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85  Degree-2 Coble Surfaces and the Coble Heegner Divisor

A mathematical research report on the established framework, precise constructions, open comparison problems, and proof strategies

85.1 Introduction and logical status

85.1.1 Purpose

This report fixes a common mathematical language for the study of the degree-2 Coble locus associated with the \((-2)\)-Heegner arrangement in the period space of numerically degree-\(2\) polarized Enriques surfaces. It has four purposes.

1. It records the definitions, constructions, lattice identifications, local calculations, and numerical formulas that have been isolated with a level of precision suitable for further research.

2. It distinguishes those statements that follow from standard results or direct calculations from those that remain comparison theorems, lifting problems, or compactification problems.

3. It formulates two principal approaches to the compactification problem: the Heegner-restriction approach and the autonomous Coble approach.

4. It gives a theorem-by-theorem research program for completing either approach and later identifying the two constructions.

The intended reader is assumed to know the theory of K3 and Enriques surfaces, integral lattices, type-IV period domains, Baily–Borel and toroidal compactifications, Kulikov models, KSBA stable pairs, Nikulin’s theory of \(2\)-elementary involutions, and the compactification framework developed by Alexeev and Engel. Standard terminology is cited rather than reproved; see Section 85.2 and the bibliography.

The report is not a proof of the final Coble compactification theorem. It is a mathematical specification of the objects and the exact statements that a proof must establish. This distinction is essential because several plausible computations discussed below are computations of finite quadratic spaces or hyperplane slices, not yet computations of arithmetic cusps or Vinberg chambers.

85.1.2 Convention on logical status

The following convention is used throughout.

• A statement labeled Theorem, Proposition, Lemma, or Corollary is either a standard cited result, a result of AEGS, or a direct calculation whose hypotheses are stated explicitly.

• A statement labeled Required theorem, Required lemma, or Problem is not presently established in the Coble setting. It records an exact unit of work needed by one of the proposed strategies.

• A Candidate construction is a mathematically defined construction whose relation to the desired moduli problem has not yet been proved.

• A Computational record is a reproducible finite or symbolic calculation. It is not promoted to an arithmetic or geometric theorem unless the relevant lifting or realization statement is separately proved.

• A Warning records a point at which an earlier line of reasoning made an invalid inference or conflated two distinct objects.

Thus the report uses formal mathematical language without disguising the logical status of unfinished comparison statements.

85.1.3 Central compactification problem

Let \(T_{\mathrm{En}} \cong U\oplus U(2)\oplus E_8(2)\) be the Enriques anti-invariant lattice in the AEGS degree-\(2\) construction, and let \(\alpha\in T_{\mathrm{En}}\) be a primitive vector with \(\alpha^2=-2\). The hyperplane \(\mathbb D(T_{\mathrm{En}})\cap\alpha^\perp\) is the period domain of the Coble Heegner locus. The final problem is to identify the normalization of the KSBA closure of that locus as an explicit semitoroidal compactification, including the arithmetic group, all Baily–Borel cusps, the cusp reflection chambers, the integral-affine and dlt models, the relevant semifans, and the toroidal versus strictly semitoroidal locus.

The ambient AEGS theorem supplies the model for the required level of specificity: the compactification is meaningful only after the lattices, groups, divisors, cusp data, Coxeter data, and stable-pair contractions have all been identified. The research prompt explicitly treats those data as part of the theorem rather than optional consequences.

85.1.4 Principal distinction between the two strategies

There are two conceptually different ways to organize the proof.

1. Heegner-restriction strategy. Begin with the AEGS Enriques moduli problem and regard the Coble locus as the normalization of a fixed \((-2)\)-Heegner component. The principal new work is to justify passage to arithmetic quotients, normalize the boundary restriction, solve the cusp lifting problem, identify the restricted semifans, and prove that the resulting KSBA family loses no moduli.

2. Autonomous Coble strategy. Define a direct moduli problem of K3 covers with a Coble involution, a commuting del Pezzo involution, a degree-\(4\) class, and a distinguished exceptional root. Reproduce direct analogues of the AEGS period, cusp, Vinberg, integral-affine, dlt, recognizable-divisor, and semitoroidal theorems. Only after this independent theory is complete does one prove that it agrees with the Enriques Heegner construction.

The second strategy avoids using the Enriques compactification as a black box, but it creates a separate comparison theorem at the end. The first strategy uses more existing geometry, but it requires careful normalization and monodromy arguments that cannot be replaced by the phrase “restrict the semifan.”

85.2 Terminology, notation, and standard background

85.2.1 General conventions

The ground field is \(\mathbb C\). All lattices are free abelian groups of finite rank equipped with a nondegenerate symmetric bilinear form. The K3 lattice is \(L_{K3}=II_{3,19}\cong U^3\oplus E_8^2\), where \(E_8\) is negative definite. If \(M\) is a lattice, then \(M(n)\) denotes the same abelian group with its bilinear form multiplied by \(n\). The orthogonal group preserving a chosen component of the positive cone is denoted \(O^+(M)\).

For an even lattice \(M\), its discriminant group and discriminant quadratic form are \(A_M=M^*/M\) and \(q_M:A_M\longrightarrow \mathbb Q/2\mathbb Z\).

A primitive embedding is an embedding whose cokernel is torsion free. The classification and gluing theory of even \(2\)-elementary lattices are used in the sense of Nikulin (Nikulin 1980).

85.2.2 Surface singularities and covers

A rational double point or ADE singularity is a Du Val surface singularity. An \(A_1\)-singularity is analytically \(\{xy-z^2=0\}\subset\mathbb A^3\).

The cyclic quotient singularity

[ (1,1) =A^2_{u,v}/_4, (u,v)=(u,v), ]

has quotient group of order \(4\) and canonical index \(2\). Its canonical index-one cover is the \(A_1\)-singularity \(\mathbb A^2/\{\pm1\}\), not the smooth plane. The smooth plane is the canonical degree-\(4\) quasi-'etale cover.

A normal surface is \(R_1\) and \(S_2\). In particular, the \(\frac{1}{4}(1,1)\)-singularity is normal. Nonnormal surfaces can nevertheless occur at the KSBA boundary through conductor self-gluings; this phenomenon is logically separate from the local quotient singularity.

The terms index-one cover, reflexive power, deminormal, semi-log-canonical, and divisorially log terminal are used in the standard MMP sense (Kollár and Mori 1998; Kollár 2023; The Stacks Project Authors 2026). In particular, an slc pair is tested by requiring the normalized pair with conductor to be log canonical, not klt.

85.2.3 K3, Enriques, and Coble terminology

A K3 surface is a smooth projective surface \(X\) with \(K_X\sim0\) and \(H^1(X,\mathcal O_X)=0\). An Enriques surface is a smooth projective surface \(Z\) with \(2K_Z\sim0\), \(K_Z\not\sim0\), and \(q(Z)=0\); see (Cossec and Dolgachev 1989; Cossec et al. 2024).

The word Coble surface is used in two closely related senses in the literature. To remove ambiguity, this report uses the following notation.

Definition 85.1 (singular Coble quotient) A singular Coble quotient is a normal rational surface \(V^\sharp\) obtained as the quotient of a K3 surface \(X^\sharp\) with one \(A_1\)-singularity by a nonsymplectic involution that fixes the singular point and is otherwise free in codimension one. The quotient has one singularity of type \(\frac{1}{4}(1,1)\).

Definition 85.2 (resolved Coble surface) The resolved Coble surface \(\widetilde V\) is the minimal resolution of \(V^\sharp\). It is a smooth rational surface satisfying

[ |-K_{V}|=, |-2K_{V}| ]

on the classical one-node locus.

The resolved surface is the object traditionally called a classical Coble surface. The singular quotient is the natural \(K\)-trivial object for the small-coefficient KSBA problem.

85.2.4 Polarizations

A quasipolarization is a nef and big line bundle. A polarization is ample. This distinction is essential in the Coble setting: the natural class on \(\widetilde V\) is orthogonal to the anti-bicanonical \((-4)\)-curve and is therefore not ample. It becomes ample after contracting the complete null locus.

85.2.5 Period domains and Heegner divisors

For a lattice \(T\) of signature \((2,n)\), a connected type-IV domain is

[ D(T) ={ []P(T_C) : (,)=0, (,)>0 }^{+}. ]

If \(\Gamma\subset O^+(T)\) is arithmetic, the quotient \(\Gamma\backslash\mathbb D(T)\) is a quasi-projective orthogonal modular variety. For a negative vector \(r\in T\), the image of \(\mathbb D(T)\cap r^\perp\) is a Heegner divisor or arrangement divisor. See (Baily and Borel 1966; Looijenga 2003) for the compactification theory and (Nikulin 1980; Namikawa 1985) for the relevant Enriques lattice geometry.

A primitive isotropic line in \(T\) determines a zero-dimensional Baily–Borel boundary component, called a \(0\)-cusp. A primitive isotropic plane determines a one-dimensional boundary component, called a \(1\)-cusp. The arithmetic orbit, not merely the rational isotropic subspace, is part of the cusp datum.

85.2.6 Reflection theory

Let \(M\) be hyperbolic. For a negative vector \(r\in M\), the reflection is

[ w_r(x)=x-r. ]

The vector \(r\) is a root if \(w_r\in O(M)\); equivalently, \(2\operatorname{div}(r)\in r^2\mathbb Z\).

A Coxeter chamber is a fundamental chamber for a reflection group acting on the positive cone. Its codimension-one walls are perpendicular to simple roots. A maximal parabolic subdiagram determines an isotropic ray and hence a Type II direction. An elliptic subdiagram determines a Type III cone. The terminology and algorithms are those of Vinberg (Vinberg 1972, 1975).

85.2.7 Degenerations and integral-affine structures

A Kulikov model is a semistable \(K\)-trivial model of a degeneration of K3 surfaces (Kulikov 1977; Persson and Pinkham 1981). Type II degenerations have dual complex a segment; Type III degenerations have dual complex a \(2\)-sphere. Type III dual complexes carry canonical integral-affine structures with singularities (Engel 2018; Engel and Friedman 2021; Gross et al. 2015a).

A divisor model is a Kulikov or dlt model carrying a nef extension of the distinguished divisor. Its integral-affine polarization is a weighted balanced graph on the dual complex (Alexeev et al. 2022; Alexeev and Engel 2022). A visible curve is a curve class represented by a path joining affine singularities with parallel monodromy. Collapsing such a path can merge two \(I_1\) affine singularities into an \(I_2\) singularity.

85.2.8 KSBA and semitoroidal compactifications

A KSBA stable pair \((X,B)\) is a projective deminormal pair such that \((X,B)\) is slc and \(K_X+B\) is ample and \(\mathbb Q\)-Cartier (Kollár 2023). For small-coefficient polarized Calabi–Yau pairs, boundedness and independence of sufficiently small \(\epsilon\) are used in the form of (Kollár and Xu 2020; Birkar 2023).

A toroidal compactification is defined by compatible rational polyhedral fans at the rational boundary components. A semitoroidal compactification is defined by compatible semifans; cones may have infinitely many rational generators (Looijenga 2003; Alexeev and Engel 2023). A generalized Coxeter semifan is obtained from a Coxeter fan by removing the walls generated entirely by roots that are invisible to the stable-pair combinatorics (Alexeev et al. 2023, 2025).

