2.1 The main diagram, general case.

Consider a surface with an involution $\tau \colon (x,y)\to (-x,-y)$ and quotient $W=Y/\tau $ . Let $B\in |-2K_Y|$ be a $\tau $ -invariant divisor with at worst $ADE$ singularities not passing through the four points with $x,y\in \{0,\infty \}$ fixed by $\tau $ . The double cover $\pi \colon X\to Y$ branched in B is a K3 surface with a del Pezzo involution so that . The involution $\tau $ on Y lifts to a basepoint-free Enriques involution on X commuting with and the quotient is an Enriques surface. The second lift of $\tau $ is a Nikulin involution and the quotient is a K3 surface with eight $A_1$ singularities and possibly more. This gives the following commutative diagram:

(2.1)

The surface is toric and the line bundle has, as its polytope, a square Q with sides of lattice length $4$ , shown in the left panel of Figure 1. The surface W is toric as well, for the same polytope but for the even sublattice shown by gray dots. It is a quartic del Pezzo surface with four $A_1$ singularities. Vectors in are in bijection with monomials $x^ay^b$ invariant under the involution $\tau \colon (x,y)\to (-x,-y)$ . Here, we freely identify monomials $1,x,\dotsc x^4$ with $x_0^4,x_0^3x_1, \dotsc , x_1^4$ , and $1,y,\dotsc ,y^4$ with $y_0^4,y_0^3y_1,\dotsc , y_1^4$ .

Figure 1 Polytopes Q and lattices for the toric surfaces Y and W.

Let $f(x,y)$ be a $\tau $ -invariant polynomial of bidegree $(4,4)$ in which $1$ , $x^4$ , $y^4$ , $x^4y^4$ have nonzero coefficients, so that the hypersurface $\{f(x,y)=0\}\subset Y=V_Q$ does not contain any of the four torus-fixed points. Let be the pyramid over with the vertex $(2,2,2)$ , to which we associate the monomial $z^2$ . Then the K3 surface X is a hypersurface defined by the equation $z^2 + f(x,y)$ in the projective toric variety $V_P$ associated with P. The polynomials $f(x,y)$ invariant under $\tau $ are linear combinations of $13$ monomials marked by gray dots. Thus, f defines a point in an open subset and in its quotient , of dimension $10$ . There are three commuting involutions on X:

which together with the identity form a Klein-four group. Both and are lifts of $\tau $ . On an affine subset of X, a nonvanishing $2$ -form is given by

$$ \begin{align*} \omega=\operatorname{Res}_X\frac{dx\wedge dy\wedge dz}{z^2 + f}. \end{align*} $$

One has and . So and are nonsymplectic and is symplectic. The Enriques surface Z is then a hypersurface in the toric variety for the polytope P but for the even sublattice . It is defined by the same polynomial $z^2+f(x,y)$ whose monomials lie in .

Let R be the ramification divisor of $\pi $ . The involution on X descends to an involution on Z, and . Let $R_Z$ and $B_W$ be the ramification and branch divisors of $\rho $ . Then $R=\psi ^*(R_Z)$ and $R_Z = \frac 12\psi _*(R)$ . Since $R=\frac 12 \pi ^* (B)$ is an ample divisor, $R_Z$ is ample as well. One has .

Horikawa [Reference Horikawa23] analyzed in some detail the sets of possible equations $f(x,y)$ and the maps from various opens subsets of to the period domain and its Baily–Borel compactification, introduced in the next section. In particular, he showed that certain mildly singular $f(x,y)$ vanishing at a torus-fixed point correspond to Coble surfaces, which are $S_2$ -quotients of nodal K3 surfaces.

The GIT compactification was described by Shah [Reference Shah43], who gave normal forms for polystable orbits. As usual for the moduli of K3 surfaces with a projective construction, the relation between the GIT and the Baily–Borel compactifications is not straightforward, cf. [Reference Looijenga34] for K3 surfaces of degree $2$ and [Reference Laza and O’Grady31] for degree $4$ K3 surfaces which are double covers of .

2.2 The main diagram, special case.

The previous section describes the general case, when the K3 cover X is non-unigonal. The special case corresponds to a Heegner divisor in for which is a singular quadric. The toric surfaces Y and W correspond to the same polytope Q shown in the right panel of Figure 1 but for different lattices: and . The surface W is a quartic del Pezzo surface with $A_3+2A_1$ singularities.

There are still $13$ even monomials giving a family over . However, in this case , the centralizer of $\tau $ in , is three-dimensional, equal to . So there is only a $9$ -dimensional family of non-isomorphic surfaces.

Remark 2.2. The entire Diagram (2.1) is intrinsic to the pair , in both the general and special cases. Indeed, , where A is the generator of , and $X = Y\times _W Z$ . One has

with the multiplications defined by the divisor $B_W \in |-2K_W| = |-2(K_W+A)|$ .

2.3 Period domains.

We follow [Reference Sterk44] for the moduli space of Enriques surfaces with a numerical degree $2$ polarization, and [Reference Alexeev and Engel2] for the moduli space of K3 surfaces of degree $4$ with a nonsymplectic involution.

Let be the K3 lattice. It is even, unimodular and has signature $(3,19)$ . Here, $U=I\!I_{1,1}$ and our $E_8=I\!I_{0,8}$ is a negative-definite unimodular lattice. Let us write L in block form:

$$ \begin{align*} L = U \oplus U \oplus U \oplus E_8 \oplus E_8 = \{ (v,u,u',e,e') \mid v,u,u'\in U,\ e,e'\in E_8 \}. \end{align*} $$

Definition 2.3. Define three involutions , and on L corresponding to the Enriques, del Pezzo, and Nikulin involutions on K3 surfaces of degree $4$ as follows (cf. [Reference Peters and Sterk40] for the nodal case):

Their $(\pm 1)$ -eigenspaces are

Here, $\Delta $ and $\Delta ^-$ denote the diagonals and anti-diagonals in $U^2$ and $E_8^2$ . As lattices, they are isomorphic to

All these lattices are even and $2$ -elementary, that is, with the discriminant group for some a. Recall, (see e.g., [Reference Nikulin38]), that an indefinite even $2$ -elementary lattice is uniquely determined by its signature and a triple $(r,a,\delta )$ , where , and $\delta \in \{0,1\}$ is the coparity: $\delta =0$ if the discriminant form , is -valued and $\delta =1$ otherwise. In our notation, $(r,a,\delta )_{n_+}$ denotes such a lattice of signature $(n_+,n_-)$ .

Definition 2.5. We have type IV period domains and , where for a lattice $\Lambda $ of signature $(2,n)$ the corresponding period domain is a connected component of

Since, , one has . The polarizations we consider in both cases are defined by the vector . Here, $\{e,f\}$ is the basis of $U(2)$ with $e^2=f^2=0$ , $e\cdot f=2$ .

Definition 2.6. Define the arithmetic group as the image in of

Additionally, define and . We have

Since and are surjective by [Reference Nikulin38, Corollary 1.5.2, Theorem 3.6.3], the homomorphisms from the centralizer groups and are surjective.

Definition 2.7. Define three quotients of period domains:

By [Reference Alexeev and Engel2], [Reference Alexeev, Engel and Han4], $F_{(2,2,0)}$ is the coarse moduli spaces of K3 surfaces with ADE singularities and a nonsymplectic involution for which the $(+1)$ -eigenspace is $(2,2,0)_1$ .

By [Reference Namikawa37, Theorem 2.13] there is a unique $(-2)$ -vector modulo . The discriminant divisor parameterizes quotients of nodal K3 surfaces by an involution fixing a node. These are rational Coble surfaces with a $\frac {(1,1)}4$ -singularity. It is well known that the points of are in a bijection with the isomorphism classes of Enriques surfaces.

By [Reference Namikawa37, Theorem 2.15] there are two -orbits of $(-4)$ -vectors $\beta $ in . The divisor corresponding to the vector with $\beta ^\perp \simeq \langle 4\rangle \oplus U\oplus E_8(2)$ parameterizes nodal Enriques surfaces, whose desingularizations contain a $(-2)$ -curve. The other $(-4)$ -vector corresponds to the unigonal Enriques surfaces which are double covers of .

By [Reference Sterk44] the complement of the discriminant divisor in is the coarse moduli space of Enriques surfaces with a numerical polarization of degree $2$ .

Proof. One has . The isometry group coincides with the image of the group

Indeed, any element of can be extended to an automorphism of that fixes h because this group of order $2$ preserves . Thus, the stabilizer of in is and so the stabilizer of in is . Thus the finite map is generically injective.

Since is a finite index subgroup, there is also an obvious morphism . It has degree $2^7\cdot 17\cdot 31$ , see [Reference Sterk44, Remark 2.12].

The boundary components of are points and modular curves, corresponding respectively to primitive isotropic lines and planes in $\Lambda $ . We call these boundary components $0$ -cusps and $1$ -cusps respectively.

2.4 KSBA stable pairs and their moduli spaces.

The idea behind KSBA spaces is very simple: they are a close generalization of Deligne–Mumford–Knudsen’s moduli spaces ${\overline M}_{g,n}$ of pointed stable curves. For a one-parameter degeneration of K3 surfaces with a distinguished ample divisor, there are often infinitely many Kulikov models that differ by flops, but there is a canonical KSBA-stable limit.

