scaling, for a bidegree $(4,4)$ polynomial in $\PP^1\times\PP^1$. In this case, a monomial $X_0^iX_1^jY_0^kY_1^\ell$ is indexed by
\[
M_{\mathrm{h}}:=\{(i,j,k,\ell)\in \ZZ_{\geq 0}^4\mid i+j=k+\ell=4\}.
\]
Let $\mathbf{U}_{\mathrm{h}}\subseteq\PP^{24}$ 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
\[
\left(\mathcal{X}_{\mathrm{h}}:=\mathbf{U}_{\mathrm{h}}\times(\PP^1\times\PP^1),\frac{1+\epsilon}{2}\mathcal{B}_{\mathrm{hyp}}\right)\rightarrow\mathbf{U}_{\mathrm{hyp}},
\]
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 $\PP^1\times\PP^1$ acts on $\mathbf{U}_{\mathrm{h}}$ identifying isomorphic fibers. In particular, $\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}(\PP^1\times\PP^1)$ 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 $18\dash$dimensional moduli space $\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}(\PP^1\times\PP^1)$. 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 $\Kthree$ surfaces. We have that $\overline{\mathbf{K}}_{\mathrm{h}}\cong\overline{\mathbf{U}}_{\mathrm{h}}$.


:::

:::{.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 $\mathbf{U}_{\mathrm{h}}/\mathrm{Aut}(\PP^1\times\PP^1)$? Recall $G=\mathbb{G}_m^2\rtimes(\ZZ/2\ZZ)$.}
:::

:::{.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}.

:::


# Enriques surfaces

## The unpolarized case

:::{.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 $\Kthree$ surface and $Y\cong X/\iota$ for $\iota$ the basepoint-free involution swapping the sheets of the cover. We write $V_{+1}(\iota^*), V_{-1}(\iota^*) \subseteq H^2(X; \ZZ)$ for the $(+1)$ and $(-1)\dash$eigenspaces respectively of the induced involution in cohomology $\iota^*\colon H^2(X; \ZZ)\rightarrow H^2(X; \ZZ)$. It is well-known that $V_{+1}(\iota)^{\perp H^2(X; \ZZ)} = V_{-1}(\iota)$.
The covering map $\pi$ induces an embedding of lattices
\[
\pi^*\colon H^2(Y; \ZZ)\hookrightarrow H^2(X; \ZZ),
\]
whose image is $V_{+1}(\iota^*)$. It is well known that

- $H^2(X;\ZZ) \cong \lkt \cong U^{\oplus3}\oplus E_8^{\oplus2}$ is the \emph{ $\Kthree$ lattice};
- $M \da H^2(Y,\ZZ)/\tors\cong \lEn \cong U\oplus E_8$ is the \emph{Enriques lattice};
- $V_{+1}(\iota^*)\cong U(2)\oplus E_8(2) \cong \lEn(2)$;
- $V_{-1}(\iota^*)\cong U\oplus U(2)\oplus E_8(2)=U\oplus \lEn(2)$.

:::

:::{.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 $U=\ZZ e \oplus \ZZ f$ with $\ts{e, f}$ the standard hyperbolic basis satisfying $e^2=f^2=e\cdot f-1=0$.

:::

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

:::

:::{.lemma title="{Torelli for Enriques surfaces, Horikawa}"}

Points in $\Munpolop$ correspond to isomorphism classes of unpolarized Enriques surfaces.
:::

:::{.theorem title="{\cite[Propositions~4.5 and 4.6]{Ste91}}"}
The Baily--Borel compactification
\[
\Munpol \da 
\bbcpt{
\leftquotient{\Orth^+(N)}{(\Omega_N^+ \setminus\cH_{-2})}
}
\]
has two $0\dash$cusps and two $1\dash$cusps. The incidences between $0\dash$cusps and $1\dash$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}

:::

## 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.

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

:::

:::{.definition}
A **numerical (resp. ample numerical) polarization** on $Y$ is a choice $L$ of a numerical equivalence class of pseudo-ample (resp. ample) line bundle $L$. 
:::