85.3 The ambient AEGS framework

85.3.1 The projective Klein-four diagram

Let \(Y=\mathbb P^1\times\mathbb P^1\) and \(\tau(x,y)=(-x,-y)\) on the dense torus. Let \(B\in|-2K_Y|=|\mathcal O_Y(4,4)|\) be \(\tau\)-invariant, with at worst ADE singularities, and assume that \(B\) avoids the four fixed points of \(\tau\). The double cover \(\pi:X\longrightarrow Y\) branched along \(B\) is a K3 surface with at worst ADE singularities. Writing \(X\) as \(z^2+f(x,y)=0\), one has three commuting involutions

[ _{}(x,y,z)=(x,y,-z), ]

[ _{}(x,y,z)=(-x,-y,-z), ]

[ {}(x,y,z)=(-x,-y,z) ={}_{}. ]

The first two are nonsymplectic and the product is symplectic. The quotient \(Z=X/\iota_{\mathrm{En}}\) is generically Enriques, while \(X/\iota_{\mathrm{Nik}}\) is a K3 surface with eight \(A_1\) singularities. The ramification divisor used in the AEGS stable-pair problem is the ramification of the del Pezzo involution, not the fixed locus of the Enriques involution.

85.3.2 The AEGS lattice package

The involutions act on \(L_{K3}=U^3\oplus E_8^2\) with eigenspaces

[ S_{}U(2), T_{}UU(2)E_8^2, ]

[ S_{}U(2)E_8(2), T_{}UU(2)E_8(2). ]

The degree-\(4\) K3 polarization is a vector \(H\in S_{\mathrm{dP}}\subset S_{\mathrm{En}}\) with \(H^2=4\).

Theorem 85.3 (AEGS embedding rigidity) The primitive embedding chain

[ T_{}T_{} L_{K3} ]

is unique up to \(O(L_{K3})\).

This theorem is the model for the stronger labeled rigidity statement needed for the Coble problem. It prevents later constructions from depending on a hidden coordinate choice.

85.3.3 AEGS arithmetic groups and period maps

AEGS define

[ {,2} =( {gO(L{K3}):gI_{}=I_{}g,  g(H)=H} O(T_{}) ). ]

They prove that this is the stabilizer of the Enriques period subdomain inside the del Pezzo arithmetic quotient. Consequently the period-domain inclusion gives a finite generically injective map \(\mathcal F_{\mathrm{En},2} \longrightarrow \mathcal F_{(2,2,0)}\).

The group definition is geometric: it is taken from the common K3 lattice, the involution, and the polarization. This is the standard that any direct Coble group must meet.

85.3.4 The AEGS compactification theorem

AEGS define stable pairs \((Z,\epsilon R_Z)\), where \(R_Z\) is the divisorial ramification of the descended del Pezzo involution. Their main theorem identifies the normalization of the KSBA closure with a semitoroidal compactification described by five explicit semifans. The proof has the following indispensable stages.

1. classify the Baily–Borel cusps;

2. compute the reflective cusp lattices;

3. prove a root-folding theorem with a converse;

4. prove that chamber intersection gives a fundamental chamber;

5. lift the cusp involutions through the discriminant gluing to the full period lattice;

6. realize monodromy by integral-affine divisor models;

7. extend the involution algebraically;

8. take dlt quotients and the relative Proj;

9. identify the generalized Coxeter semifans; and

10. prove that the classifying map is finite and that no further coarsening occurs.

This architecture will be used repeatedly below as a list of theorem types, not as a black-box proof of the Coble case.

85.4 Direct Coble geometry from invariant \((4,4)\) curves

85.4.1 The invariant linear system

The vector space \(H^0\bigl(Y,\mathcal O_Y(4,4)\bigr)^\tau\) has basis the monomials whose two affine exponents have even sum. It has dimension \(13\), so its projectivization is \(\mathbb P^{12}\).

Fix a \(\tau\)-fixed point \(p\in Y\). Choose local coordinates \((x,y)\) centered at \(p\). An invariant local equation has the form

[ f(x,y) =a_{00}+a_{20}x^{2+a_{11}xy+a_{02}y}2 +4. ]

There are no linear terms.

Lemma 85.4 (node condition) The curve \(B=(f=0)\) has an ordinary node at \(p\) if and only if \(a_{00}=0\) and \(a_{11}^2-4a_{20}a_{02}\ne0\).

Proof. The first condition says that \(p\in B\). Invariance kills the linear part. The second condition says that the quadratic tangent cone is nondegenerate. This is the analytic criterion for an ordinary plane node.

A general member satisfying these conditions is smooth away from \(p\). This is an open condition, and it is enough to exhibit one member having no other singularities. A symbolic example is recorded in Section 85.22.

85.4.2 Toric interpretation

Let \(Q\) be the square polytope of \(\mathcal O_Y(4,4)\). The quotient \(W=Y/\tau\) is the toric surface defined by the same rational polygon but with cocharacter lattice \(\mathbb Z^2_{\mathrm{ev}} =\{(a,b)\in\mathbb Z^2:a+b\in2\mathbb Z\}\).

The invariant monomials are precisely the lattice points of \(Q\) in this even sublattice. If \(P\) is the pyramid over \(Q\) with apex corresponding to \(z^2\), the K3 hypersurface is defined in the toric threefold \(V_P\). The quotient by the sign involution uses the lattice

[ Z^3_{} ={(a,b,c)Z^3:a+b+cZ}. ]

Passing through a chosen torus-fixed point is the vanishing of the coefficient of the corresponding vertex monomial. The nondegeneracy of the quadratic part is the local node condition of the preceding lemma.

85.4.3 The nodal K3 cover and the Coble quotient

Let \(X^\sharp=\{z^2=f(x,y)\}\longrightarrow Y\).

Then \(X^\sharp\) has an \(A_1\)-singularity over \(p\). The involutions

[ {}(x,y,z)=(x,y,-z), {}(x,y,z)=(-x,-y,-z) ]

commute. The first is the del Pezzo deck involution. The second fixes the \(A_1\)-point.

Proposition 85.5 (local Coble quotient) The quotient \(X^\sharp/\iota_{\mathrm{Co}}\) has a singularity of type \(\frac{1}{4}(1,1)\) at the image of the node.

Proof. Write

[ X^{} {xy-z^2=0} A^2{u,v}/{} ]

using \(x=u^2\), \(y=v^2\), and \(z=uv\). The involution on the node lifts to \((u,v)\mapsto(iu,iv)\). The composite quotient is therefore \(\mathbb A^2/\mu_4\) with scalar weights \((1,1)\).

85.4.4 Equivariant smoothing

Vary the constant coefficient \(a_{00}\) in a one-parameter family. For \(a_{00}\ne0\), the branch curve avoids the fixed point \(p\), the K3 cover is smooth near the corresponding fiber, and the involution \(\iota_{\mathrm{Co}}\) has no point over \(p\). If the branch curve avoids all four \(\tau\)-fixed points, then the same fixed-point calculation shows that \(\iota_{\mathrm{Co}}\) is fixed-point free. Its quotient is an Enriques surface.

Proposition 85.6 (explicit Coble-to-Enriques smoothing) The invariant \((4,4)\) family obtained by varying \(a_{00}\) gives an explicit equivariant smoothing of the singular Coble quotient to Enriques quotients.

This proposition proves existence of one smoothing mechanism. It does not imply that every abstract smoothing of the K3 cover carries an extension of the node-fixing involution.

85.4.5 Dimension and coverage

The projective invariant linear system has dimension \(12\). Imposing \(a_{00}=0\) gives a hyperplane, and the connected centralizer of \(\tau\) in \(\operatorname{Aut}(Y)\) has dimension \(2\). Thus the expected moduli dimension is \(12-1-2=9\).

The finite symmetry group permutes the four fixed points.

Problem 85.7 (Required theorem (projective coverage and intrinsic reconstruction)) The quotient of the open nodal invariant \((4,4)\) locus by the full centralizer of \(\tau\) is the complete direct moduli stack of generic degree-\(2\) Coble data. Equivalently, the projective diagram can be reconstructed intrinsically from the Coble pair, and the resulting period map has image the full relevant period-domain component.

A dimension count and one explicit family do not prove this theorem. A proof must establish irreducibility at the level of the actual arithmetic group and recover the K3 cover, the two involutions, and the degree-\(4\) class from the Coble datum.

85.5 The Coble lattice package

85.5.1 The orthogonal complement lattice

Write \(T_{\mathrm{En}} \cong U\oplus E(2)\) and \(E=U\oplus E_8\).

Choose a standard basis \(e,f\) of the first copy of \(U\), and define \(\alpha=e-f\) and \(\eta=e+f\).

Then \(\alpha^2=-2\), \(\eta^2=2\), and \((\alpha,\eta)=0\).

The notation \(\eta\) is deliberately used here: the letter \(H\) is reserved for the degree-\(4\) K3 polarization in the AEGS construction.

Proposition 85.8 (Coble anti-invariant lattice) The orthogonal complement of \(\alpha\) in \(T_{\mathrm{En}}\) is

[ T_{} :=^{T{}} U(2)E_8(2). ]

It has signature \((2,9)\), rank \(11\), discriminant group \((\mathbb Z/2)^{11}\), and \(2\)-elementary invariant \(\delta=1\). Equivalently, \(T_{\mathrm{Co}}\cong I_{2,9}(2)\).

Proof. The first copy of \(U\) decomposes over the sublattice \(\mathbb Z\alpha\oplus\mathbb Z\eta\), and the orthogonal complement of \(\alpha\) there is \(\mathbb Z\eta\cong\langle2\rangle\). The remaining summand is \(E(2)=U(2)\oplus E_8(2)\). The stated invariants follow directly.

85.5.2 The invariant lattice and the reflection twist

Let \(I_{\mathrm{En}}\) be the Enriques involution on the K3 lattice, acting as \(+1\) on \(S_{\mathrm{En}}\) and as \(-1\) on \(T_{\mathrm{En}}\). Let \(w_\alpha\) be the reflection in the root \(\alpha\).

Construction 85.9 (reflection twist) Define

[ I_{}:=w_I_{}. ]

Since \(I_{\mathrm{En}}(\alpha)=-\alpha\), the two factors commute.

Proposition 85.10 (eigenspaces of the reflection twist) One has

[ L_{K3}^{I_{}=1} =S_{}Z U(2)E_8(2), ]

and

[ L_{K3}^{I_{}=-1} =^{T{}} =T_{}. ]

Thus the invariant lattice is

[ S_{} U(2)E_8(2) I_{1,10}(2). ]

Proof. The reflection fixes \(\alpha^\perp\) and sends \(\alpha\) to \(-\alpha\). The formula follows by separating \(S_{\mathrm{En}}\), \(\mathbb Z\alpha\), and \(\alpha^\perp_{T_{\mathrm{En}}}\).

The abstract eigenspace computation agrees with the lattice type appearing in the period theory of classical Coble surfaces (Dolgachev and Kondō 2013). It does not, by itself, prove that every family of direct Coble data realizes the same labeled embedding in \(L_{K3}\).

85.5.3 The local geometric meaning of the root

Let \(\widetilde X\longrightarrow X^\sharp\) be the minimal resolution of the \(A_1\)-point, and let \(E\subset\widetilde X\) be the exceptional \((-2)\)-curve. In the explicit local model, the lift of \(\iota_{\mathrm{Co}}\) fixes \(E\) pointwise, while the lift of \(\iota_{\mathrm{dP}}\) acts nontrivially on \(E\) with two fixed points. The cohomology class \([E]\) is the geometric realization of \(\alpha\) after a choice of marking.