In brief, a KSBA stable pair $(X,B=\sum _{i=1}^n b_iB_i)$ consists of a projective variety X which is deminormal: seminormal with only double crossings in codimension $1$ , $B_i$ are effective Weil divisors not containing any components of the double locus of X, $0<b_i\le 1$ are rational numbers, all satisfying two conditions:

1. 1. (on singularities) the pair $(X,B)$ has semi log canonical (slc) singularities, the generalization of the log canonical singularities appearing in the MMP to the nonnormal case, and

2. 2. (numerical) the divisor $K_X+B$ is an ample -Cartier divisor.

The main result is that in characteristic zero for the fixed dimension $d=\dim X$ , number n, coefficient vector $(b_1,\dotsc , b_n)$ and degree $(K_X+B)^d$ there is a (carefully defined) moduli functor for families of KSBA stable pairs, the moduli stack is Deligne–Mumford, and the coarse moduli space is projective. We refer the reader to [Reference Kollár25] for complete details.

We need a version of this definition when $K_X$ is numerically trivial, R is an ample Cartier divisor and the pair is $(X,\epsilon R)$ with $0<\epsilon \ll 1$ allowed to be arbitrarily small. By [Reference Birkar8], [Reference Kollár and Xu28] in any dimension d for fixed degree $R^d$ there exists $\epsilon _0>0$ such that the moduli space for any $0<\epsilon <\epsilon _0$ is the same. We only need this result for K3 surfaces, in which case the construction and the proof were given in [Reference Alexeev, Engel and Thompson5] and [Reference Alexeev, Brunyate and Engel1]. The Enriques case then immediately follows.

In [Reference Alexeev and Engel2], [Reference Alexeev, Engel and Thompson5] this general construction was applied to describe a geometric compactification for the pairs $(X,\epsilon R)$ where X is a K3 surface with a non-symplectic involution and ADE singularities, and R is a connected component of genus $g\ge 2$ of the ramification divisor for the induced double cover.

In this paper we apply it to the pairs $(Z,\epsilon R_Z)$ , where Z is an Enriques surface with ADE singularities and with a numerical degree $2$ polarization, an $S_2$ -quotient of a K3 surface $X\in F_{(2,2,0)}$ with ADE singularities, and $R_Z$ is the ramification divisor of the induced involution as in the introduction. For the KSBA-stable limits, $R_Z$ will be the divisorial part of the ramification divisor of that is not contained in the double locus of Z.

Our main goal is to describe the normalization of and the surfaces appearing on the boundary.

3.1 Cusp diagram of $F_{(2,2,0)}$ .

Figure 2 reproduces the cusp diagram of given in the last section of [Reference Alexeev and Engel2]. An equivalent diagram is found in [Reference Laza and O’Grady31].

Figure 2 Cusp diagram of $F_{(2,2,0)}$ , for $T_{\mathrm {dP}}=U\oplus U(2)\oplus E_8^2$ .

There are two $0$ -cusps which are in bijection with the primitive isotropic lines , distinguished by the divisibility $\operatorname {div}(e)\in \{1,2\}$ of e in the dual lattice . For a primitive vector in a $2$ -elementary lattice one must have $\operatorname {div}(e)\in \{1,2\}$ .

The lattices $e^\perp /e$ are hyperbolic and $2$ -elementary, and here are of the form $U\oplus E_8^2 = (18,0,0)_1$ and $U(2)\oplus E_8^2 = (18,2,0)_1$ depending on whether $\operatorname {div}(e)=2$ or $1$ respectively. Similarly, the eight $1$ -cusps are in bijection with the primitive isotropic planes . For each of them there is a negative-definite lattice $\Pi ^\perp /\Pi $ which is $2$ -elementary but is no longer uniquely determined by the triple $(r,a,\delta )$ . The label denotes the root sublattice of $\Pi ^\perp /\Pi $ . Here, $D_{16}$ is a root lattice with determinant $4$ and $D_{16}^+$ is its unique even unimodular extension.

The bipartite diagram in Figure 2 depicts all $0$ - and $1$ -cusps added to compactify $F_{(2,2,0)}$ . An arrow indicates that a $0$ -cusp lies in the closure of a $1$ -cusp. Equivalently, there is, up to the group action, an inclusion of the corresponding isotropic subspaces. The single versus double arrow indicates, respectively, that the rank of the discriminant group of $e^\perp /e$ and $\Pi ^\perp /\Pi $ stays the same, or drops by $2$ . See [Reference Alexeev and Engel2, Sections 5C-5D] for more details.

3.2 Cusp diagram of $F_{(10,10,0)}$ .

The cusp diagram for is well known. It can also be easily found by [Reference Alexeev and Engel2, Section 5]. We give it in Figure 3, keeping the same notation as above. There are two $0$ -cusps distinguished by the divisibility $\operatorname {div}(e)=1$ or $2$ .

Figure 3 Cusp diagram of $F_{(10,10,0)}$ , for $T_{\mathrm {En}}=U\oplus U(2)\oplus E_8(2)$ .

A geometric interpretation of these cusps is as follows. Let be a Kulikov model and consider the completed period mapping

Suppose that on the generic fiber extends to an involution on the central fiber. If $0\in C$ is sent to the double-circled cusp $(10,10,0)_1$ or $(8,8,0)_0$ then is basepoint free. Otherwise, has a nonempty finite set of fixed points.

Furthermore, the dual complex is a $2$ -sphere and the induced action of on in the $(10,10,0)_1$ case is an antipodal involution, while in the $(10,8,0)_1$ case it is a hemispherical involution, see [Reference Alexeev and Engel2, Sections 8F]. So the quotients of by the Enriques involution are, respectively, the real projective plane and a disk . In Type II, is a segment. In the $(8,8,0)_0$ case, the action of flips the segment, whereas in the $(8,6,0)_0$ case it fixes the segment.

3.3 Cusp diagram of .

Sterk [Reference Sterk44] computed the cusp diagram for . There are five $0$ -cusps for which we use Sterk’s numbering $1,2,3,4,5$ . There are also $9$ distinct $1$ -cusps.

Lemma 3.2. The morphisms and extend to the Baily–Borel compactifications, mapping $0$ -cusps to $0$ -cusps and $1$ -cusps to $1$ -cusps in the manner shown in Figure 4.

Figure 4 Cusps of with images in and .

The images in are shown by labels from Figure 4, and in by the corresponding border shapes (oval, double oval, rectangle, double rectangle).

3.4 Vinberg’s theory and Coxeter diagrams.

We refer to [Reference Vinberg46], [Reference Vinberg47] for Vinberg’s theory of reflection groups of hyperbolic lattices, saying just enough to fix the notations.

Let $\Lambda $ be a hyperbolic lattice. Let be the component of the set , containing a fixed class h with $h^2>0$ . Let be the corresponding hyperbolic space. A vector $v\ne 0$ with $v^2=0$ in the closure of defines a point on the sphere at infinity of . Let denote the closure of .

A reflection in a vector $\alpha \in \Lambda $ is the isometry

$$ \begin{align*}w_\alpha(v) = v - \frac{2(\alpha\cdot v)}{\alpha\cdot \alpha} \alpha.\end{align*} $$

A root is a vector $\alpha $ with $\alpha ^2<0$ such that $w_\alpha (\Lambda )=\Lambda $ , equivalently such that . For a group of isometries $\Gamma \subset O(\Lambda )$ we denote by $W(\Gamma )$ its subgroup generated by reflections.

We denote by the fundamental chamber for $W(\Gamma )$ . Equivalently, one can treat it as the (possibly infinite) polyhedron . One has

(3.1)

(3.2) $$ \begin{align} &O(\Lambda) = S\ltimes W(\Gamma) \quad\text{for some } S\subset\operatorname{Sym}(P). \end{align} $$

The fundamental chamber is encoded in a Coxeter diagram. The vertices correspond to the simple roots $\alpha _i$ and the edges show the angles between them as follows. Let $g_{ij} = (\alpha _i\cdot \alpha _j) / \sqrt {(\alpha _i\cdot \alpha _i) (\alpha _j\cdot \alpha _j)}$ . One connects i and j by

• • an m-tuple line if $g_{ij} = \cos \frac {\pi }{m+2}$ . The hyperplanes $\alpha _i^\perp $ , $\alpha _j^\perp $ intersect in .

• • a thick line if $g_{ij}=1$ . $\alpha _i^\perp $ , $\alpha _j^\perp $ are parallel, meet at an infinite point of .

• • a dotted line if $g_{ij}> 1$ . $\alpha _i^\perp $ , $\alpha _j^\perp $ do not meet in or its closure.

The lattices and are even $2$ -elementary. For such lattices the roots are the $(-2)$ -vectors and the $(-4)$ -vectors with $\operatorname {div}(\alpha )=2$ . We denote the roots with $\alpha ^2=-2$ by white vertices and those with $\alpha ^2=-4$ by black vertices.

3.5 Coxeter diagrams for the $0$ -cusps of $F_{(2,2,0)}$ .

The Coxeter diagrams for the lattices $(18,0,0)_1 = U\oplus E_8^2$ and $(18,2,0)_1 = U(2)\oplus E_8^2$ , (cf. [Reference Alexeev and Engel2]), are shown in Figure 5. To describe Kulikov models and KSBA stable models, it is important to keep track of the even and odd nodes on the boundaries of these diagrams. We assign even numbers to the even nodes; in Figure 5 they are shown as double-circled nodes. For typographical reasons, in the diagrams that follow we skip these double circles. The corners are always even.