:::{.definition}
A **numerically polarized Enriques surface** is a pair $(Y, L)$ where $Y$ is an Enriques surface and $L$ is a numerical polarization on $Y$.^[Why introduce **numerical** polarizations? Recall that $A$ is a polarized abelian variety if it is equipped with an isogeny $\lambda: A\to A\dual$. 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.]
:::


:::{.remark}
Let $L\da U\oplus \lEn\sumpower{2}$, 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 $\mu: H^2(X; \ZZ)\to L$ such that $I\circ \mu = \mu \circ I^*$ and produces an embedding
\begin{align*}
M &\to L \\
m &\mapsto (m, m, 0)
.\end{align*}

> 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

Define
\[
\Gamma'\coloneqq\{g\in \Orth(L)\mid g\circ I=I\circ g~\textrm{and}~g(e+f,e+f,0)=(e+f,e+f,0)\}
,\] 
automorphisms in the centralizer of $I$ in $\Orth(L)$ fixing the point $(e+f, e+f, 0)$. If $g\in \Gamma'$ then $\ro{g}{N} \in \Orth(N)$.
\dzg{this is what Sterk claims, needs proof.} 
So define 
\[
\Gamma\da \{\ro{g}{N} \mid g\in\Gamma'\}\leq O(N)
,\]
which is the image of $\Gamma'$ in $\Orth(L_-)$.
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
\[
\Mpolop \cong 
F(N, \Gamma) =
\leftquotient{\Gamma}{\Omega_N^+}
.\]
:::

:::{.theorem title="{\cite[\S 4.3]{Ste91}}"}
The Baily--Borel compactification 
\[
\Mpol \da
\bbcpt{
\leftquotient{\Gamma}{\Omega_N^+}
}
\] has five $0\dash$cusps and nine $1\dash$cusps. The incidences between $0\dash$cusps and $1\dash$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}

:::

## Numerically polarized
\cite{GH16}

## The family of degree $2$ polarized Enriques surfaces

:::{.remark}
We review the construction of degree $2$ polarized Enriques surfaces following \cite[Chapter~V, \S\,23]{BHPV04}. Let us consider the involution on $\PP^1\times\PP^1$ 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 $B\subseteq\PP^1\times\PP^1$ be a general $\iota\dash$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 $\iota$.


:::

:::{.remark}
The double cover $\pi\colon T\rightarrow\PP^1\times\PP^1$ branched along $B$ is a well known to be a $\Kthree$ surface: $T$ is smooth and minimal, $K_T\sim\pi^*\left(K_{\PP^1\times\PP^1}+\frac{1}{2}B\right)\sim0$, and $\pi_*\OO_T=\OO_{\PP^1\times\PP^1}\oplus\OO_{\PP^1\times\PP^1}\left(-\frac{1}{2}B\right)$, which gives $h^1(\OO_T)=0$.

:::

:::{.remark}
Let $\mathcal{L}^{\otimes2}=\OO_{\PP^1\times\PP^1}(4,4)$ and let $p\colon L\rightarrow\PP^1\times\PP^1$ be the total space of the line bundle $\mathcal{L}$. Then the double cover $T$ of $\PP^1\times\PP^1$ 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 $\OO_{\PP^1\times\PP^1}(1,1)$.

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:
\[
K_T+\epsilon R\sim_{\QQ}\pi^*\left(K_{\PP^1\times\PP^1}+\frac{1+\epsilon}{2}B\right),
\]
\[
K_T+\epsilon R\sim_{\QQ}q^*\left(K_S+\frac{\epsilon}{2}\overline{R}\right).
\]

:::

:::{.lemma}
With the notation introduced above, we have the following self-intersection numbers:

- $\left(K_{\PP^1\times\PP^1}+\frac{1+\epsilon}{2}B\right)^2=8\epsilon^2$;

- $(K_T+\epsilon R)^2=16\epsilon^2$;

- $\left(K_S+\frac{\epsilon}{2}\overline{R}\right)^2=8\epsilon^2$.