The involution \(I_{\mathrm{Co}}\) is therefore the expected cohomological involution on the resolved cover. Turning this expectation into a global statement in families requires a simultaneous resolution and a rigidity theorem; see Section 85.5.4.

85.5.4 Labeled lattice data

The abstract isomorphism type of \(T_{\mathrm{Co}}\) does not determine the geometric moduli problem. The relevant datum is a labeled tuple

[ L_{,2} =( L_{K3}, I_{}, I_{}, H, , ), ]

where

• \(I_{\mathrm{Co}}\) and \(I_{\mathrm{dP}}\) are commuting nonsymplectic involutions;

• \(H^2=4\) is fixed by both involutions;

• \(\alpha^2=-2\), \(H\cdot\alpha=0\), and \(\alpha\) is fixed by \(I_{\mathrm{Co}}\);

• \(\mcc\) is the chosen positive/nef chamber in which \(\alpha\) is effective and \(H\) is nef and big;

• the product \(I_{\mathrm{Co}}I_{\mathrm{dP}}\) has the required symplectic character.

Problem 85.11 (Required theorem (labeled Coble rigidity)) Any two tuples \(\mathfrak L_{\mathrm{Co},2}\) arising from the direct projective construction are conjugate by an element of \(O(L_{K3})\) preserving the labels, the degree-\(4\) class, the effective root, and the chosen chamber.

This is stronger than uniqueness of a primitive embedding of \(S_{\mathrm{Co}}\) into \(L_{K3}\). It is the Coble analogue of the AEGS embedding-rigidity lemma and is needed before any direct period quotient can be declared coordinate independent.

85.5.5 The embedded comparison chain

The Heegner strategy uses the chain

[ T_{} =^{T{}} T_{} T_{} L_{K3}. ]

The autonomous strategy begins with the leftmost lattice as the anti-invariant lattice of \(I_{\mathrm{Co}}\) and must later construct and identify the two larger lattices. The comparison theorem must show that the two chains are conjugate as labeled embedded chains, not merely termwise isometric.

85.6 The resolved surface, the quasipolarization, and the stable divisor

85.6.1 The resolved quotient diagram

Let \(\psi:\widetilde X\longrightarrow\widetilde V\) be the quotient of the resolved K3 cover by \(\iota_{\mathrm{Co}}\). Let \(E\subset\widetilde X\) be the fixed exceptional curve and let \(C:=\psi(E)\subset\widetilde V\).

Then \(\psi^*C=2E\).

Let \(H=\pi^*\mathcal O_Y(1,1)\) on \(\widetilde X\). The class \(H\) satisfies \(H^2=4\) and \(H\cdot E=0\).

With the natural descent linearization, there is a class \(L\in\operatorname{Pic}(\widetilde V)\) such that \(\psi^*L=H\).

Proposition 85.12 (degree-$2$ quasipolarization) The class \(L\) satisfies \(L^2=2\) and \(L\cdot C=0\).

Hence \(L\) is not ample on \(\widetilde V\). On a general one-node member it is nef and big and its only null curve is \(C\). After contracting the full \(L\)-null configuration, its descent is ample.

The final sentence must be interpreted with care on intersections with other Heegner divisors: additional curves can become \(L\)-trivial.

85.6.2 The anti-bicanonical curve

Riemann–Hurwitz for the double cover \(\psi\) gives

[ K_{X} =^*(K_{V}+12C). ]

Since \(K_{\widetilde X}\sim0\) and \(\widetilde V\) is rational, \(C\sim-2K_{\widetilde V}\).

Moreover, \(2C^2=(2E)^2=-8\) and \(C^2=-4\).

Proposition 85.13 (anti-bicanonical system on the one-node locus) For the resolved one-node Coble surface,

[ |-K_{V}|=, |-2K_{V}|={C}. ]

Proof. If \(D\in|-2K_{\widetilde V}|\), then \(\psi^*D\in|2E|\). Since \(E^2=-2\), every effective divisor linearly equivalent to \(2E\) contains \(E\); iterating gives \(|2E|=\{2E\}\). Thus \(D=C\). If \(D\in|-K_{\widetilde V}|\), then \(\psi^*D\in|E|=\{E\}\). The pullback of a divisor from the quotient has even multiplicity along a branch component, so this is impossible.

If the invariant branch curve has ordinary nodes at \(k\) distinct fixed points, the same argument gives \(|-2K_{\widetilde V}| =\{C_1+\cdots+C_k\}\), where the \(C_i\) are disjoint rational \((-4)\)-curves. Thus \(n=1\) on the generic Heegner divisor but not on its higher-codimension multiple-node loci.

85.6.3 The del Pezzo ramification divisor

Let \(\widetilde R\subset\widetilde X\) be the strict transform of the fixed curve of \(\iota_{\mathrm{dP}}\). At the node, the total ramification divisor contains the exceptional curve with multiplicity one, and \(\widetilde R+E\sim2H\).

Let \(R_{\widetilde V}\) be the quotient divisor on \(\widetilde V\). Then \(\psi^*R_{\widetilde V}=\widetilde R\).

Proposition 85.14 (ramification class and numerical invariants) One has

[ R_{V}2L+K_{V}, ]

and

[ R_{V}^2=7, R_{V}C=2, K_{V}R_{V}=-1, p_a(R_{V})=4. ]

Proof. Pulling back \(2L+K_{\widetilde V}\) gives \(2H-E=\widetilde R\). Moreover, \(\widetilde R^2=(2H-E)^2=14\), so \(R_{\widetilde V}^2=7\). The remaining formulas follow from \(C=-2K_{\widetilde V}\), \(R_{\widetilde V}\cdot C=2\), and adjunction.

The divisor \(R_{\widetilde V}\) is not the anti-bicanonical curve \(C\). It is the divisor relevant to the KSBA polarization.

85.6.4 Contraction to the singular Coble surface

Let \(\mu:\widetilde V\longrightarrow V^\sharp\) contract \(C\). The discrepancy formula is

[ K_{V} =^*K_{V}-12C. ]

Consequently \(K_{V^\sharp}\sim_{\mathbb Q}0\).

Let \(R^\sharp=\mu_*R_{\widetilde V}\). Then

[ ^*R =R_{V}+12C =2L. ]

It follows that \((R^\sharp)^2=8\).

Locally, \(R^\sharp\) is the image of the union of the coordinate axes \((uv=0)\) in the quotient singularity \(\frac{1}{4}(1,1)\). Its local class is \(2\) in \(\operatorname{Cl}(\frac{1}{4}(1,1))\cong\mathbb Z/4\), so it is \(\mathbb Q\)-Cartier of index \(2\).

Definition 85.15 (open stable Coble pair) An open degree-\(2\) stable Coble pair is

[ (V^,R^), <, ]

where \(V^\sharp\) is the singular canonical contraction and \(R^\sharp\) is the divisorial ramification of the descended del Pezzo involution.

The log-crepant pullback is

[ ^*(K_{V}+R^) =K_{V} +C +R_{V}. ]

For a general member, \(C+R_{\widetilde V}\) is simple normal crossing and the coefficients are less than one when \(0<\epsilon<1\). Thus the resolved pair is klt, and so is the singular pair. If the descent of \(L\) is ample, then \(K_{V^\sharp}+\epsilon R^\sharp\) is ample.

Warning 85.16 (index-one cover) The canonical index-one cover of the local surface singularity \(\frac{1}{4}(1,1)\) is the \(A_1\) singularity. Pulling back further to \(\mathbb A^2\) is useful for local log calculations, but it is not the index-one cover.

85.7 Direct moduli functors, arithmetic groups, and period spaces

85.7.1 The resolved-cover moduli problem

The autonomous strategy should begin with a stack of labeled K3-cover data. The following definition is a mathematical specification; representability and equivalence with the singular pair moduli are required theorems.

Definition 85.17 (direct resolved Coble cover datum)

Over a scheme \(S\), a direct resolved degree-\(2\) Coble cover datum consists of

[ (/S, E, {}, {}, H, ) ]

with the following properties.

1. \(\widetilde{\mathcal X}/S\) is a family of K3 surfaces, or a family with the controlled ADE singularities allowed by the period problem.

2. \(\mathcal E\) is a relative Cartier divisor whose geometric fibers are smooth \((-2)\)-curves.

3. \(\iota_{\mathrm{Co}}\) is a nonsymplectic involution whose divisorial fixed locus is \(\mathcal E\).

4. \(\iota_{\mathrm{dP}}\) is a commuting nonsymplectic involution whose divisorial fixed locus is the ramification divisor \(\mathcal R\).

5. \(\mathcal H\) is a relatively nef and big line bundle with \(\mathcal H_s^2=4\) and \(\mathcal H_s\cdot\mathcal E_s=0\).

6. \(\mcc\) specifies the positive/nef chamber in which \(\mathcal E\) is effective and \(\mathcal H\) gives the desired contraction.

7. Fiberwise contraction of \(\mathcal E\) gives an \(A_1\)-surface on which the Coble involution descends and fixes the node.

The chamber and the effective root are part of the data because an abstract lattice marking does not determine which root is represented by the exceptional curve or which degree-\(4\) class gives the projective model.

85.7.2 The singular-pair moduli problem

Definition 85.18 (direct singular Coble datum)

Over \(S\), a direct singular Coble datum consists of

[ (V^/S, L^, _{}, R^) ]

such that each geometric fiber has the following properties.

1. \(V^\sharp\) is a normal rational surface with one \(\frac{1}{4}(1,1)\) singularity.

2. \(K_{V^\sharp}\sim_{\mathbb Q}0\).

3. \(L^\sharp\) is an ample degree-\(2\) class.

4. \(\tau_{\mathrm{dP}}\) is an involution whose divisorial fixed locus is \(R^\sharp\).

5. \(R^\sharp\) is ample, \(\mathbb Q\)-Cartier, and \((R^\sharp)^2=8\).

6. The canonical index-one cover is a K3 surface with one \(A_1\) singularity.

Problem 85.19 (Required theorem (equivalence of the two open moduli problems)) After accounting for the Weyl ambiguity of the exceptional root, contraction, canonical cover, and quotient define inverse equivalences between the direct resolved-cover stack and the direct singular-pair stack. The equivalence is compatible with base change and automorphisms.

This theorem is the direct replacement for silently identifying a Heegner period point with a geometric Coble pair.

85.7.3 Four arithmetic groups that must not be conflated

Fix an embedded root \(\alpha\in T_{\mathrm{En}}\) and put \(T_{\mathrm{Co}}=\alpha^\perp\). There are at least four natural groups.

Definition 85.20 (direct unpolarized Coble group) Dropping the degree-\(4\) polarization and the del Pezzo involution from the labeled datum gives the candidate direct unpolarized group

[ {}^{} :=( Z{O(L_{K3})}(I_{}){,} O^+(T{}) ). ]

The corresponding geometric monodromy group is defined from the direct unpolarized Coble moduli stack. The expected equality \(\Gamma_{\mathrm{Co}}^{\mathrm{dir}}=O^+(T_{\mathrm{Co}})\) is compatible with the full stabilizer lemma below, but it remains a geometric monodromy statement until the direct period theorem is proved.

Definition 85.21 (Hodge-theoretic Heegner group) [ {,2}^{} :=( {{,2}}(Z) O^+(T{}) ). ]

This group is the natural group of a fixed normalized Heegner component once the Enriques construction has been fixed.

Definition 85.22 (direct labeled centralizer group) [ {,2}^{} :=( Z{O(L_{K3})}(I_{},I_{}){H,,} O^+(T{}) ). ]

This is the arithmetic group naturally attached to the autonomous lattice datum, provided the labeled rigidity theorem holds.