Figure 5 Coxeter diagrams for $(18,2,0)_1$ and $(18,0,0)_1$ .

The lattice $U\oplus E_8^2$ is generated by $19$ roots $\alpha _1,\dotsc ,\alpha _{19}$ with a single relation

(3.3) $$ \begin{align}\begin{aligned} v &=3\alpha_1 + 2\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6 + 3\alpha_7 + 2\alpha_8 + \alpha_9 \\ &= 3\alpha_{19} + 2\alpha_{18} + 4\alpha_{17} + 6\alpha_{16} + 5\alpha_{15} + 4\alpha_{14} + 3\alpha_{13} + 2\alpha_{12} + \alpha_{11}. \end{aligned}\end{align} $$

The lattice $U(2)\oplus E_8^2$ is generated by $22$ roots $\alpha _0,\dotsc ,\alpha _{21}$ . The relations come from maximal parabolic subdiagrams with more than one connected component. Maximal parabolic subdiagrams correspond to parabolic sublattices with a unique isotropic line; the generator of this line is a linear combination of roots in each connected component, which gives a linear relation. For example, the following relation results from ${\widetilde E}_7^2{\widetilde C}_2$ :

(3.4) $$ \begin{align} \alpha_{16}+\alpha_{20}+\alpha_{18} = \alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 +3\alpha_5 + 2\alpha_6 + \alpha_7 + 2\alpha_{17}. \end{align} $$

3.6 Coxeter diagrams for the $0$ -cusps of $F_{(10,10,0)}$ .

The Coxeter diagrams for the lattices $(10,10,0)_1 = U(2)\oplus E_8(2)$ and $(10,8,0)_1 = U\oplus E_8(2)$ are well-known. They are shown in Figure 6.

Figure 6 Coxeter diagrams for $(10,10,0)_1$ and $(10,8,0)_1$ .

3.7 Folding Coxeter diagrams by involutions.

Definition 3.3. Let $\Lambda $ be a lattice with an involution and let $\alpha \in \Lambda $ be a vector. We call the following vector $\alpha _I\in \Lambda ^{I=1}$ a folded vector:

$$ \begin{align*} \alpha_I = \begin{cases} \alpha &\text{if } I(\alpha)=\alpha\\ \alpha + I(\alpha) &\text{if } I(\alpha)\ne\alpha.\\ \end{cases} \end{align*} $$

Proof. If , the claim is clear. Now suppose that , so $\alpha _I = \alpha +I(\alpha )$ . Write in the block form as in Definition 2.3. Then

$$ \begin{align*} &\alpha = (v,u,-u,e,e') \quad\text{and}\quad I(\alpha) = (v,u,-u,-e',-e)\\ &I(\alpha) = \alpha - (0,0,0,e+e',e+e'), \quad\\ &\alpha\cdot I(\alpha) = \alpha^2 - (e+e')^2,\quad \alpha_I^2 = 2\alpha^2 + 2\alpha\cdot I(\alpha). \end{align*} $$

Since, $\alpha \ne I(\alpha )$ , $e+e'$ is a nonzero vector in $E_8$ . Therefore, $\alpha \cdot I(\alpha )> \alpha ^2$ .

For $\alpha ^2=-2$ this leaves the only possibility $\alpha \cdot I(\alpha )=0$ and $\alpha _I^2=-4$ . Clearly, $\operatorname {div}(\alpha _I)=2$ in so $\alpha _I$ is a root of . But $\operatorname {div}(\alpha _I)\ne 2$ in . Otherwise, $e-e'\in 2E_8$ , which implies that $(e+e')^2\equiv 0\pmod 4$ , $\alpha \cdot I(\alpha )\ge 2$ and $\alpha _I^2\ge 0$ .

Now let $\alpha $ be a $(-4)$ -root in . Since the divisibility of $\alpha $ is $2$ , one must have $e,e'\in 2E_8$ , so also $e+e'\in 2E_8$ . But then $-(e+e')^2\ge 8$ , $\alpha \cdot I(\alpha )\ge 4$ and $\alpha _I^2\ge 0$ , a contradiction. This completes the forward direction.

The converse follows from [Reference Namikawa37, Theorem 2.13 and Theorem 2.15]: up to acting on there is only one type of $(-2)$ -vector and two types of $(-4)$ -vectors.

Definition 3.5. Consider a primitive vector with $e^2=0$ . We get two hyperbolic lattices

There are induced involutions and on these hyperbolic lattices. We denote , which is an involution on , for which the $(+1)$ -eigenspace of J in is .

Definition 3.6. Let be primitive isotropic. The stabilizer of e in fits into an exact sequence

where $U_e$ is the unipotent subgroup, which acts trivially on . We define and similarly.

Denote by the reflection subgroup of . Its Coxeter diagram is one of the two Coxeter diagrams in Figure 5. Denote by the reflection subgroup of ; it is generated by reflections in the roots with $\alpha \cdot e=0$ .

3.8 Coxeter diagrams for the $0$ -cusps of by folding.

We now find five involutions of the lattices $U\oplus E_8^2$ and $U(2)\oplus E_8^2$ and compute folded diagrams for them. We prove that they are the Coxeter diagrams for the groups for some isotropic vectors . These turn out to be the same as the Coxeter diagrams computed in [Reference Sterk44] by Vinberg’s method [Reference Vinberg47]. We keep Sterk’s numbering for the $0$ -cusps. In the order of appearance, they are 2, 1, 3, 4, 5.

Lemma 3.9. On the Coxeter diagram for the lattice , consider the reflection $ J\colon \alpha _k \to \alpha _{20-k}$ about the vertical line. Then and the folded diagram is shown in Figure 7.

Figure 7 Folded diagram for cusp 2.

Lemma 3.10. Consider the following involutions J on the lattice :

1. 1. rotation of the diagram by $180$ degrees.

2. 2. reflection of the diagram about the diagonal, followed by a lattice reflection in the root $\alpha _{20}$ .

3. 3. reflection of the diagram about a horizontal line.

4. 4. the composition of $8$ commuting reflections in the roots $\alpha _1, \alpha _3, \dotsc , \alpha _{15}$ .

The fixed sublattice is isomorphic to $U(2)\oplus E_8(2)$ in case (1) and to $U\oplus E_8(2)$ in cases (3,4,5). The folded diagrams are shown in Figure 8.

Figure 8 Folded diagrams for cusps 1, 3, 4, 5.

Proof. For the involution of Lemma 3.9 the statement is obvious: we simply define $ {I}$ to be the identity on the first summand of . Similarly for cusp (1) in Lemma 3.10 one has and we extend $ {I}$ to U as the identity.

In the cases (3,4,5) we have an exact sequence of abelian groups

with , , and the trivial extension does not work.

Write $U=\langle e,f\rangle $ using the standard basis with $e^2=f^2=0$ , $e\cdot f=1$ . A section is the same as an element , so that . The orthogonal complement of is $\langle e, f-a\rangle \simeq U$ if , and $\langle e, 2f-2a\rangle $ if . One has $(2f-2a)^2 = 4a^2$ . From this, we see that the last lattice is isomorphic to $U(2)$ if the discriminant form of satisfies , and it is isomorphic to $I_{1,1}(2)=\langle 2\rangle \oplus \langle -2\rangle $ otherwise.

The discriminant form of is the same as for . We choose $a=\frac 12 e'$ and define the involution I on to be ${J}$ on and the identity on its orthogonal complement $U(2)$ . Then

We complete the proof by Lemma 2.4.

Indeed, our Coxeter diagrams, obtained by folding, coincide with the ones found by Sterk in [Reference Sterk44] who used the Vinberg algorithm [Reference Vinberg47] to compute them.

3.9 $1$ -cusps of by folding.

Indeed, both correspond to the isotropic planes contained in the sublattice of T fixed by the involution. We list these $1$ -cusps in Figures 9 and 10. They agree with Sterk’s computations in [Reference Sterk44], and the entire cusp diagram agrees with Figure 4.

Figure 9 $1$ -cusps of passing through $0$ -cusp 2.

Figure 10 $1$ -cusps of passing through $0$ -cusps 1, 3, 4, 5.

Figures 9 and 10 contain the information of all cusp incidences of . The figures are read as follows: the first numeral indicates one of the five folding symmetries $1,2,3,4,5$ of the relevant hyperbolic lattice , and this symmetry is also depicted on the Coxeter diagram for , with an $\times $ indicating that we reflect in the corresponding root. These correspond to the five $0$ -cusps added to .

In blue is highlighted a maximal parabolic subdiagram invariant under the given folding symmetry. Necessarily, all $\times $ -ed vertices are contained in this diagram, since only these diagrams can be invariant under the corresponding composition of root reflections. Such blue diagrams are in bijection with the $1$ -cusps incident upon the corresponding $0$ -cusp. The collection of all numerals, including the first label, indicate the corresponding $1$ -cusp, see Notation 3.1.

Finally, adjacent to each maximal parabolic diagram for is the corresponding maximal parabolic subdiagram of the folded lattice .

4.1 General theory.

The general theory of integral affine spheres, $\operatorname {IAS^2}$ for short, in the form that we need it here is detailed in [Reference Alexeev, Brunyate and Engel1], [Reference Alexeev and Engel2], [Reference Alexeev, Engel and Thompson5], [Reference Engel14], [Reference Engel and Friedman15]. We refer the reader to the above papers for the necessary background, and give a broad summary now.