:::

:::{.remark}
We now relativize the above construction. Let $\PP^{12}$ 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 $[\ldots:c_{ijk\ell}:\ldots]\in\PP^{12}$ with $(i,j,k,\ell)$ within the following set:
\[
M\da\{(i,j,k,\ell)\in \ZZ_{\geq0}^4\mid i+j=k+\ell=4,~i+k\equiv j+\ell\equiv0~\mathrm{mod}~2\}.
\]
Let $\mathbf{U}\subseteq\PP^{12}$ be the dense open subset of coefficients such that the corresponding $\iota\dash$invariant $(4,4)$ curve $B\subseteq\PP^1\times\PP^1$ is smooth and does not pass through the torus fixed points of $\PP^1\times\PP^1$. Define $\mathcal{X}\da\mathbf{U}\times(\PP^1\times\PP^1)$ and let $\mathcal{X}\rightarrow\mathbf{U}$ be the projection. Let
\[
\mathcal{B}\da V\left(\sum_{(i,j,k,\ell)\in M}c_{ijk\ell}X_0^{i}X_1^{j}Y_0^{k}Y_1^{\ell}\right)\subseteq\mathcal{X}.
\]
Then $\left(\mathcal{X},\frac{1+\epsilon}{2}\mathcal{B}\right)\rightarrow\mathbf{U}$ is a family of stable pairs with fibers given by $\left(\PP^1\times\PP^1,\frac{1+\epsilon}{2}B\right)$ 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}.

:::

:::{.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 $\mathrm{Aut}(\PP^1\times\PP^1)\cong(\mathrm{PGL}_2\times\mathrm{PGL}_2)\rtimes \ZZ/2 \ZZ$ (see \cite{Dol12}) on $H^0(\OO(4,4))$. 
More precisely, we want to look at the subgroup $G$ which preserves $\iota\dash$invariant $(4,4)\dash$curves not passing through the torus fixed points. 
Note that the $\ZZ_2\dash$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 $G\cong\mathbb{G}_m^2\rtimes(\ZZ/2\ZZ)$. 
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}

:::

:::{.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 $\Kthree$ and Enriques surfaces.
:::

:::{.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.


:::


# Morphisms of moduli and cusps

## Mapping the boundaries: matching cusp diagrams

:::{.remark}
Let $\cF_{2d}$ be the moduli space of polarized $\Kthree$ 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 $\Kthree$ surfaces? This is actually quite subtle, and it works as follows:

- \cite[\S\,5]{Ste91} matches cusps for degree $2$ Enriques surfaces and degree $4$  $\Kthree$ surfaces,
\[
\Mpol \mapstofrom \bbcpt{\cF_4}
.\]

- Using Scattone's method in \cite[\S\,6]{Sca87}, we can match the cusps for degree $4$ hyperelliptic  $\Kthree$ surfaces and degree $4$  $\Kthree$ surfaces,
\[
\Mhe \mapstofrom \bbcpt{\cF_4}
.\]

- The above two points imply the matching we need.

:::

:::{.remark}
What is Scattone's method uses the following observations:

- $1\dash$cusps of $\bbcpt{ \cF_4 }$ are in one-to-one correspondence with the orthogonal complements of $D_8$ in the Niemeier lattices, and
- The $1\dash$cusps of $\Mhe$ are in one-to-one correspondence with the orthogonal complements of $D_7$ in the Niemeier lattices.

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

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

- Polarized $\ts{X_i, p_i}$ to unpolarized $\ts{Y_i, q_i}$:

\input{tikz/cusp_correspondence_one}

- Polarized $\ts{X_i, p_i}$ to hyperelliptic $\ts{Z_i, r_i}$:
    
\input{tikz/cusp_correspondence_two}

:::

## Type III KPP model at the (18,2,0) odd 0-cusp

:::{.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 $\Kthree$ surface with a nonsymplectic involution $\iota$ with induced involution $i^*$ on $H^2(X; \ZZ)$. We define $S$ to be the $(+1)\dash$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 $H^2(X;\ZZ)$.