Definition 85.23 (geometric monodromy group) \(\Gamma_{\mathrm{Co},2}^{\mathrm{geom}}\) is the image of the orbifold monodromy representation of the connected direct moduli stack of polarized Coble data on the local system \(T_{\mathrm{Co}}\).

Finally, the degree-\(2\) marking gives an orthogonal decomposition \(A_{T_{\mathrm{En}}} \cong A_{U(2)}\perp A_{E_8(2)}\), and hence the finite subgroup \(P:=O(q_{U(2)})\times O(q_{E_8(2)})\).

Using the characteristic class \(\eta/2\) to identify the remaining finite quadratic space with the Enriques discriminant space, discriminant gluing suggests the congruence subgroup

[ {,2}^{} :={T_{}}^{-1}(P). ]

Problem 85.24 (Required theorem (arithmetic and monodromy identification)) Prove

[ {,2}^{} ={,2}^{} ={,2}^{} ={,2}^{}. ]

Each equality requires a separate argument: global Torelli and parallel transport for the first, labeled embedding comparison for the second, and discriminant-form gluing for the third.

Defining one of these groups to be “the group for which the desired semifan works” is circular.

85.7.4 A lattice-theoretic stabilizer sequence

There is a useful pure lattice calculation. Let \(L_0=\mathbb Z\alpha\oplus T_{\mathrm{Co}}\).

The lattice \(T_{\mathrm{En}}\) is an index-\(2\) overlattice of \(L_0\), obtained by adjoining \((\alpha+\eta)/2\). The gluing subgroup in \(A_{\mathbb Z\alpha}\oplus A_{T_{\mathrm{Co}}}\) is generated by

[ (,). ]

The class \(\eta/2\) is the characteristic element of the discriminant form of \(T_{\mathrm{Co}}\cong I_{2,9}(2)\) and is fixed by every isometry.

Lemma 85.25 (full orthogonal stabilizer) Restriction gives an exact sequence

[ 1w_ {O^+(T{})}(Z) O^+(T_{}) . ]

Proof. Every isometry of \(T_{\mathrm{Co}}\) fixes the characteristic class \(\eta/2\), hence preserves the gluing subgroup and extends to the index-\(2\) overlattice. The two extensions differ by the root reflection \(w_\alpha\).

This lemma is a lattice statement. It does not identify the degree-\(2\) monodromy group and does not solve the integral isotropic-subspace lifting problem.

85.7.5 Period domains

Define \(\mathbb D_{\mathrm{Co}}:=\mathbb D(T_{\mathrm{Co}})\).

The unpolarized Hodge quotient is naturally

[ F_{}^{} :=O^+(T_{})D_{}. ]

The direct and polarized quotients are, provisionally,

[ F_{,2}^{} :={,2}^{} D{}, ]

and

[ F_{,2}^{} :={,2}^{} D{}. ]

The period theorem for classical Coble surfaces identifies an open subset of an orthogonal modular variety of this lattice type (Dolgachev and Kondō 2013). The direct polarized period theorem and the equality of the two polarized quotients are required results, not consequences of the abstract lattice isomorphism.

85.7.6 The required diagram of period quotients

At the level of fixed embedded period domains, one has inclusions

[ D(T_{}) D(T_{}) D(T_{}). ]

The desired quotient diagram is

[

]

A diagram of rational domains is not automatically a diagram of arithmetic quotients. To construct each arrow, one must prove that the relevant group acts through a compatible subgroup of a common \(O(L_{K3})\), and one must identify the image group. The direct arrow to \(\mathcal F_{(2,2,0)}\) must later be shown to factor through the Enriques Heegner component.

Problem 85.26 (Required theorem (commuting period diagram)) The direct and Heegner arithmetic groups are compatible with the common labeled embedding chain; all arrows in the displayed diagram are well defined on quotients; and the direct Coble construction factors through the normalized \((-2)\)-Heegner divisor in \(\mathcal F_{\mathrm{En},2}\).

85.8 Baily–Borel boundary and integral cusp problems

85.8.1 Standard cusp data

Let \(T\) have signature \((2,n)\) and let \(\Gamma\subset O^+(T)\) be arithmetic. A \(0\)-cusp is a \(\Gamma\)-orbit of primitive isotropic lines \(\mathbb Ze\subset T\). Its hyperbolic cusp lattice is \(\overline{T}_e:=e^\perp/\mathbb Ze\).

A \(1\)-cusp is a \(\Gamma\)-orbit of primitive isotropic planes \(J\subset T\). Its negative-definite quotient is \(K_J:=J^\perp/J\).

For a fixed line \(e\), the stabilizer has an exact sequence

[ 1U_e _(e) _e , ]

where \(U_e\) is the unipotent radical and \(\Gamma_e\) acts on \(\overline{T}_e\). The arithmetic group \(\Gamma_e\), not the full orthogonal group of \(\overline{T}_e\), is the group relevant to the cusp reflection problem.

85.8.2 The unpolarized Coble boundary

Since scaling a form does not change its integral orthogonal group, \(O(T_{\mathrm{Co}})=O(I_{2,9})\).

Standard transitivity results for indefinite odd unimodular forms imply the following (Wall 1962; Dolgachev and Kondō 2013).

Theorem 85.27 (unpolarized Coble cusps) The quotient \(O^+(T_{\mathrm{Co}})\backslash\mathbb D_{\mathrm{Co}}\) has one orbit of primitive isotropic lines and one orbit of primitive isotropic planes. For representatives \(e\) and \(J\),

[ e^/eI_{1,8}(2), J^/JI_{0,7}(2). ]

This theorem concerns the full orthogonal group. It does not determine the cusps for the polarized subgroup.

85.8.3 Reduction to the discriminant form

For a \(2\)-elementary lattice, a primitive isotropic vector of divisibility \(2\) determines a nonzero isotropic class in the discriminant group. Thus reduction modulo the lattice gives maps

[ { {,2} } { ({,2})A_T }, ]

and analogous maps for isotropic planes and incident flags.

Warning 85.28 (finite shadow versus integral cusp) These maps need not be bijective. The integral parabolic stabilizer can have a proper image in the finite stabilizer of the reduced flag. Consequently, an orbit calculation in \(O(A_{T_{\mathrm{Co}}})\) does not classify Baily–Borel cusps without a separate integral lifting theorem.

The standard ways to solve this problem are:

1. integral normal forms and explicit Eichler transvections;

2. computation of the images of integral line and plane stabilizers followed by finite double-coset calculations; or

3. a certified integral orbit algorithm, for example of the type developed in (Dawes 2022a).

85.8.4 Computational record in the finite quadratic space

A finite calculation was carried out in a \(10\)-dimensional plus-type quadratic space over \(\mathbb F_2\) for the subgroup associated with the proposed degree-\(2\) congruence condition. It produced four orbits of nonzero singular vectors and five orbits of totally singular planes. The labels below are internal finite labels; they are not cusp labels.

Table 85.1: Finite singular-vector orbits.

Finite vector type	Orbit size
\(A\)	\(135\)
\(B\)	\(2\)
\(C\)	\(270\)
\(D\)	\(120\)

Table 85.2: Finite totally singular plane orbits.

Finite plane type	Orbit size
\(AAA\)	\(1575\)
\(ABC\)	\(270\)
\(ACC\)	\(9450\)
\(ADD\)	\(3780\)
\(CCD\)	\(8640\)

A later enhanced calculation decorated a finite plane by one of eight ``stars’’ in the natural \(S_8\)-set attached to its \(6\)-dimensional quotient. For a prescribed \(S_7\) subgroup it produced seven enhanced plane orbits and thirteen enhanced flag orbits. This remains a computational record only: the required identification of that \(S_7\) with the image of the integral plane stabilizer has not been proved.

Problem 85.29 (Required theorem (integral cusp lifting)) Determine the exact images of the integral isotropic-line and isotropic-plane stabilizers in the finite orthogonal group and use them to classify

[ _{,2} {}, ]

[ _{,2} {}, ]

and the incident flags. Give explicit integral representatives and stabilizers.

85.8.5 Necessary restrictions on maps to Enriques cusps

Suppose the comparison map to the Enriques period quotient has been constructed. Let \(e\) be a Coble isotropic line. The root \(\alpha\) lies in \(e^\perp\) and gives a \((-2)\) class in the Enriques cusp lattice. Therefore an Enriques \(0\)-cusp whose hyperbolic lattice has all norms divisible by \(4\) cannot contain a Coble branch. In the AEGS numbering, this excludes cusp \(1\) and leaves only cusps \(2,3,4,5\) as possible images.

Similarly, if \(J\) is a Coble isotropic plane, the class of \(\alpha\) gives a \((-2)\) vector in \(J^\perp/J\). Hence a Type II Enriques cusp whose negative quotient is \(E_8(2)\) cannot contain a Coble branch. Reading the border types in AEGS Figure 4, the double-rectangle cusps \(12,\ 13,\ 14,\ 15,\ 245\) map to the segment-flipping \(E_8(2)\) cusp and are excluded. The only possible Enriques \(1\)-cusp images are therefore \(34,\ 35,\ 45,\ 55\).

Remark 85.30 This is a necessary-condition argument, not an integral orbit classification. It does not determine how many Coble cusps lie above any of the four possible image cusps.

An earlier finite-orbit calibration that assigned a Coble branch to cusp \(245\) is therefore not compatible with the embedded root criterion and must not be used.

85.9 Reflection theory and Coxeter data

85.9.1 The unpolarized cusp chamber

Write

[ I_{1,8}(2) =ZH_0_{i=1}^8ZE_i, H_0^2=2, E_i^2=-2. ]

Define

[ r_0=H_0-E_1-E_2-E_3, ]

[ r_i=E_i-E_{i+1}(1i), r_8=E_8. ]

Then

[ r_0^2==r_72=-4, r_8^2=-2. ]

Theorem 85.31 (unpolarized Coble Coxeter chamber) A fundamental chamber for the reflection group of \(I_{1,8}(2)\) is defined, for

[ x=aH_0-_{i=1}^8b_iE_i, ]

by

[ b_1b_2b_8, ab_1+b_2+b_3. ]

The simple roots are \(r_0,\ldots,r_8\). The diagram is the chain

[ r_1-r_2-r_3-r_4-r_5-r_6-r_7r_8 ]

with \(r_0\) attached to \(r_3\) by a single edge.

Proof. Signed permutations of the \(E_i\) reduce to the ordered region, and the Cremona reflection in \(r_0\) reduces the final inequality. This is the standard Vinberg chamber for \(I_{1,8}\), with the form scaled by \(2\).

The unique maximal parabolic subdiagram is obtained by deleting \(r_1\) and has type \(\widetilde B_7\). The rank-\(8\) elliptic subdiagrams are listed in Table 85.3.

Table 85.3: Rank-\(8\) elliptic subdiagrams of the unpolarized chamber.

Deleted node	Elliptic type
\(r_0\)	\(B_8\)
\(r_2\)	\(A_1+B_7\)
\(r_3\)	\(A_1+A_2+B_5\)
\(r_4\)	\(A_4+B_4\)
\(r_5\)	\(D_5+B_3\)
\(r_6\)	\(E_6+B_2\)
\(r_7\)	\(E_7+A_1\)
\(r_8\)	\(E_8\)

85.9.2 Orthogonal wall links

Let \(a\) be a \((-2)\) root in a hyperbolic lattice \(M\), and let \(b\) be another negative wall normal. The normal inside \(a^\perp\) to the intersection \(b^\perp\cap a^\perp\) is the primitive vector proportional to the orthogonal projection of \(b\) to \(a^\perp\).