A Kulikov model is a K-trivial semistable model of a degeneration of K3 surfaces over a pointed curve [Reference Kulikov29], [Reference Persson and Pinkham39]. For Type III degenerations, the dual complex of the central fiber is the $2$ -sphere $S^2$ , and for Type II degenerations is a segment. By [Reference Gross, Hacking and Keel20, Remark 1.1v1], [Reference Engel14, Proposition 3.10] there is a natural integral-affine structure on , with singularities. The correct notion of singularities is detailed in [Reference Alexeev, Brunyate and Engel1, Section 5].

Fixing one Kulikov model , we get Kulikov models for all other degenerations with the same Picard–Lefschetz transform, of the same combinatorial type [Reference Friedman and Scattone18, Lemma 5.6], [Reference Alexeev and Engel3, Definition 7.14] by deforming the gluings and moduli of components. We can extract the KSBA stable limit of a degeneration of K3 pairs, if we can describe the integral-affine polarization , a certain weighted balanced graph [Reference Alexeev, Brunyate and Engel1, Definition 5.17]. This weighted graph encodes the line bundle on a divisor model : a Kulikov model which admits a nef extension of , $t\in C\setminus 0$ , containing no singular strata of [Reference Alexeev, Engel and Thompson5, Theorem 3.12], [Reference Laza30, Theorem 2.11].

By [Reference Alexeev, Engel and Han4, Theorem 3.24], our chosen divisor R, as the fixed locus of an automorphism on a general Enriques K3 surface, is recognizable, see [Reference Alexeev and Engel3, Definition 6.2]. By the main theorem on recognizable divisors [Reference Alexeev and Engel3, Theorem 1], there is a unique semifan whose corresponding semitoroidal compactification [Reference Looijenga35], [Reference Alexeev and Engel3, Section 5C] normalizes the KSBA compactification of . By [Reference Alexeev and Engel3, Theorem 8.11(5)], can be chosen to be the same for all degenerations with fixed Picard–Lefschetz transform.

In turn, the combinatorial data of determines the combinatorial type of the KSBA stable limit of the degeneration by [Reference Alexeev and Engel3, Corollary 8.13]. Then [Reference Alexeev and Engel3, Theorem 9.3] gives an algorithm to determine : Its cones are given by collections of Picard–Lefschetz transformations for which determines a KSBA-stable pair of a fixed combinatorial type. This is the natural notion of combinatorial constancy of such pairs.

The possible Picard–Lefschetz transformations of Kulikov degenerations in are encoded by a vector called the monodromy invariant. It is valued in the fundamental chamber for one of the five folded diagrams of Figures 7 and 8 as in Lemma 3.8 because $\lambda $ must be invariant under the involution J on . An algorithm (albeit a complicated one), is provided in [Reference Alexeev and Engel2, Theorem 8.3] to build for all monodromy invariants in the fundamental chamber for the Weyl group action, for either hyperbolic lattice or corresponding to a $0$ -cusp of $F_{(2,2,0)}$ .

Restricting $\operatorname {IAS^2}$ for $F_{(2,2,0)}$ to the involution-invariant sublattice exhibits a polarized $\operatorname {IAS^2}$ for any Type III degeneration in . Then, one can hope (and it is indeed the case, as shown below), that on these subloci, the corresponding divisor models admit a second involution identified with the limit of the Enriques involution. Thus, these polarized $\operatorname {IAS^2}$ will provide a method to compute the Kulikov and KSBA-stable models of all degenerations of both the Enriques surfaces and their corresponding double covers, the Enriques K3 surfaces.

Definition 4.1. Let , for the one of the two $0$ -cusps of $F_{(2,2,0)}$ . We define

$$ \begin{align*}\ell := (\ell_i)_{i\in G} = (\lambda\cdot \alpha_i)_{i\in G}\end{align*} $$

where $\alpha _i$ are the roots of either diagram in Figure 5. Thus, for the cusp $(18,2,0)_1$ and for the cusp $(18,0,0)_1$ .

4.2 $\operatorname {IAS^2}$ for $F_{(2,2,0)}$ .

We now identify the polarized $\operatorname {IAS^2}$ for degenerations in $F_{(2,2,0)}$ following the instructions of [Reference Alexeev and Engel2, Theorems 7.4, 8.3]. We treat each of the two $0$ -cusps individually.

Cusp : We are to first take a K3 surface ${\widehat X}$ in the mirror moduli space for this $0$ -cusp—these are $U(2)\oplus E_8^{\oplus 2}$ -polarized K3 surfaces. Then we are to consider the anticanonical pair quotient

by the mirror involution and, for each ${\widehat L}$ in the nef cone of ${\widehat Y}$ , we must build a Symington polytope $P(\ell )$ for the line bundle ${\widehat L}\to ({\widehat Y},{\widehat D})$ corresponding to $\ell $ , see [Reference Symington45], [Reference Alexeev and Engel2, Construction 6.16]. We build a sphere by identifying two copies of this integral-affine disk along their common boundary, to form the equator of the sphere. Then for a monodromy-invariant $\lambda \leftrightarrow \ell \leftrightarrow {\widehat L}$ and the integral-affine polarization corresponding to the flat limit is the equator of the sphere, with weights alternating $2$ and $1$ .

The anticanonical pair $({\widehat Y},{\widehat D})$ is a rational elliptic surface with an anticanonical cycle of $16$ curves, of alternating self-intersections $-1$ and $-4$ , which result from blowing up the corners of an $I_8$ Kodaira fiber. This pair admits a toric model

$$ \begin{align*}({\widehat Y},{\widehat D})\to (\overline{{\widehat Y}},\overline{{\widehat D}})\end{align*} $$

whose fan is depicted on the left-hand side of Figure 17. The rays going to the four corners correspond to components of the toric model receiving an internal blow-up, that is, a blow-up at a smooth point of the anticanonical boundary.

A moment polytope ${\overline P}(\ell )$ for the line bundle $\overline {{\widehat L}}\to (\overline {{\widehat Y}},\overline {{\widehat D}})$ is depicted on the left of Figure 11 and a Symington polytope $P(\ell )$ for the line bundle ${\widehat L}\to ({\widehat Y},{\widehat D})$ corresponding to $\ell $ is depicted on the right of Figure 11. The right hand-side also serves as a visualization of each hemisphere of the integral-affine sphere , with the equator in blue and integral-affine singularities in red. The quantities $\ell _{20}$ and $\ell _{21}$ are, respectively, twice the lattice length between the singularities introduced by Symington surgeries on opposite sides of the figure.

Figure 11 Moment and Symington polytopes for cusp $(18,2,0)_1$ .

Cusp : The procedure for constructing polarized $\operatorname {IAS^2}$ at this cusp is essentially the same as the above, instead taking the mirror moduli space to be $U\oplus E_8^{\oplus 2}$ -polarized. The fan of a toric model of the mirror is provided by the right hand side of Figure 17. The integral-affine structures are similar to those depicted in [Reference Alexeev, Brunyate and Engel1, Figure 4], with an important difference: The cusp $(18,0,0)_1$ corresponds to a non-simple mirror of $F_{(2,2,0)}$ . This means that the integral-affine polarization has no support on the bottom edge of the Symington polytope P, and is empty on the corresponding components of . See the discussion of a “B-move” in [Reference Alexeev and Engel2, Section 8D] for further details.

The $\operatorname {IAS^2}$ we need is the result of taking the $\operatorname {IAS^2}$ of [Reference Alexeev, Brunyate and Engel1, Figure 4], splitting the $I_2$ singularity at the bottom into two $I_1$ singularities traveling in opposite directions, and colliding each one with a corner. This produces singularities in the bottom left and right corners of charge $2$ , depicted with a larger red triangle, see Figure 12.

Figure 12 Symington polytope for cusp $(18,0,0)_1$ .

To summarize, by [Reference Alexeev and Engel2, Theorem 8.3], we have:

Theorem 4.4. Let be the polarized $\operatorname {IAS^2}$ built above, from or . Then, upon triangulation into lattice simplices,

is the dual complex of the central fiber of a divisor model , whose monodromy invariant satisfies $\ell = (\lambda \cdot \alpha _i)_{i\in G}$ .

4.3 and for .

Now suppose that is a Type III divisor model as in Theorem 4.4, whose period map $C^*\to F_{(2,2,0)}$ factors through . Then, the general fiber is an Enriques K3 surface with degree $4$ polarization, and is a divisor model for the degeneration. The quotient

of the general fiber by the Enriques involution is a degenerating family of Enriques surfaces. The monodromy invariant then necessarily lies in the fundamental chamber for one of the five $0$ -cusps of . Equivalently, $\ell $ must be invariant under one of the five folding symmetries.

Proof. For each $0$ -cusp, we directly inspect the $\operatorname {IAS^2}$ for the parameters $\ell $ corresponding to and see that there is an additional symmetry of $B(\ell )$ .

Cusp 2: We have if and only if $\ell _i=\ell _{20-i}$ for all $i=1,\dots ,9$ . The $\operatorname {IAS^2}$ in Figure 12 then has a visible symmetry, which is to act on the both the hemisphere P, and its opposite hemisphere , by a flip across the vertical line bisecting the bottom and top edges.