:::

:::{.definition}
What makes an **odd** $0\dash$cusp different from an **even** $0\dash$cusp?
:::

:::{.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 $\PP^1\dash$fibration $V_i\rightarrow\PP^1$. 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).
:::

:::{.theorem title="{\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,

- If $\overline{T}$ is an odd $0\dash$cusp of $F_S$, then $\overline{X}_0$ is of pumpkin type.

- If $\overline{T}$ is an even $0\dash$cusp of $F_S$, then $\overline{X}_0$ is of smashed pumpkin type. The surfaces $V_i$ of the last type in definition Definition~9.8, 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 "$+$".

:::

## ADE surfaces

:::{.definition}
An **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. $L\da -2(K_Y+C)$ is referred to as the **polarization** of the ADE surface $(Y,C)$.
:::

:::{.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 **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 $M\tensor_\ZZ \RR$.
Given a surface $(Y, C)$ of pure shape, the irreducible components of $C$ are called **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 **long** or **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~1,~2,~3 therein). Here are some basic examples:

- The ADE surface $(Y, C)$ corresponding to $D_4$ is $Y=\PP^1\times\PP^1$ and $C$ is the sum of two incident torus fixed curves.

- The ADE surface $(Y, C)$ corresponding to $A_1$ is $Y=\PP^2$ and $C$ is the sum of two torus fixed curves. The polarization is $\OO(2)$.

:::

:::{.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 $(-2)\dash$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}.


:::

# Our new results

## Enriques strategy

:::{.remark}
This story suggests the following approach to Enriques surfaces:

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

## Lemmas/theorems

:::{.lemma}
If $\signature(L) = (p, q)$ and $e\in L$ is isotropic, then $\signature(\ZZ e) = (?, ?)$ and $\signature(\ZZ e^\perp) = (?, ?)$.
:::

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

:::{.proposition}
Let $\lkt$ be the K3 lattice and let $h$ be an ample class of degree $d$. Then
\[
L_{2d} \da h^{\perp \lkt } \cong \gens{-2d} \oplus U\sumpower{2}\oplus E_8\sumpower{2}
\]
and $\signature L_{2d} = (2, 19)$.
Thus $F_{2d}$ arises as the Hodge-theoretic moduli space associated with the period domain $D_L$ for the lattice $L \da L_{2d}$.

:::

:::{.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
\[
I(m,m',h)=(m,m',-h).
\]

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

\[
\Omega_-=\{[v]\in\PP(L_-\otimes\mathbb{C})\mid v^2=0,~v\cdot\overline{v}>0\}
\]

:::

:::{.definition}
Consider the following subgroup of $O(\lkt)$:
\[
\Gamma'=\{g\in O(\lkt)\mid g\circ I=I\circ g,~g(e+f,e+f,0) = (e+f,e+f,0)\}
\]
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 $\Gamma'\rightarrow O(L_-)$.

:::

:::{.remark}
\[
E_2=\Omega_-/\Gamma
\]

\[
\Omega_{4,\mathrm{h}}=\{[v]\in\PP(\Lambda_{18}\otimes\mathbb{C})\mid v^2=0,~v\cdot\overline{v}>0\}
\]

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

\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}

:::

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

:::{.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 $\pi_i\colon T_i\rightarrow\PP^1\times\PP^1$ 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
\[
L_-=U\oplus U(2)\oplus E_8(2)=\langle e,f\rangle\oplus\langle e',f'\rangle\oplus\langle\alpha_1,\ldots,\alpha_8\rangle.
\]

1. $\delta_1=e$;

2. $\delta_2=e'$;

3. $\delta_3=e'+f'+\overline{\alpha}_8$;

4. $\delta_4=2e'+f'+\overline{\alpha}_1$;

5. $\delta_5=2e+2f+\overline{\alpha}_1$.


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
\[
\delta_1^{\perp\Lambda_{18}}/\delta_1=U(2)\oplus E_8(2).
\]

:::

:::{.lemma}
> Cusp correspondence 1

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


\input{tikz/cusp_correspondence_two}
:::