Construction 85.32 (orthogonal wall link) Define

[ b_{} :=(2b+(b,a)a). ]

The formula is integral and defines the correct hyperplane in \(a^\perp\). It is a linear-algebra statement. It does not imply that \(b_{\operatorname{link}}\) is a root of the orthogonal lattice.

Warning 85.33 (restricted wall versus reflection wall) A hyperplane obtained by slicing an ambient Coxeter chamber can fail to be the mirror of an integral reflection in the sublattice. Conversely, the orthogonal sublattice can have reflective roots whose mirrors are invisible in the ambient root system.

A Coble analogue of the AEGS root-folding theorem must prove both directions: that every relevant restricted wall is a Coble root wall and that every Coble root wall meeting the chamber is obtained in the prescribed way.

85.9.3 Candidate reflection twist at a cusp

At the full K3-lattice level one has the reflection twist \(I_{\mathrm{Co}}=w_\alpha I_{\mathrm{En}}\). If a primitive isotropic line \(e\) is orthogonal to \(\alpha\), the involutions induce actions on the cusp quotient \(e^\perp/e\). This suggests the cusp-level candidate \(J_{\mathrm{Co}}=w_{\bar\alpha}J_{\mathrm{En}}\), where \(\bar\alpha\) is the image of \(\alpha\) in the cusp lattice.

This formula is not yet a direct folding theorem. One must prove that the cusp action lifts through the full discriminant gluing and corresponds to the geometric involution, just as AEGS separately prove their lifting lemma.

85.9.4 Corrected ambient square Gram data

In the square ambient K3 cusp diagram, the two black roots \(\alpha_{20}\) and \(\alpha_{21}\) both have square \(-4\). The edge joining them is thick. Therefore \((\alpha_{20},\alpha_{21})=4\).

The previously used value \(2\) gives a Gram matrix of rank \(20\), incompatible with the rank-\(18\) cusp lattice. The value \(4\) gives the correct rank \(18\).

85.9.5 Candidate wall-slice diagrams

Correcting the Gram entry above, one can intersect selected Enriques chambers with selected white-root hyperplanes. This produces four explicit hyperplane arrangements, denoted

[ G_{}^2, G_{}^3, G_{}^4, G_{}^5. ]

The chosen roots in the previously used labeling were \(\alpha_{10}\), \(\alpha_0\), \(\alpha_6\), and \(\alpha_{16}\), respectively. Each displayed sliced wall is a genuine root in the orthogonal lattice in the cases checked symbolically. No proof was given that the displayed walls form a complete simple system.

The raw numbers of maximal parabolic faces in the four sliced arrangements were \(1\), \(3\), \(3\), \(9\).

These counts are diagnostic data only. They disagree with some finite-shadow orbit counts, which indicates that chamber completeness, arithmetic diagram automorphisms, and the calibration of finite orbits must all be resolved before boundary strata are enumerated.

Problem 85.34 (Required theorem (Coble Vinberg and root completeness))

For every polarized Coble \(0\)-cusp:

1. determine the arithmetic cusp lattice and the image of its stabilizer;

2. run Vinberg’s algorithm or prove a two-sided root-link theorem;

3. prove that the resulting simple roots define a fundamental chamber;

4. compute the chamber automorphism group and reflection index; and

5. enumerate all maximal parabolic and elliptic subdiagrams modulo the full arithmetic group.

A list of vectors or a finite Gram-matrix computation is not a Coxeter theorem without these steps.

85.10 Integral-affine structures, dlt quotients, and stable models

85.10.1 The AEGS dictionary

AEGS relate five forms of degeneration data:

[ . ]

For a primitive isotropic line \(e\), a Type III monodromy invariant is a vector \(\lambda\in\overline{T}_e\otimes_\ZZ\mathbb R\) in the closure of a positive chamber. If \(\{r_i\}\) is a simple-root system, the root coordinates are \(\ell_i=(\lambda,r_i)\ge0\).

They become edge lengths, internal blowup parameters, and nodal-surgery sizes in a moment or Symington polygon. A zero coordinate places the monodromy on a wall. If the wall corresponds to a visible curve, the zero can collapse a path joining two focus-focus singularities.

The direct Coble theory should preserve this dictionary but must add the data of the fixed exceptional root and its quotient branch curve.

85.10.2 The Coble integral-affine package

Definition 85.35 (candidate polarized Coble integral-affine package)

A Coble integral-affine package is a tuple

[ (B(),R_{},E_{}, {},{}) ]

where

1. \(B(\lambda)\) is the integral-affine dual sphere or Type II segment of a K3 divisor model;

2. \(R_{\mathrm{IA}}\) is the weighted balanced graph recording the del Pezzo ramification divisor;

3. \(E_{\mathrm{IA}}\) records the visible class, collapsed path, or marked affine stratum associated with the exceptional root \(\alpha\);

4. the two affine involutions commute and preserve the full polarized package;

5. the quotient data predicts the branch curve \(C\) and the divisor \(R_{\widetilde V}\) on a Coble dlt model.

The notation \(E_{\mathrm{IA}}\) is schematic. Depending on the cusp, the exceptional root can be represented by a visible path, a collapsed edge, an \(I_2\) affine singularity, or equivalent marked data in the triangulation. A direct theory must specify the correct object cusp by cusp.

85.10.3 Heegner-wall specialization of an ambient model

Suppose a degeneration lies in the \(\alpha\)-Heegner locus and approaches a Type III cusp represented by \(e\). After a suitable marking and base change, one expects \((\alpha,e)=0\) and \((\alpha,\lambda)=0\).

If \(\alpha\) is represented by a visible path in the chosen integral-affine model, the second equation says that the path has zero affine length. The corresponding pair of parallel \(I_1\) singularities coalesces to an \(I_2\) singularity. This is the same local operation used by AEGS for crossed nodes, but in the Coble problem it records the persistent \(A_1\) singularity of the K3 cover.

Remark 85.36 The local collision picture is reliable once the relevant root has been identified in the genuine cusp lattice. It cannot be used to prove that a candidate wall slice is the full Coxeter chamber.

85.10.4 Required realization theorem

Problem 85.37 (Required theorem (direct integral-affine realization)) For every sufficiently divisible monodromy invariant in every direct Coble cusp chamber, construct a K3 divisor model \((\mathcal X,\mathcal R,\mathcal E)\longrightarrow(C,0)\) whose dual complex is the prescribed Coble integral-affine package. Conversely, show that every direct Coble degeneration admits such a model after finite base change and allowed dlt modification.

The proof must include polygon closure, parity and divisibility conditions, triangulation, component periods, gluing parameters, and the realization of the exceptional class as a Cartier divisor or contractible ADE configuration.

85.11 Required algebraic extension theorem

An affine symmetry is not automatically an algebraic involution. AEGS use limiting-period conditions and invariant component/gluing parameters to extend their involution over a general divisor model. The Coble version must also retain the exceptional root.

Problem 85.38 (Required theorem (algebraic extension of the Coble package)) A general divisor model with direct Coble monodromy admits commuting regular involutions \(\iota_{\mathrm{Co}}\) and \(\iota_{\mathrm{dP}}\) such that

[ {}(E)=E, {}(R)=R. ]

The cohomological action is the prescribed labeled lattice action, the involutions commute on the total space, and equivariant simultaneous resolution and contraction are compatible with the family.

A proof must not infer regularity solely from the action on the dual complex. For special degenerations the initial action can be birational and may require contraction of ADE configurations before it becomes regular.

85.12 The direct dlt Coble model

Assume the algebraic extension theorem. Form the quotient \(\widetilde{\mathcal V}:=\mathcal X/\iota_{\mathrm{Co}}\).

Let \(\mathcal C\) be the divisorial branch locus descending from \(\mathcal E\), and let \(\mathcal R_{\widetilde V}\) be the quotient of the del Pezzo ramification divisor. The natural crepant pair is

[ ( , C +R_{V} ). ]

Problem 85.39 (Required theorem (direct dlt quotient)) After the prescribed dlt modification, the above pair is dlt over the base; its fibers are slc; the boundary contains no prohibited log canonical center; and

[ K_{} +C +R_{V} ]

is relatively big and nef.

The local analysis differs from the Enriques case because the Coble involution has a divisorial fixed locus on the resolved K3 model. One must analyze fixed components, fixed double curves, triple points, and the interaction between \(\mathcal C\) and the conductor.

85.13 Relative Proj and the stable model

The dlt model is not the KSBA stable model. The stable model is the relative log canonical model

[ C {m} H^0!( , m( K_{} +C +R_{V} ) ). ]

After contracting \(\mathcal C\) to the singular Coble family, this is expected to agree with the relative Proj for a sufficiently divisible multiple of \(\mathcal R^\sharp\).

Problem 85.40 (Required theorem (stable contraction)) Determine exactly which curves, components, and conductor strata are contracted by the above relative Proj, and prove that the resulting family is the KSBA stable family of pairs \((V^\sharp,\epsilon R^\sharp)\).

85.14 Component classification problem

The direct Coble quotient can produce surface components with branch boundary \(C\), conductor boundary \(D\), and ramification divisor \(R\). The appropriate normalized component datum is

[ ( V, D+C+R ). ]

Problem 85.41 (Required theorem (Coble component classification)) Classify all finite Type III and affine Type II component pairs arising in direct Coble stable limits. Give explicit equations or toric models, involutions, quotient singularities, branch allocation, decorations, and stable contractions; prove that every component occurs in the list and every listed type occurs.

This is the Coble analogue of the AEGS ABCDE surface classification. It cannot be obtained merely by replacing the Enriques involution in the existing tables: the resolved Coble involution has a fixed curve.

85.15 Status of adaptations of AEGS examples

Several candidate recipes were identified:

• start with the AEGS cusp-\(3\) integral-affine model and impose one additional Coble-root equation;

• start with the cusp-\(5\) model, where several crossed-root coordinates already vanish, and impose an independent Coble-root condition;

• start with the non-simple cusp-\(2\) model and impose the appropriate white-root wall while retaining the fact that the bottom fixed edge has no ramification support.

These are recipes, not completed examples. A complete example must specify the integral root, the visible path, the compatible triangulation, the component periods and gluings, the two algebraic involutions, all quotient singularities, and the final \(R^\sharp\)-trivial contraction.

85.16 Fans, semifans, and semitoroidal compactifications

85.16.1 Coxeter fans and generalized Coxeter semifans

Let \(M\) be a hyperbolic cusp lattice, \(W\) a reflection group, and \(C\) a fundamental chamber. The Coxeter fan consists of the \(W\)-translates of \(C\) in the rational closure of the positive cone. If the chamber has finite volume, this is a rational polyhedral fan.

Suppose the simple roots are partitioned into relevant and irrelevant roots. Let \(W_{\mathrm{irr}}\) be generated by the irrelevant reflections. The generalized chamber is

[ C_{} ={wW{}}wC. ]

Its \(W\)-translates form the generalized Coxeter semifan. Geometrically, a root is relevant only if crossing its wall changes the stable pair after the relative Proj.

Warning 85.42 (relevance is geometric) In the autonomous Coble theory, a root cannot be declared relevant merely because it lies on the boundary of an ambient Enriques or K3 diagram. Relevance must be characterized by the Coble package \((R_{\mathrm{IA}},E_{\mathrm{IA}})\) and the resulting stable contraction.