Cusp 1: We have if and only if $\ell _i=\ell _{8+i}$ for all $i=0,\dots , 7$ , $\ell _{16}=\ell _{18}$ and $\ell _{17}=\ell _{19}$ . Then the corresponding involution of the $\operatorname {IAS^2}$ is to rotate each hemisphere, shown in Figure 11, by $180$ degrees, and then flip the two hemispheres P and .

Cusp 3: We have if and only if $\ell _i=\ell _{16-i}$ for all $i=1,\dots , 7$ , $\ell _{17}=\ell _{19}$ , and $\ell _{20}=0$ . This is because the folding symmetry also reflects in the root $\alpha _{20}$ . So if $w_{\alpha _{20}}(\lambda )=\lambda $ , then $\ell _{20}=\lambda \cdot \alpha _{20}=0$ .

Recall that $\ell _{20}$ is the lattice distance between the singularities introduced by the Symington surgeries resting on the edges parallel to $(1,-1)$ and $(-1,1)$ , on the right-hand side of Figure 11. We construct $B(\ell )$ in such a way that the two singularities introduced by these Symington surgeries coincide. The involution of the $\operatorname {IAS^2}$ acts by flipping each hemisphere P, diagonally.

Cusp 4: Similar to Cusp 2, we have if and only if each hemisphere of $B(\ell )$ is symmetric with respect to flipping along a horizontal line bisecting the edges $\ell _6(0,1)$ and $\ell _{14}(0,-1)$ .

Cusp 5: We have if and only if $\ell _{2i+1}=0$ for $i=0,\dots ,7$ . We declare that act in the same manner as the extension of to : It flips the two hemispheres P and . The eight $\times $ -ed nodes correspond to eight collisions of pairs of $I_1$ singularities along the equator.

By [Reference Alexeev and Engel2, Proposition 6.17], the mirror K3 surface ${\widehat X}$ admits a symplectic form $\omega $ and Lagrangian torus fibration

$$ \begin{align*}\mu\colon ({\widehat X},\omega) \to B(\ell),\end{align*} $$

for generic . Note that while some of the $24\ I_1$ -singularities collide on $B(\ell )$ for Cusps 3 and 5, we only ever get, for generic $\ell $ , a collision of two $I_1$ -singularities with parallel -monodromies. So the fibration $\mu $ still exists, but has $I_2$ fibers over these collisions.

The involution acting on $B(\ell )$ induces an involution of the Lagrangian torus fibration $({\widehat X},\mu )$ and in turn on or $(18,0,0)_1$ which is generated by classes of visible curves, cf. [Reference Alexeev and Engel2, Section 6G]. In the current setting, the visible curves (which correspond to the roots $\alpha _i$ ) are all of the following simple form: A path connecting two $I_1$ -singularities with parallel -monodromies. For Cusps 1, 2, 4, the involution acts on the classes of visible curves by the Enriques involution on and thus, by the Mirror/Monodromy theorem [Reference Engel and Friedman15, Proposition 3.14], [Reference Alexeev and Engel2, Theorems 6.19, 7.6], $B(\ell )$ is the dual complex of a degeneration with a monodromy invariant in .

Some additional care must be taken for Cusps 3 and 5, where $B(\ell )$ is a limit of $\operatorname {IAS^2}$ with $24$ distinct $I_1$ -singularities. For each $\times $ -ed node, the involution J acts on by reflecting along $\alpha _i$ and so the class $[\omega ]$ of the symplectic form should satisfy $[\omega ]\cdot \alpha _i=0$ . Equivalently, there should be a nodal slide, see [Reference Alexeev and Engel2, Section 6E], which collides the two $I_1$ singularities of the visible curve corresponding to $\alpha _i$ into an $I_2$ singularity. This is indeed the case for the $\operatorname {IAS^2}$ described above. To summarize, invariance under reflection of an $\times $ -ed node $\alpha _i$ corresponds, on the $\operatorname {IAS^2}$ , to colliding the two $I_1$ singularities bounding the corresponding visible curve.

We conclude that an -invariant polarized $\operatorname {IAS^2}$ , which has a coalescence to an $I_2$ -singularity for each $\times $ -ed node, is the dual complex of a divisor model for degree $4$ Enriques K3 surfaces whose monodromy invariant is generic. The passage from the result for generic to all is a standard trick involving a limit procedure on the corresponding $\operatorname {IAS^2}$ , examining $B(\ell )$ as some $\ell _i\to 0$ , see [Reference Alexeev, Engel and Thompson5, Theorem 6.29], [Reference Alexeev and Engel2, Section 6G].

More generally, every degeneration of Enriques surfaces admits a divisor model for which defines a birational involution, and for which the union of the fixed locus and the locus of indeterminacy contains .

Definition 4.7. Let , be a Kulikov, resp. divisor, model of Enriques K3 surfaces for which defines a regular involution on , resp. preserving . We define the dlt model, resp. the half-divisor model, to be the quotient by the Enriques involution:

Proof. In Type III, the homeomorphism type of follows directly from the description of the action of in Proposition 4.5. In Type II, we can construct divisor models by taking limits of $B(\ell )$ as it collapses to a segment, or equivalently as $\lambda $ approaches a rational isotropic ray at the boundary of .

The resulting central fiber is a chain of surfaces and by arguments similar to Theorem 4.6, a general degeneration to the given Type II boundary divisor admits an additional involution which acts on by the limit of the action of on the segment $B(\ell )$ . This gives the claimed action on , by direct examination of the limiting dual segment $B(\ell )$ for all entries of Figures 9 and 10.

If permutes two irreducible components, it is clear that the normalization of the quotient agrees with the normalization of either component.

So suppose preserves the pair (necessarily we are at a Cusp 2–5). Possibly assuming further divisibility of $\lambda $ , and choosing our triangulation of $B(\ell )\simeq S^2$ appropriately, we may assume that the fixed locus of is a circle $S^1\subset B(\ell )$ formed from a collection of vertices $\widetilde {v}_i$ and edges $\widetilde {e}_{ij}$ of .

Denote the corresponding collection of components the Enriques equator. For Cusps 2, 3, 4 the Enriques equator is distinct from the del Pezzo equator, which corresponds to the common glued boundary of P or and supports . For Cusp 5, the Enriques and del Pezzo equators coincide since .

The logarithmic $2$ -form on is of the form $dx\wedge dy$ , $\frac {dx}{x}\wedge dy$ , or $\frac {dx}{x}\wedge \frac {dy}{y}$ depending, respectively, on whether $(x,y)$ are local coordinates (in a component) at a smooth point, a point in a double curve, or a triple point of . Since has no divisorial fixed locus and is non-symplectic, it fixes at most a finite subset of contained in the double locus, where is non-symplectic.

So in Type III, the only fixed points of are points in some along the Enriques equator. Being an involution of , there must be exactly $2$ such fixed points. We recover then a similar phenomenon as for the Kulikov models of Enriques degenerations which were described in [Reference Alexeev and Engel2, Section 8F].

In Type II, the analysis is similar: If preserves a component ${\widetilde V}_i$ , then the involution on a preserved double curve ${\widetilde D}_{ij}\simeq E$ is locally given by negation on E. So the induced action on this (and in turn any) double curve is an elliptic involution. On the other hand, suppose permutes the two components containing E. Then, since the residues from the two components of the logarithmic two-form are negatives of each other, we must have by non-symplecticness that preserves a holomorphic one-form on E. So it acts by a $2$ -torsion translation, nontrivial because the fixed locus is finite.

Having analyzed the action of on , we see the quotient has SNC singularities at all points, except the images of the fixed points along the double curves of the Enriques equator. Here the local equation of the quotient is

(4.1)

which is slc, with the only log canonical center being the image of the double locus $x=z=0$ . So has slc singularities.

Since the fixed locus is finite, is numerically trivial. Furthermore, inherits the property of containing no log canonical centers, and being relatively big and nef, from —it is important here that no log canonical center was introduced at $(0,0,0)$ in the above quotient (4.1). The proposition follows.

Corollary 4.9. The KSBA-stable limit of a degeneration of can be computed from the half-divisor model of Proposition 4.7 as

This also furnishes a somewhat inexplicit description of the components of the KSBA-stable limit : First, we take a component of the divisor model for the degeneration in $F_{(2,2,0)}$ . This is, up to corner blow-ups, the minimal resolution of an ADE surface of [Reference Alexeev and Thompson6]. Then, we impose the condition that the periods and dual complex of are involution invariant. Now, the Torelli theorem for anticanonical pairs [Reference Gross, Hacking and Keel21, Theorem 1.8], [Reference Friedman17, Section 8] implies that $({\widetilde V}_i,{\widetilde D}_i,{\widetilde R}_i)$ admits an involution which acts in an orientation-reversing manner on the cycle ${\widetilde D}_i$ . Then the quotient

is a log Calabi–Yau pair of index $\leq 2$ with $R_i$ big and nef. The stable component

is (up to normalization) the result of contracting all curves which intersect $R_i$ to be zero. Alternatively, we can reverse the order, taking first the stable model of $({\widetilde V}_i,{\widetilde D}_i+\epsilon {\widetilde R}_i)$ to get an ADE surface of [Reference Alexeev and Thompson6] forming a component of and then taking the quotient by the induced involution . These stable surfaces and their quotients are described further in Section 6.

4.4 Examples.

We give some examples of divisor and half-divisor models. To distinguish notationally between different $0$ -cusps, we write $B_k(\ell )$ , $k\in \{1,\dotsc , 5\}$ for the folding-symmetric polarized $\operatorname {IAS^2}$ at Cusp k, from Proposition 4.5.

Example 4.13. $B_3(2,0^{15},2,4,6,4,0,4)$ : Consider Cusp 3, with the diagonal folding symmetry, and set . Then from Section 4.2, the moment polygon ${\overline P}(\ell )$ for the toric model of the mirror is the sequence of vectors $(3,-3)$ , $(2,2)$ , $(-3,3)$ , $(-2,-2)$ put successively end-to-end.

We perform Symington surgeries of size $1, 2, 3, 2$ along the four edges $(3,-3)$ , $(2,2)$ , $(-3,3)$ , $(-2,-2)$ respectively, because $(\ell _{16},\ell _{17},\ell _{18},\ell _{19})=(2,4,6,4)$ . The result is the Symington polytope $P(\ell )$ . Glue $P(\ell )$ and to produce $B_3(\ell )$ , which is depicted on the left of Figure 13 (of course, only a fundamental domain of the sphere $S^2$ can be depicted on flat paper). Five red triangles depict the integral-affine singularities, with their charge [Reference Alexeev, Brunyate and Engel1, Definition 5.3] shown in red.

Figure 13 $B_3(\ell )$ and central fibers for $\ell =(2,0^{15},2,4,6,4,0,4)$ .

The $\operatorname {IAS^2}$ is then admits two involutions, Enriques and del Pezzo, whose actions are shown in orange and blue, respectively. The corresponding Enriques and del Pezzo equators are shown in the respective colors. A triangulation into (green) lattice triangles is chosen, subordinate to both equators. The blue del Pezzo equator, with integer weight $2$ , forms the integral-affine polarization .

The middle image of Figure 13 depicts the corresponding Kulikov model of Enriques K3 degeneration. Triple points ${\widetilde T}_{ijk}= {\widetilde V}_i\cap {\widetilde V}_j\cap {\widetilde V}_k$ are depicted in yellow, double curves ${\widetilde D}_{ij} = {\widetilde V}_i\cap {\widetilde V}_j$ are depicted in black. The self-intersection numbers

$$ \begin{align*}{\widetilde D}_{ij}\big{|}_{{\widetilde V}_i}^2+{\widetilde D}_{ij}\big{|}_{{\widetilde V}_j}^2=-2\end{align*} $$

are written in red (suppressed when both equal $-1$ ). The faces, including an outer face, represent the components ${\widetilde V}_i$ with their anticanonical cycles ${\widetilde D}_i = \sum _j {\widetilde D}_{ij}$ .

The righthand of Figure 13 depicts the dlt model. It consists of eight components $V_i$ , $i=1,\dotsc ,8$ . Double loci and triple points are still depicted in black and yellow. Successive components along the image of the Enriques equator are

$$ \begin{align*}V_1\cup V_2\cup V_4 \cup V_6\cup V_8\cup V_7\cup V_5\cup V_3\cup V_1\end{align*} $$

and the double curves between these two components contain two $A_1$ -singularities of either containing surface, depicted by orange diamonds.

The double covers $({\widetilde V}_6,{\widetilde D}_6) \simeq ({\widetilde V}_7,{\widetilde D}_7)$ are toric, isomorphic to a two-fold corner blowup of as is $({\widetilde V}_1,{\widetilde D}_1)$ , which is the blow-up of the four corners of an anticanonical square in .

The double covers $({\widetilde V}_2,{\widetilde D}_2)\simeq ({\widetilde V}_3,{\widetilde D}_3)$ are the internal blow-ups of at two points $p,q$ on opposite components of an anticanonical square. The Enriques involution acts in the corresponding toric coordinates by , and thus for this involution to lift to the internal blow-up, the two blow-up points must be interchanged: $y(p)=-y(q)$ . This corresponds to choosing the involution anti-invariant periods on for the unique $\times $ -ed node at Cusp 3.

The double covers $({\widetilde V}_4,{\widetilde D}_4)\simeq ({\widetilde V}_5,{\widetilde D}_5)$ are both isomorphic to . Finally, $({\widetilde V}_8,{\widetilde D}_8)$ is a minimal resolution of the $A_{15}$ surface of [Reference Alexeev and Thompson6]. It is the $16$ -fold internal blowup of at $16$ points on a section s, $s^2=8$ . These $16$ points are placed symmetrically with respect to an involution of s and , giving rise to an Enriques involution on $({\widetilde V}_8,{\widetilde D}_8)$ .

The divisor is entirely supported on $V_1\cup _{D_{18}} V_8$ and has intersection number . We have that $R_1^2=0$ and $R_1\subset V_1$ is the image of two fibers of a toric ruling on ${\widetilde V}_1$ while $R_8\subset V_8$ satisfies $R_8^2=8$ as it is the reduced image of the fixed locus ${\widetilde R}_8\subset {\widetilde V}_8$ satisfying ${\widetilde R}_8^2=16$ .

The map to the stable model contracts all components except $V_1$ and $V_8$ to points and contracts $V_1$ along a ruling, leaving the image of $V_8$ as the only component. The normalization

has, as anticanonical boundary , which is self-glued in along an involution fixing $0,\infty \in {\overline D}_8$ . The singularities at $0,\infty \in {\overline V}_8$ are rather complicated.

Example 4.14. $B_5(0,0,2,0^7,1,0^3,1,0,0,0,2,0,2,6)$ : Consider Cusp 5, whose folding symmetry is the same as . This value of $\ell $ dictates that we should put $(2,0)$ , $(-1,1)$ , $(-1,0)$ , $(0,-1)$ end-to-end, then perform a surgery of size $1$ along the edge $(-1,1)$ , to construct $P(\ell )$ . The corresponding sphere $B(\ell )$ is shown in Figure 14, together with a Kulikov and dlt model, following the conventions of Example 4.13.

Figure 14 $B_5(\ell )$ and central fibers for $\ell =(0,0,2,0^7,1,0^3,1,0,0,0,2,0,2,6)$ .

There are four components of the Kulikov model, all of them preserved by the Enriques involution. Both $({\widetilde V}_3,{\widetilde D}_3)$ and $({\widetilde V}_4,{\widetilde D}_4)$ are corner blow-ups of $D_4$ involution pairs. The surface $({\widetilde V}_1,{\widetilde D}_1)$ is the internal blow-up at points on two opposite fibers of an anticanonical square in and the Enriques involution interchanges the blow-up points. Finally, $({\widetilde V}_2,{\widetilde D}_2)$ is a corner blow-up of a $D_8$ involution pair. We have ${\widetilde R}_1^2=0$ , ${\widetilde R}_2^2=8$ , ${\widetilde R}_3^2={\widetilde R}_4^2=4$ .

The components of the stable limit of Enriques surfaces $({\overline V}_3,{\overline D}_3+\epsilon {\overline R}_3)\simeq ({\overline V}_4,{\overline D}_4+\epsilon {\overline R}_4)$ are denoted and $({\overline V}_2,{\overline D}_2+\epsilon {\overline R}_2)$ is denoted by in Section 6, where these surfaces are described further. Only $V_1$ is contracted (along a ruling) in the stable limit .

Example 4.15. $B_2(0,1,0^7, 2, 0^7, 1,0)$ : Note here that since we are at Cusp 2, . To form ${\overline P}(\ell )$ , we put $(0,1)$ , $(-2,0)$ , $(0,-1)$ end-to-end, and then close the base of the polygon by a horizontal line. No Symington surgeries of positive size are made, so $P(\ell )={\overline P}(\ell )$ . We glue to get $B(\ell )$ as in Figure 15. Even though the central horizontal segment is fixed by it does not form part of the support of , see Section 4.2.

Figure 15 $B_2(\ell )$ and central fibers for $\ell =(0,1,0^7, 2, 0^7, 1,0)$ .

Then ${\widetilde V}_2$ and ${\widetilde V}_3$ are each two disjoint copies of $V_2$ and $V_3$ . where ${\widehat L}$ is the strict transform of a line in and C is a conic. The surface $(V_3,D_3)$ is, up to two corner blow-ups, the $D_8$ involution pair as in Example 4.14, but since $({\widetilde V}_3,{\widetilde D}_3)$ is two disjoint copies of such, there is no period-theoretic restriction. Finally, $(V_1,D_1)$ and $(V_4,D_4)$ form the image of the Enriques equator. They are both quotients of smooth toric surfaces by an involution .

Only $V_3$ survives as a component $({\overline V}_3, {\overline D}_3+\epsilon {\overline R}_3)$ . The double locus is a banana curve. In the stable model , each is self-glued by an involution fixing the two nodes in ${\overline D}_3$ . We have $({\overline R}_3)^2=8$ .

5.1 Toroidal compactification for the Coxeter fans.

In Section 3.4, we reviewed the basic results of [Reference Vinberg46], [Reference Vinberg47] on reflection groups acting on hyperbolic lattices. Now we recall applications of this theory to toroidal compactifications.

Let $\Lambda $ be a hyperbolic lattice of rank r and signature $(1,r-1)$ , and let be the positive cone, one of the two halves of the set . In the applications to (semi)toroidal compactifications, instead of the closure one operates with the rational closure , obtained by adding only rational vectors at infinity.

Let W be a group acting on $\Lambda $ , generated by reflections in a set of vectors of $\Lambda $ . Its fundamental domain is

for a set of simple roots $\alpha _i$ with $\alpha _i^2<0$ which is encoded in a Coxeter diagram G. The chamber can be identified with a polyhedron P in the hyperbolic space . The vectors with $v^2=0$ are treated as points at infinity of .