85.16.2 Direct Coble semifans

Assume that the direct cusp groups and Coxeter chambers have been computed. For each direct Coble \(0\)-cusp, define a root to be relevant if a general crossing of its wall changes at least one of the following stable data:

1. the support or degree distribution of \(R^\sharp\);

2. the branch curve or exceptional-root contraction;

3. the normalization components or conductor gluing;

4. the number of surviving double curves or components; or

5. the isomorphism class of the relative Proj.

Problem 85.43 (Required theorem (direct Coble semifan)) The maximal regions of combinatorial constancy of direct Coble stable pairs are exactly the cones of the generalized Coxeter semifan obtained from the verified direct Coble Coxeter diagram and the intrinsic relevance marking.

The theorem must determine cusp by cusp whether the irrelevant reflection group is finite, whether the semifan is an actual fan, and whether any Type II parabolic component consists entirely of irrelevant roots.

85.16.3 Normalized closure of a subtorus

The Heegner-restriction strategy uses the following standard toric fact.

Lemma 85.44 (normalization of a subtorus closure) Let \(N\subset M\) be a saturated sublattice of cocharacter lattices, let \(T_N\subset T_M\) be the corresponding subtorus, and let \(X_\Sigma\) be a toric variety for a fan \(\Sigma\subset M_\mathbb R\). The normalization of the closure of \(T_N\) in \(X_\Sigma\) is the toric variety associated with the induced fan \(\Sigma|_{N_\mathbb R} =\{\sigma\cap N_\mathbb R:\sigma\in\Sigma\}\), with the induced lattices saturated.

This lemma is local and toric. It does not by itself identify the normalized closure of a Heegner divisor in a semitoroidal arithmetic quotient.

85.16.4 Why semitoroidal restriction is not formal

To apply the subtorus lemma to the Coble Heegner locus, one must prove all of the following.

1. The local Heegner branch is the asserted saturated subtorus rather than a translated subtorus or a union of branches.

2. The integral unipotent-center lattice for the Coble cusp is the expected sublattice of the Enriques cusp lattice.

3. The arithmetic cusp stabilizer acts compatibly on the induced fan.

4. The induced local constructions agree along every \(1\)-cusp.

5. Normalization commutes with the prescribed semitoroidal contractions.

6. The ambient contraction does not identify distinct normalized Coble branches.

7. No additional coarsening occurs after passing to Coble stable pairs.

Problem 85.45 (Required theorem (semitoroidal restriction)) After passage to a common toroidal refinement, the normalization of the Heegner closure is obtained by saturated intersection with the Coble cusp subspaces, and the induced semitoroidal contractions coincide with combinatorial constancy for the Coble stable-pair family.

Without this theorem, “intersect the AEGS semifans” is a candidate construction rather than a compactification theorem.

85.17 The KSBA moduli space and the two target theorems

85.17.1 The open KSBA moduli functor

Let \(\mathfrak M_{\mathrm{Co},2}^{\circ}\) denote the direct open moduli stack of singular Coble data for which the pair \((V^\sharp,\epsilon R^\sharp)\) is klt and \(K_{V^\sharp}+\epsilon R^\sharp\) is ample. Define \(\overline{\mathfrak M}_{\mathrm{Co},2}^{\mathrm{KSBA}}\) to be its closure in the appropriate KSBA stack with fixed coefficient and volume.

The divisor \(R^\sharp\) must be defined universally as the divisorial fixed locus of the descended del Pezzo involution. Choosing a divisor separately on each fiber does not define a moduli problem.

85.17.2 Autonomous target theorem

Problem 85.46 (Target theorem A (autonomous Coble compactification)) The direct open period map identifies

[ M_{,2}^{} {,2}^{} D{}^{}, ]

for an explicit hyperplane complement. The normalization of \(\overline{\mathfrak M}_{\mathrm{Co},2}^{\mathrm{KSBA}}\) is the semitoroidal compactification defined by the explicit direct Coble semifans. All cusps, Coxeter diagrams, Type II and Type III strata, and fan-versus-semifan loci are explicitly determined.

This theorem is logically independent of the Enriques Heegner comparison. It requires direct analogues of the entire AEGS proof architecture.

85.17.3 Heegner comparison target

Let \(\Delta_\alpha\) be the \((-2)\)-Heegner divisor in the degree-\(2\) Enriques period quotient, defined using a fixed embedded root.

Problem 85.47 (Target theorem B (comparison with the Enriques Heegner divisor))

The direct Coble period quotient is canonically isomorphic to the normalization of \(\Delta_\alpha\). Under this isomorphism:

1. the direct and Heegner arithmetic groups agree;

2. the period maps to \(\mathcal F_{\mathrm{En}}\), \(\mathcal F_{\mathrm{En},2}\), \(\mathcal F_{(2,2,0)}\), and \(\mathcal F_4\) commute;

3. the direct projective construction agrees with the Heegner specialization;

4. the direct cusps and Coxeter diagrams map to the corresponding Enriques and K3 data;

5. the direct stable divisor agrees with the restriction of the ambient ramification divisor; and

6. the direct semitoroidal compactification is the normalization of the KSBA closure of the Heegner divisor inside the AEGS compactification.

Target theorem B is the theorem requested by the original problem. Target theorem A is a substantial independent theorem and an intermediate strategy, but it becomes a proof of the Heegner statement only after Target theorem B is established.

85.18 Proof strategies and research programs

This section records the proof strategies that emerged, the precise theorem chains each requires, and the points at which earlier attempts failed. It is not an attempt to advance any of the proofs.

85.18.1 Strategy A: normalize the Heegner closure inside the AEGS compactification

85.18.1.1 Outline

Fix a root \(\alpha\in T_{\mathrm{En}}\) and let \(\Delta_\alpha\) be the corresponding degree-\(2\) Heegner component. Let

[ {}: F{,2}^{{}} {,2}^{} ]

be the AEGS normalization map. Take the normalization \(H_\alpha\) of the closure of \(\Delta_\alpha\) in the source. The strategy is to identify \(H_\alpha\) with the normalization of the Coble KSBA closure and compute its local semitoroidal structure.

85.18.1.2 Required chain of results

The strategy requires the following sequence.

Problem 85.48 (A1. Root-orbit and embedded-component lemma) The relevant roots form one orbit under the actual degree-\(2\) Enriques group, or the chosen component is otherwise specified intrinsically. The stabilizer image on \(T_{\mathrm{Co}}\) is computed.

Problem 85.49 (A2. Root-marked universal-family lemma) Over the stack-theoretic normalization of the Heegner component, there is a universal family in which the fixed \(A_1\) point, its simultaneous resolution after the required Weyl cover, the exceptional root, and the ambient ramification divisor are compatible with base change.

Problem 85.50 (A3. Coble-pair identification lemma) The restriction of the AEGS stable pair to the generic Heegner point is precisely \((V^\sharp,\epsilon R^\sharp)\) of Section 85.6.4, with the same universal divisor.

Problem 85.51 (A4. Generic reconstruction lemma) A general stable Coble pair recovers its canonical K3 cover, the distinguished \(A_1\) point, the degree-\(4\) class, the del Pezzo involution, and the Heegner period point. Consequently the restricted finite map is generically injective.

Problem 85.52 (A5. Arithmetic cusp-lifting theorem) The Coble cusp orbits, stabilizers, and maps to Enriques cusps are computed integrally.

Problem 85.53 (A6. Local toroidal normalization theorem) In every cusp chart, the normalized Heegner closure is the saturated toric subvariety associated with the actual Coble unipotent-center lattice and the induced arithmetic action.

Problem 85.54 (A7. Semitoroidal compatibility theorem) The local normalized toric closures glue along the Type II boundary and are compatible with the AEGS semitoroidal contractions.

Problem 85.55 (A8. No-further-coarsening theorem) Every wall retained by the induced Coble semifan changes the Coble stable pair; equivalently, the classifying map from the normalized Heegner closure to the Coble KSBA closure is finite and has no positive-dimensional fibers.

With A1–A8, the conclusion follows by normality and finite birationality.

85.18.1.3 What worked

The following ingredients are useful and survive scrutiny.

• The root complement \(T_{\mathrm{Co}}\) and reflection twist are explicit.

• The open local Coble quotient and stable divisor are explicit.

• The parent AEGS normalization is already finite.

• The normalization of a saturated subtorus closure in an ordinary toric chart is understood.

• Parent charts that are honest fans remain rational polyhedral after intersection with a rational subspace.

85.18.1.4 What failed

The earlier restriction argument failed at the following points.

1. It treated the stabilizer-image group as automatically equal to geometric monodromy.

2. It inferred a quotient map from an inclusion of rational domains without proving group compatibility.

3. It used generic index-one-cover reconstruction without proving recovery of the degree-\(4\) projective model and the labeled lattice tuple.

4. It replaced the integral cusp problem by an \(O(A)\) computation.

5. It called hyperplane slices Coxeter diagrams without a root-completeness theorem.

6. It assumed that normalization of a toric subtorus closure automatically survives arithmetic quotienting and semitoroidal contraction.

These failures do not discredit Strategy A. They identify the exact new lemmas it needs.

85.18.2 Strategy B: autonomous Coble theory in the style of AEGS

85.18.2.1 Autonomous outline

This strategy never defines the direct Coble objects as a locus in Enriques moduli. It begins with the direct cover datum of Definition (direct resolved Coble cover datum) (Definition 85.17), computes its period theory, and reproduces Coble analogues of the AEGS constructions. Only after the autonomous compactification theorem is proved does one compare it with the Heegner divisor.

85.18.2.2 Open moduli and periods

The following results are required in order.

Problem 85.56 (B2.1. Direct projective-model theorem) The nodal invariant \((4,4)\) construction defines the complete generic direct Coble family and its universal divisors.

Problem 85.57 (B2.2. Intrinsic reconstruction theorem) The singular Coble pair recovers the quartic del Pezzo quotient, the two double-cover algebras, the nodal K3 cover, both involutions, and the degree-\(4\) class.

Problem 85.58 (B2.3. Labeled lattice classification theorem) Compute the simultaneous eigenspaces of the Klein-four action, their primitive closures and discriminant forms, and prove the labeled rigidity theorem of Section 85.5.4.

Problem 85.59 (B2.4. Direct global Torelli theorem) The marked direct Coble moduli stack is an explicit arrangement complement \(\mathbb D_{\mathrm{Co}}^\circ\), with the ample/nef chamber and exceptional root correctly encoded.

Problem 85.60 (B2.5. Direct monodromy theorem) The deck group of the marking cover and the geometric monodromy group equal the labeled centralizer group \(\Gamma_{\mathrm{Co},2}^{\mathrm{dir}}\).

Problem 85.61 (B2.6. Direct stable-pair theorem) The universal divisor \(R^\sharp\) is flat, uniformly \(\mathbb Q\)-Cartier and ample, and the pairs \((V^\sharp,\epsilon R^\sharp)\) define a separated bounded KSBA moduli problem for \(0<\epsilon\ll1\).

85.18.2.3 Cusps and reflection chambers

Problem 85.62 (B3.1. Integral Baily--Borel classification) Classify all primitive isotropic lines, planes, and flags under \(\Gamma_{\mathrm{Co},2}^{\mathrm{dir}}\), with explicit representatives and stabilizers.

Problem 85.63 (B3.2. Vinberg theorem at each $0$-cusp) Run Vinberg’s algorithm for each actual cusp lattice and arithmetic reflection group. Prove termination, completeness of the simple roots, and compute the chamber automorphism group.