The subgroup $O^+(\Lambda )$ of the isometry group $O(\Lambda )$ is the subgroup of index $2$ that preserves . One has $O^+(\Lambda ) = S.W$ , where S is a subgroup of symmetries of P.

It is a fan iff P has finite volume, which is equivalent to W having finite index in $O(\Lambda )$ . If this condition is satisfied then the faces of are of two types:

1. 1. Type II rays generated by vectors with $v^2=0$ on the boundary of . These are in bijection with the maximal parabolic subdiagrams of G.

2. 2. Type III cones. These are in bijection with elliptic subdiagrams of G.

By [Reference Alexeev and Engel2, Section 3B] the moduli space $F_{(2,2,0)}$ admits a toroidal compactification defined by the collection of fans , one for each $0$ -cusp. These fans are Coxeter fans for the hyperbolic lattices $(18,0,0)_1$ , $(18,2,0)_1$ for the full reflection groups $W_r$ , generated by reflections in the $(-2)$ -roots and in the $(-4)$ -roots of divisibility $2$ . The Coxeter diagrams $G_r(18,0,0)$ and $G_r(18,2,0)$ are given in Figure 5.

By Lemma 2.8 there is an immersion whose image is a Noether–Lefschetz locus in $F_{(2,2,0)}$ . The normalization of the closure of in is then a toroidal compactification for the fans , one for each of $0$ -cusp of . The fans are the intersections of the above fans in the lattices and $(18,2,0)_1$ with the sublattices and $(10,8,0)_1$ , as in Section 3. By Lemma 3.8 and Corollary 3.12 these five fans are the Coxeter fans for the folded Coxeter diagrams $G_r^k$ of Figures 7 and 8. By [Reference Sterk44] the induced groups acting on $e^\perp /e$ are of the form .

5.2 Semitoroidal compactification for the generalized Coxeter fans.

Looijenga’s semitoric, or semitoroidal compactifications of Type IV domains [Reference Looijenga35] generalize toroidal compactifications in several ways. By [Reference Alexeev and Engel3, Theorem 7.18] these are the normal compactifications dominating the Baily–Borel compactification and dominated by some toroidal compactification. They are defined by collections of compatible semifans, one for each Baily–Borel $0$ -cusp. The data for the $1$ -cusps is then uniquely determined. The cones in semifans have rational generators but, unlike in fans, there could be infinitely many generators, and the stabilizer groups of the Type III cones may be infinite.

The generalized Coxeter semifans were defined in [Reference Alexeev, Engel and Thompson5, Section 4D] using the Wythoff construction [Reference Coxeter12], as follows. As above, let W be a reflection group with a fundamental chamber and $G=\{\alpha _i\}$ be the corresponding Coxeter diagram. Divide the vertices of G into two complementary sets $I\sqcup J$ of relevant and irrelevant roots. Let be the subgroup of W generated by the irrelevant roots and let The maximal dimensional cones in the semifan are the chamber and its images under W. Another way to describe is that it is the coarsening of the Coxeter fan obtained by removing the faces of the form in which $\{\alpha _j,\ j\in J'\subset J\}$ is a collection of irrelevant roots.

In [Reference Alexeev and Engel2, Section 9A] the authors defined a specific semitoroidal compactification of the moduli space $F_{(2,2,0)}$ by the collection of two semifans. (Here, ram stands for the ramification divisor.) These are the generalized Coxeter semifans for the Coxeter diagrams of Figure 5 in which the irrelevant roots are those that do not lie on the boundary of the square, resp. the triangle, numbered respectively $0$ – $15$ and $2$ – $18$ . The main theorem of [Reference Alexeev and Engel2] for the moduli space $F_{(2,2,0)}$ says that the normalization of the KSBA moduli compactification ${\overline F}_{(2,2,0)}$ for the pairs $(X,\epsilon R)$ is this semitoroidal compactification.

5.3 The main theorem.

By Section 2.4 there exists a compact moduli space whose points correspond to the pairs $(Z,\epsilon R_Z)$ of Enriques surfaces with numerical polarization of degree $2$ and their KSBA stable limits, for any $0<\epsilon \ll 1$ . This is the closure of in the KSBA moduli space of stable pairs.

6.1 Type III ADE surfaces.

The only ADE surfaces needed in this paper are the ones that appear on the boundary of the KSBA compactification ${\overline F}_{(2,2,0)}$ . They are described in detail in the last section of [Reference Alexeev and Engel2]. Most of them are easy: they are hypersurfaces in projective toric varieties in a way very similar to the construction in Section 2.1.

Figure 16 Some ADE surfaces.

As an example, consider one of the lattice polytopes Q in Figure 16 with an ADE Dynkin diagram fitted into it. The polytopes are in , and the gray dots indicate the sublattice . The Type III polytopes for the ordinary elliptic ADE diagrams have a distinguished vertex with two bold blue sides emanating from it. In the Type II polytopes for the extended ${\widetilde D}{\widetilde E}$ diagram there is a distinguished point in the interior of the bold blue segment. Together with the ends of this segment, it makes three special vertices.

By the standard construction, one associates to Q a toric variety $V_Q$ with an ample line bundle $L_Q$ . Let us define a section of $L_Q$ as the following sum of monomials in . For Type III, each of the three special vertices above gets coefficient $1$ . The coefficients of the vertices in the highlighted Dynkin diagrams are arbitrary numbers . The other coefficients are zero. Concretely:

1. 1. ( $A_5$ ) $f= (1 + y^2 + x^6) + \sum _{i=1}^5 a_ix^i$ ,

2. 2. () $f=(x + y^2 + x^3) + a_1x^2$ ,

3. 3. ( $A_0^-$ ) $f=1 + y^2 + x$ ,

4. 4. () $f=(y + x^2y^2 + x^3) + a_1xy+a_2 + a_3x+a_4x^2$ ,

5. 5. ( $D_5^-$ ) $f=(y^2 + x^2y^2 + x^3) + a_1xy + a_2y + a_3+a_4x+a_5x^2$ ,

6. 6. ( $D^{\prime }_8$ ) $f=(y^2 + x^2y^2 + x^4y) + a_1xy + a_2y + \sum _{i=0}^4 a_{i+3}x^i + a_8x^3y$ ,

7. 7. () $f=(y^3 + x^2y^2 + x^4) + a_1xy + a_2y^2 + a_3y+ a_4 + a_5x + a_6x^2 + a_7x^3$ .

The corresponding del Pezzo ADE surface is the toric variety $Y=V_Q$ together with the boundary $C=C_1+C_2$ for the two blue sides, and the divisor B is $(f)$ . Combinatorially the condition $B \sim -2(K_Y+C)$ is equivalent to the condition that the other sides of Q have lattice distance $2$ from the distinguished point.

We now put these polytopes in . Let $p_0$ be the position of the distinguished vertex, we then add another vertex at the point $p_0 + (0,0,2)$ to which we associate the monomial $z^2$ . Let P be the pyramid with the apex at the new vertex and with base Q. Associated with it we have a $3$ -dimensional polarized toric variety $(V_P,L_P)$ and a section $z^2 + f$ of $L_P$ . An anticanonical ADE surface X is the zero set of this section, so it is a hypersurface in $V_P$ . It comes with a del Pezzo involution the quotient map is , the boundary is $D=\pi ^{-1}(C)$ , and the ramification divisor is $R=\pi ^{-1}(B)$ .

Varying the free coefficients $a_i$ we get a family over , where n is the rank of the Dynkin diagram. This is the quotient of the algebraic torus by the Weyl group $W(\Lambda )$ for the ADE root lattice $\Lambda $ with the weight lattice $\Lambda ^*$ . So it naturally comes from a family over a torus.

6.2 Decorations.

Because of $z^2$ and the double cover, the vertices in are clearly distinguished; let’s call them even. When the end of a bold blue edge is even, this edge is long, of lattice length $2$ . Then we use no decorations. When this end is odd, the edge is short, of lattice length $1$ . To indicate that it is short, we use a minus or a prime sign. We also use primes to distinguish shapes where the long leg pokes into the interior of Q.

The classification of del Pezzo ADE surfaces $(Y,C+\textstyle \frac {1+\epsilon }2 B)$ in [Reference Alexeev and Thompson6] is divided into pure and primed shapes. The surfaces for the pure shapes are all toric. The surfaces for some of the primed shapes are toric, but not in general. They are obtained from pure shapes by making a blow up at a point $x\in D\cap R$ on X, resp. a weighted blowup at a point $y\in C\cap B$ in Y. For each side $D_1, D_2$ , the set $D_i\cap R$ is either a single point (if the side is short) or two points (if it is long). For example priming $A_n$ on a long side once gives $A^{\prime }_n$ and twice gives $A^{\prime \prime }_n$ . Priming $A^-_n$ on a short side is denoted by $A^+_n$ .

The blow up disconnects $D_i$ from R at that point. If all points in $D_i\cap R$ are blown up, for the strict preimages we have $D^{\prime }_i \cdot R'=0$ . In this case the linear system $|mR'|$ for $m\gg 0$ contracts $D^{\prime }_i$ and the corresponding ADE surface has fewer boundary components. Thus, the surfaces for example for the shapes $A^{\prime \prime }_n$ and $A^+$ have only one boundary component, and for the shapes , , have zero boundary components.