Problem 85.64 (B3.3. Full-lattice lift theorem) Lift every cusp-lattice involution or reflection-twist description through the discriminant gluing to the labeled K3 lattice datum.

Problem 85.65 (B3.4. Parabolic and elliptic exhaustion theorem) Enumerate all maximal parabolic and relevant elliptic subdiagrams modulo the arithmetic chamber automorphism group and verify agreement with the independently computed isotropic-plane orbits.

85.18.2.4 Integral-affine and dlt models

Problem 85.66 (B4.1. Mirror and polygon theorem) Construct, for every direct Coble cusp, the mirror anticanonical pair, moment polygons, Symington surgeries, and root-coordinate formulas.

Problem 85.67 (B4.2. Integral-affine realization theorem) Realize every sufficiently divisible chamber point by a divisor model carrying the full Coble affine package.

Problem 85.68 (B4.3. Affine-symmetry characterization theorem) Characterize the direct Coble period locus exactly by the presence of the prescribed affine involutions and exceptional-root data.

Problem 85.69 (B4.4. Algebraic involution theorem) Promote the affine symmetries to commuting regular involutions on general divisor models and control the birational-to-regular modification for special models.

Problem 85.70 (B4.5. dlt quotient and relative-Proj theorem) Prove the dlt and slc properties, canonical bundle formula, positivity, and exact stable contractions.

85.18.2.5 Sections 5–7 analogues: semifans and boundary classification

Problem 85.71 (B5.1. Coble recognizability theorem) Prove that the direct divisor package is recognizable from limiting Hodge and combinatorial data in the sense required by the Alexeev–Engel compactification theory.

Problem 85.72 (B5.2. Geometric relevance theorem) Determine exactly which direct Coxeter walls change the stable pair and hence which roots are relevant.

Problem 85.73 (B5.3. Generalized Coxeter semifan theorem) Prove that the direct regions of combinatorial constancy are the generalized Coxeter semifans of the verified direct chambers.

Problem 85.74 (B5.4. Direct KSBA normalization theorem) Construct the classifying map, prove properness, finiteness, generic injectivity, and absence of further coarsening.

Problem 85.75 (B6.1. Coble component classification) Classify all finite and affine component pairs, including equations, quotient involutions, branch curves, ramification divisors, and contractions.

Problem 85.76 (B7.1. Complete boundary theorem) List every Type II and Type III stable Coble surface and prove compatibility with the Baily–Borel and semitoroidal incidence diagrams.

85.18.2.6 Advantages and principal difficulties

The autonomous strategy has three advantages.

1. The arithmetic group is defined geometrically rather than inherited.

2. Relevance and the stable divisor are intrinsic to Coble pairs.

3. The final comparison with Enriques moduli becomes a discrete rigidity theorem rather than part of every local construction.

Its principal difficulties are equally clear: direct monodromy, complete Vinberg calculations, algebraic realization of the fixed exceptional curve in degenerations, and a Coble-specific recognizable-divisor theorem.

85.18.3 Strategy C: direct K3-to-Coble folding or reflection twisting

85.18.3.1 Candidate mechanism

The identity \(I_{\mathrm{Co}}=w_\alpha I_{\mathrm{En}}\) suggests that the Coble theory can be obtained from the ambient K3 involution space by a direct reflection-twisted folding, rather than by first folding to Enriques and then slicing by \(\alpha^\perp\).

At a cusp, the candidate operation is \(J_{\mathrm{Co}}=w_{\bar\alpha}J_{\mathrm{En}}\).

85.18.3.2 Required chain

Problem 85.77 (C1. Classification of admissible ambient actions) Classify involutions of the two reflective K3 cusp lattices whose fixed lattices have the Coble cusp types and whose full-lattice lifts preserve the labeled direct Coble data.

Problem 85.78 (C2. Direct root-folding theorem) Classify all ambient root orbits under the direct Coble involution and prove that every negative folded vector is a Coble root and every Coble root arises in this way.

Problem 85.79 (C3. Chamber-intersection theorem) Prove that intersection of an ambient K3 chamber with the direct Coble fixed subspace is a fundamental chamber for the direct Coble reflection group.

Problem 85.80 (C4. Full-lattice and geometric realization theorem) Lift the cusp action to \(L_{K3}\) and identify it with the geometric pair of involutions on divisor models.

Problem 85.81 (C5. Comparison with the wall-link construction) Prove that the direct folded chamber agrees with the orthogonal link of the corresponding Enriques root wall after the comparison of labeled embeddings.

85.18.3.3 Failure mode in the earlier attempt

The previous argument verified only the orthogonal-projection formula for selected walls. It did not prove the root converse, chamber completeness, or full-lattice lifting. Thus it produced wall-slice diagrams rather than Coxeter diagrams. Strategy C remains viable only if C1–C5 are proved.

85.18.4 Strategy D: computation-first arithmetic and Vinberg analysis

85.18.4.1 Legitimate uses

Computation is well suited to the following tasks.

• verifying lattice signatures, discriminant groups, and primitive closures;

• computing centralizers of explicit lattice involutions;

• enumerating finite discriminant-form orbits once the integral stabilizer images are known;

• running Vinberg’s algorithm in explicit cusp lattices;

• certifying parabolic and elliptic subdiagram exhaustion;

• checking integral-affine polygon closure and involution symmetry;

• computing local invariant rings, divisor classes, and intersection numbers.

85.18.4.2 Invalid substitutions

Computation does not replace the following parametric statements.

• A finite \(O(A)\) orbit computation does not identify integral cusps.

• A Gram matrix does not certify that a root list is a complete simple system.

• A finite list of monodromy vectors does not prove an integral-affine realization theorem.

• A family of explicit curves does not prove coverage of the moduli space.

• A local quotient calculation does not identify the universal KSBA divisor.

Each script must therefore state a verification contract: the exact finite or symbolic assertion it certifies and the theorem it does not prove.

85.18.5 Strategy E: hybrid direct-open and inherited-boundary approach

A potentially efficient hybrid is to prove the direct open period and monodromy theorem first, then use the AEGS ambient stable family only to construct boundary models. The comparison with the Heegner divisor is made at the open level before the boundary is analyzed.

The required chain is:

1. direct projective coverage and labeled rigidity;

2. equality of direct monodromy with the Heegner stabilizer group;

3. open-period factorization through \(\mathcal F_{\mathrm{En},2}\);

4. restriction of the AEGS family to the now identified direct period space;

5. integral cusp and Vinberg calculations for the direct group;

6. normalized toroidal restriction and semitoroidal compatibility;

7. generic reconstruction/no-further-coarsening.

This strategy reduces duplication of integral-affine existence theorems but still requires all arithmetic and comparison steps. It is not equivalent to simply declaring the Coble compactification to be a closed subspace of the AEGS compactification.

85.19 Common errors, caveats, and dependency-sensitive warnings

The following warnings are stated in self-contained form and cross-reference the constructions they affect.

85.19.1 Surface-theoretic warnings

Warning 85.82 (Coble is not Enriques) The smooth resolution \(\widetilde V\) is rational and has non-torsion canonical class. The singular surface \(V^\sharp\) is \(\mathbb Q\)-Calabi–Yau only after contracting the anti-bicanonical \((-4)\)-curve. Arguments using \(K_Z\) torsion on an Enriques surface cannot be transferred to \(\widetilde V\).

Warning 85.83 (order versus canonical index) The notation \(\frac{1}{4}(1,1)\) records a quotient group of order \(4\). The Cartier index of the canonical class is \(2\).

Warning 85.84 (normality versus boundary nonnormality) The generic \(\frac{1}{4}(1,1)\) Coble quotient is normal. Irreducible nonnormal stable surfaces can occur only through conductor self-identifications or other boundary phenomena.

Warning 85.85 (anti-bicanonical curve versus stable divisor) The divisor \(C\in|-2K_{\widetilde V}|\) is the branch curve of the Coble cover on the resolution. The stable divisor is \(R^\sharp\), the descended del Pezzo ramification divisor. The two are related by the crepant formula in Section 85.6.4 but are not equal.

Warning 85.86 (quasipolarization versus polarization) The natural degree-\(2\) class on \(\widetilde V\) is orthogonal to \(C\) and is not ample. It becomes ample only after contracting the complete null locus. Any moduli group that purports to remember an ample class on \(\widetilde V\) is using the wrong object.

Warning 85.87 (special Heegner intersections) On the generic one-node locus, \(C\) is the only null curve. On higher Heegner intersections additional curves may be null and must also be contracted before the singular degree-\(2\) polarization becomes ample.

Warning 85.88 (Coble versus unigonal Heegner divisors) The Coble locus is the \((-2)\)-Heegner arrangement associated with a fixed node and a \(\frac{1}{4}(1,1)\) quotient. The unigonal locus is a distinct \((-4)\)-Heegner divisor whose projective model uses \(\mathbb P(1,1,2)\). The two constructions must not be interchanged.

Warning 85.89 (which involution defines the stable divisor) In AEGS and in the proposed Coble theory, the stable divisor is the divisorial fixed locus of the del Pezzo involution. The Enriques involution is fixed-point free on the generic Enriques K3 cover, and the Coble involution fixes the exceptional curve only on the resolution. Neither of these is the source of \(R^\sharp\).

Warning 85.90 (the Coble singularity is not ADE) The singularity \(\frac{1}{4}(1,1)\) is a non-Gorenstein cyclic quotient singularity. ADE arguments apply only after passing to the canonical \(A_1\) cover or to the resolution, and the boundary coefficients must be tracked through that passage.

85.19.2 Group and period warnings

Warning 85.91 (group by fiat) The Hodge stabilizer image, centralizer image, congruence inverse image, and geometric monodromy group are distinct definitions; see Section 85.7.3. Equality must be proved. Choosing one definition because it makes the desired compactification statement true is circular.

Warning 85.92 (abstract isometry is not labeled rigidity) An isomorphism \(T_{\mathrm{Co}}\cong I_{2,9}(2)\) does not determine the embedding into \(L_{K3}\), the degree-\(4\) class, the exceptional root, the second involution, or the chamber. Strategy B requires the labeled rigidity theorem of Section 85.5.4.

Warning 85.93 (domain inclusion versus quotient morphism) An inclusion of type-IV domains does not by itself descend to arithmetic quotients. The groups must arise from compatible subgroups of a common orthogonal group. This issue controls the diagram in Section 85.7.6.

Warning 85.94 (direct construction versus Heegner theorem) A direct Coble period quotient inside the K3 involution space is a separate theorem until its group, labeled embedding, and universal family are shown to factor through the Enriques Heegner construction. Strategy B must be followed by Target theorem B.

Warning 85.95 (smoothing the cover versus smoothing the quotient) An arbitrary smoothing of the nodal K3 cover need not carry the node-fixing involution. The explicit invariant \((4,4)\) family of Section 85.4.4 does give an equivariant smoothing; no general conclusion should be drawn without a family-level extension theorem.

85.19.3 Cusp and Coxeter warnings

Warning 85.96 (four orbit problems) Orbits under \(O(T_{\mathrm{Co}})\), \(O^+(T_{\mathrm{Co}})\), \(\Gamma_{\mathrm{Co},2}\), and \(O(A_{T_{\mathrm{Co}}})\) can all differ. A claim about one group cannot be transferred to another without an exact sequence, a stabilizer-image calculation, or an integral transitivity theorem.