6.3 Type II ADE surfaces.

The construction for the Type II polytopes is similar. The ends of the bold blue edge have coefficients $1$ in f, and the distinguished interior point has coefficient . For clarity, in Figure 16 one has

1. 1. ( ${\widetilde E}_7$ ) $f= (y^4 + \lambda x^2y^2 + x^4) + a_1 xy + a_2y^3+a_3y^2 + a_4y+ a_5 + a_6x + a_7x^2 + a_8x^3$ ,

2. 2. ( ${\widetilde E}_8^-$ ) $f = (y^3+\lambda x^2y^2 + x^6) + a_1xy + a_2y^2+a_3y + \sum _{i=0}^5 a_{4+i}x^i$ ,

3. 3. ( ${\widetilde D}_8^-$ ) $f = (y^2+\lambda x^2y^2 + x^4y^2) + a_1xy + a_2x^3y + a_4x^4y + \sum _{i=0}^4 a_{i+5}x^i$ .

The coefficients for the nodes of the extended Dynkin diagram are arbitrary numbers , not all of them zero, and they are now treated as homogeneous coordinates of weight equal to the lattice distance from the bold blue edge. Thus, for a fixed $\lambda $ one gets a family of sections $z^2+f$ of $L_P$ and a family of anticanonical KSBA stable pairs $(X,D+\epsilon R)$ parameterized by a weighted projective space. For ${\widetilde E}_7$ it is , for ${\widetilde E}_8$ it is , and for ${\widetilde D}_{2n}$ it is . The weight of the coordinate $a_i$ is the fundamental weight of the Dynkin diagram on the associated monomial, shown in Figure 16.

The restriction of $z^2+f$ to the divisor corresponding to the bold blue line gives a double cover of which is an elliptic curve. Varying $\lambda $ we get a family of ${\widetilde A}{\widetilde D}{\widetilde E}$ surfaces parameterized by a bundle of weighted projective spaces over the j-line. This is the same bundle of weighted projective spaces that appeared in [Reference Looijenga32], [Reference Looijenga33], [Reference Pinkham41], whose fiber over $j(E)$ is the Weyl group quotient of for the relevant root lattice $\Lambda $ .

The ${\widetilde A}_{2n-1}$ surfaces do not directly correspond to polytopes. These surfaces are double covers of cones over elliptic curves branched in a bisection. The easiest description, closest to toric is to use the Tate curve. For each define the theta function $\theta _i$ as the formal power series

It converges for any with $|q|<1$ and defines a section of $L^2$ , where L is an ample line bundle of degree n on the elliptic curve . For any not all zero, $g(x) = \sum c_i\theta _i$ is a nonzero section of $L^2$ and $f(x,y) = y^2 + g(x)$ is a section on the square of the tautological line bundle on . It also defines a section of a line bundle on the surface Y that is a cone over E, obtained by contracting an exceptional section of ${\widetilde Y}$ . Finally, $z^2 + f(x,y)$ defines a double cover $X\to Y$ and the covering involution is $ (x,y,z)\to (x,y,-z). $

6.4 Anticanonical ADE surfaces with two commuting involutions.

Let $(X,D)$ be a log canonical pair with $K_X+D\sim 0$ . Pick a generator $\omega $ of the space . Just as for K3 surfaces, an involution $\iota $ is called symplectic if $\iota ^*\omega =\omega $ and nonsymplectic if $\iota ^*\omega =-\omega $ . By looking at a local equation $dx\wedge \frac {dy}y$ of $\omega $ near the boundary, it is easy to see that for a nonsymplectic involution the quotient map $X\to X/\iota $ is not ramified along any irreducible component of D.

Proposition 6.1. Let $\pi \colon (X,D+\epsilon R)\to (Y,C+\textstyle \frac {1+\epsilon }2 B)$ be the anticanonical and del Pezzo ADE surfaces, and be the anticanonical involution such that . Suppose that is another nonsymplectic involution commuting with such that and the induced involution $\tau \colon Y\to Y$ both have finite fixed sets. Then there exists a diagram of log canonical pairs

(6.2)

in which

1. 1. $\psi \colon X\to Z$ is the quotient by and $\psi '\colon X\to Z'$ is the quotient by the symplectic involution .

2. 2. $R_Z=\frac 12\rho ^*(B_W)$ and $R_{Z'}=\frac 12\rho '{}^*(B_W)$ are reduced divisors and one has $R = \psi ^*(R_Z) = \psi '{}^*(R_{Z'})$ .

3. 3. $D_Z=\rho ^*(C_W)$ and $D_{Z'}=\rho '{}^*(C_W)$ are reduced divisors and one has $D = \psi ^*(D_Z) = \psi '{}^*(D_{Z'})$ .

4. 4. $(W,C_W+\textstyle \frac {1+\epsilon }2 B_W)$ is a del Pezzo ADE surface, and $(Z',D_{Z'}+\epsilon R_{Z'})$ is an anticanonical ADE surface which is its index- $1$ cover.

5. 5. For any $\epsilon $ one has

   $$ \begin{align*} &K_X + D + \epsilon R = \psi^*(K_Z + D_Z + \epsilon R_Z) = \psi'{}^*(K_{Z'} + D_{Z'} + \epsilon R_{Z'}) \\ &K_Z + D_Z + \epsilon R_Z = \rho^*(K_W + C_W + \textstyle\frac{1+\epsilon}2 B_W)\\ &K_{Z'} + D_{Z'} + \epsilon R_{Z'} = \rho'{}^*(K_W + C_W + \textstyle\frac{1+\epsilon}2 B_W). \end{align*} $$

6. 6. $2(K_Z+D_Z)\sim 0$ but $K_Z+D_Z\not \sim 0$ .

7. 7. $\rho '$ is branched in $C_W$ and a finite subset of $\operatorname {Branch}(\varphi )$ .

8. 8. $\rho $ is branched in $C_W$ , a finite subset of $\operatorname {Branch}(\varphi )$ , and the irreducible components of $C_W$ which are part of the branch locus of $Y\to W$ .

9. 9. For any $p\in \operatorname {Branch}(\varphi )\setminus C_W$ , one has $p\in \operatorname {Branch}(\rho )$ iff $p\notin \operatorname {Branch}(\rho ')$ .

One could say that the ADE surfaces $Z'\to W$ are obtained by folding the ADE surfaces $X\to Y$ by the symplectic involution , and $Z\to W$ are obtained from $X\to Y$ by folding by the nonsymplectic involution . The index- $1$ cover $\rho '\colon Z'\to W$ and the index $2$ cover $\rho \colon Z\to W$ are dual in a similar way to Remark 2.2.

In the next two sections we find several examples of such foldings, naturally corresponding to foldings of ADE Dynkin diagrams, producing some non simply laced Dynkin diagrams of B and C types. The smaller versions of these examples can be found in Figures 9, 10, 18, and 19. The involutions appearing at Cusp 5 are described in Section 6.5, and those appearing at other cusps in Section 6.6 below.

For the parabolic diagrams, we follow Vinberg’s conventions [Reference Vinberg46]: the ${\widetilde D}_n$ diagram has two forks, ${\widetilde B}_n$ has one, and ${\widetilde C}_n$ is a chain without forks.

6.5 Quotients by $\pm 1$ in the torus.

We first consider the Enriques involution on an ADE anticanonical surface $X = \{z^2 + f(x,y)=0\}$ that is given by the same formula as in Section 2.1. The pairs $(X,D)$ of this type appear very naturally in Horikawa’s construction, when degenerates to a stable surface $Y=\cup (Y_i,D_i)$ . As in Section 2.1, let . We have .

Let Q be one of the ADE polytopes of Sections 6.1 and 6.3 above and assume that the monomials of $f(x,y)$ lie in . This means that the bold blue edges are long, the Dynkin diagram ends in odd vertices on the boundary, and there are no minus or prime decorations.

We then have four surfaces as in Diagram (6.2). Our notation for the covers will be $\alpha :2={}_2\beta \subset \gamma $ , where $\alpha $ is the ADE type of $X\to Y$ , $\gamma $ is the ADE type of $Z'\to W$ , and ${}_2\beta $ is the ABCDE type of the index- $2$ cover $Z\to W$ ; or simply $\alpha :2={}_2\beta $ if $\beta =\gamma $ .

Thus, we describe the index- $2$ anticanonical surface $(Z,D_Z)$ in two ways:

1. 1. as the quotient of $(X,D)$ by , and

2. 2. as an index- $2$ cover of a del Pezzo ADE surface $(W,C_W)$ .

The first way presents Z as a hypersurface in the toric variety $V_P$ for the same polytope P as X but for a new lattice .

The branch locus of $\varphi \colon Y\to W$ consists of:

1. 1. The torus-fixed points corresponding to the vertices of Q. Let us denote the distinguished vertex of Q by c and the adjacent corners of Q by $v_i$ .

2. 2. The boundary divisors corresponding to the sides $(c,v_i)$ of Q which are long with respect to the lattice . We number them by i with $i\equiv 0\pmod 4$ .

By Proposition 6.1, $\rho \colon Z\to W$ is branched at the point for the distinguished vertex c and in the boundary divisors for the sides $C_i = (c,v_i)$ with $i\equiv 0\pmod 4$ . There are two $A_1$ singularities over each point in $C_i\cap B_W$ . Also, $\rho $ is unramified over the points for $v_i$ with $i\equiv 2\pmod 4$ .