Warning 85.97 (cusp maps require embedded representatives) A permutation of cusp labels is not a proof of a Baily–Borel map. One must exhibit an isotropic line or plane inside the common embedded lattice chain and compute its divisibility and arithmetic orbit in both groups.

Warning 85.98 (AEGS Figure 4 border types) The double-rectangle Enriques \(1\)-cusps are \(12,13,14,15,245\); the single-rectangle cusps are \(34,35,45,55\). The former map to the segment-flipping quotient lattice \(E_8(2)\) and cannot contain a \((-2)\) Coble root in the negative quotient. Earlier assignments that treated \(245\) as a Coble image are incompatible with this lattice obstruction.

Warning 85.99 (wall slice is not Coxeter diagram) The construction in Section 85.9.2 gives an arrangement of restricted hyperplanes. It becomes a Coxeter chamber only after root integrality, simplicity, completeness, and arithmetic-index statements are proved.

Warning 85.100 (parabolic inspection is not exhaustion) Type II rays are arithmetic orbits of maximal parabolic subdiagrams of the complete chamber. Listing visually apparent affine subdiagrams is insufficient unless a Vinberg computation or classification proves that no others occur.

Warning 85.101 (square-diagram Gram entry) The black nodes \(\alpha_{20}\) and \(\alpha_{21}\) are joined by a thick edge and have inner product \(4\), not \(2\). Any computation based on the latter value has the wrong rank.

Warning 85.102 (the correct reflective ambient space) The reflective cusp diagrams used by AEGS belong to the hyperelliptic degree-\(4\) K3 involution space \(\mathcal F_{(2,2,0)}\), not to the full quartic K3 moduli space \(\mathcal F_4\). Any direct folding strategy must explain its relation to \(\mathcal F_{(2,2,0)}\) before using the ambient diagrams.

85.19.4 Integral-affine and stable-pair warnings

Warning 85.103 (affine symmetry versus algebraic involution) An involution of a dual complex predicts but does not prove an involution of the degeneration. One must impose the correct limiting-period condition and choose invariant component and gluing moduli; see Section 85.11.

Warning 85.104 (crossed node) A crossed Coxeter node is not merely a diagram symmetry. It includes a root reflection, forces the corresponding root coordinate to vanish, and is realized by an \(I_1\)–\(I_1\) collision to an \(I_2\) affine singularity.

Warning 85.105 (fixed affine edge need not support ramification) In the non-simple AEGS cusp, a fixed bottom edge has no support in the ramification polarization. Coble relevance must likewise be read from the divisor package, not from fixed loci alone.

Warning 85.106 (dlt model versus stable model) The quotient dlt model can contain \(R^\sharp\)-trivial curves, components, and conductor strata. The KSBA model is obtained only after the relative Proj of Section 85.13.

Warning 85.107 (restriction versus compactification of the restriction) The intersection of an ambient semifan with a Heegner subspace is not automatically the semitoroidal normalization of the Heegner closure. The arithmetic, normalization, gluing, and no-further-coarsening statements of Section 85.16.4 are indispensable.

85.20 Dependency graph for further work

The following diagram records the logical order of the autonomous strategy. An arrow means that the lower statement uses the upper statement.

[

]

The later comparison with the Heegner divisor requires, in addition,

[

]

The arithmetic cusp classification precedes the Coxeter theorem because the lattice and group acting at a cusp must be known before a chamber is computed. The stable-model analysis precedes the relevance marking because relevance is defined by the relative Proj. Any proof order that reverses either dependency risks classifying the wrong objects.

85.21 Consolidated status ledger

The following table summarizes the present mathematical status.

Table 85.4: Consolidated status of the Coble compactification project.

Item	Status	Controlling section
Local invariant \((4,4)\) node construction	established	Section 85.4.1
Local \(\frac{1}{4}(1,1)\) quotient	established	Section 85.4.3
Explicit equivariant smoothing	established for the projective family	Section 85.4.4
Coverage of the complete Coble locus	required theorem	Section 85.4.5
\(T_{\mathrm{Co}}\cong\langle2\rangle\oplus U(2)\oplus E_8(2)\)	established	Section 85.5.1
\(S_{\mathrm{Co}}\) and reflection twist	established as lattice calculations	Section 85.5.2
Labeled embedding uniqueness	required theorem	Section 85.5.4
Degree-\(2\) class is quasipolarized on \(\widetilde V\)	established	Section 85.6.1
\(\lvert-2K_{\widetilde V}\rvert=\{C\}\) generically	established	Section 85.6.2
Stable divisor \(R^\sharp\) and numerical formulas	established on the projective one-node locus	Section 85.6.3
Open KSBA pair \((V^\sharp,\epsilon R^\sharp)\)	established fiberwise on the general locus	Section 85.6.4
Direct moduli equivalence in families	required theorem	Section 85.7.2
Equality of four arithmetic groups	required theorem	Section 85.7.3
Full orthogonal stabilizer exact sequence	established lattice lemma	Section 85.7.4
Unpolarized cusp count	established with standard lattice input	Section 85.8.2
Polarized cusp count	not established	Section 85.8.3
Finite \(O(A)\) orbit tables	established computation	Section 85.8.4
Unpolarized Coxeter chamber	established	Section 85.9.1
Polarized wall-slice arrangements	established computations	Section 85.9.5
Polarized Coble Coxeter diagrams	not established	Section 85.9
Integral-affine Coble package	defined; realization open	Section 85.10.2
Direct dlt and stable-model theorem	open	Section 85.12
Exact Coble semifans	open	Section 85.16.2
Normalization of the Heegner KSBA closure	target theorem	Section 85.17.3

85.22 Computational record and verification contracts

This section records calculations that may be reused after the corresponding parametric theorems are proved.

85.22.1 Explicit invariant branch curve

On the affine chart \(x_0=y_0=1\), one checked the invariant polynomial

[ \[\begin{aligned} f(x,y)={}&29x^4y^4+23x^4y^2+19x^4 +17x^3y^3+13x^3y\\ &+11x^2y^4+7x^2y^2+x^2 +5xy^3+xy+3y^4+2y^2. \end{aligned}\]

]

Its quadratic part is \(x^2+xy+2y^2\), with discriminant \(-7\). Gr"obner-basis calculations on the four standard affine charts give one singular point, at \((0,0)\) on the displayed chart, and no singularities on the other charts.

Remark 85.108 (Verification contract) This calculation proves nonemptiness of the open one-node invariant family. It does not prove that the family is the complete Coble moduli space.

85.22.2 Lattice calculations

Symbolic Gram-matrix computations verify:

• the signature and determinant of the displayed \(T_{\mathrm{Co}}\) and \(S_{\mathrm{Co}}\) decompositions;

• the eigenspaces of the reflection twist on an explicit model;

• the rank correction for the ambient square diagram when \((\alpha_{20},\alpha_{21})\) is changed from \(2\) to \(4\).

Remark 85.109 (Verification contract) These computations certify explicit lattice models. They do not prove labeled embedding uniqueness or geometric monodromy.

85.22.3 Finite cusp computations

The finite-orbit data of Section 85.8.4 and the enhanced star calculation are reproducible. They are inputs to, not substitutes for, the integral cusp-lifting theorem.

Remark 85.110 (Verification contract) The scripts compute orbits under explicitly specified finite matrix groups. They do not identify those groups with the images of integral parabolic stabilizers.

85.22.4 Wall-slice computations

The corrected ambient Gram matrix reproduces the selected wall-link roots and the four candidate arrangements. The scripts test root integrality for the displayed vectors and enumerate parabolic and elliptic subgraphs of the candidate graphs.

Remark 85.111 (Verification contract) The scripts verify the finite graph calculations for the input root list. They do not prove that the list is a complete Vinberg simple system or that graph automorphisms equal arithmetic chamber automorphisms.

85.23 Appendix: corrected candidate wall-slice arrangements

This appendix records the finite arrangements obtained by slicing selected AEGS Enriques chambers by selected white-root hyperplanes. The data are retained because they are useful starting points for a direct Vinberg calculation. They are not asserted to be complete Coxeter diagrams.

White vertices below have square \(-2\) and black vertices have square \(-4\). A double arrow denotes the non-simply-laced Coxeter edge between roots of unequal length; \(==\) denotes a thick edge; and \(\cdots\) denotes a dotted edge.

85.23.1 Slice over the triangular ambient cusp

The arrangement \(G_{\mathrm{slice}}^2\) has nine vertices. The vertices \(c_0,\ldots,c_7\) are black and \(c_8\) is white. Its edges are

[ c_0-c_3, c_1-c_2-c_3-c_4-c_5-c_6-c_7c_8. ]

This arrangement agrees with the standard unpolarized chamber after a relabelling, but its identification with a polarized cusp requires the integral cusp theorem.

85.23.2 First square-cusp slice

The white vertices are \(c_0,c_7,c_8\) and the black vertices are \(c_1,\ldots,c_6,c_9,c_{10}\). The nonzero edges are

[ c_0c_1, c_0==c_8, ]

[ c_1-c_2-c_3-c_4-c_5-c_6c_7, ]

[ c_3-c_9, c_7c_{10}, c_8c_{10}. ]

85.23.3 Second square-cusp slice

The white vertices are \(c_5,c_7\) and all other displayed vertices are black. The edges are

[ c_0-c_1-c_2-c_3-c_4c_5, ]

[ c_0-c_6c_7, c_0-c_8, c_4-c_9. ]

85.23.4 Third square-cusp slice

The white vertices are \(c_0,c_8,c_9,c_{10},c_{11}\) and the black vertices are \(c_1,\ldots,c_7,c_{12}\). The edges are

[ c_0c_1, c_0c_7, c_0==c_{11}, ]

[ c_1-c_2-c_3-c_4-c_5-c_6-c_7, ]

[ c_2c_8, c_4c_9, c_6c_{10}, ]

[ c_8c_{12}, c_{10}c_{12}, c_9==c_{11}, c_{11}c_{12}. ]

85.23.5 Discarded cusp calibration

One later finite calculation decorated each totally singular plane by one of eight stars and assigned the resulting seven finite orbits the labels

[ 245,,,^{(2)},{(6)},^{(2)},{(6)}. ]

This calibration is not retained. The integral parabolic image used to define the star action was not proved, and the label \(245\) is incompatible with the necessary \((-2)\)-class condition in Section 85.8.5. The finite enhanced orbit calculation can still be reused after the actual integral stabilizer image is known, but the displayed labels have no current arithmetic meaning.

85.24 Appendix: adversarial consistency audit

The report was checked against the principal failure modes identified in the research discussions. The purpose of this appendix is not to add new mathematics, but to make explicit where each ambiguity is resolved or retained as an open theorem.

Table 85.5: Adversarial consistency audit.

Audit question	Resolution in this report
Are the ambient Enriques and del Pezzo lattices stated with all summands?	Yes; see Section 85.3.2.
Is the Coble surface distinguished from its smooth rational resolution?	Yes; see Definitions in Section 85.2.3.
Is \(\frac{1}{4}(1,1)\) treated as a normal non-Gorenstein quotient with canonical index \(2\)?	Yes; see Section 85.2.2 and the warning on order versus canonical index (Warning 85.83).
Is the canonical index-one cover correctly identified as \(A_1\)?	Yes; see the warning on the index-one cover (Warning 85.16).
Is the stable divisor distinguished from the anti-bicanonical curve?	Yes; see Section 85.6.3 and the warning distinguishing the stable divisor (Warning 85.85).
Is the degree-\(2\) class identified as a quasipolarization before contraction?	Yes; see Section 85.6.1.
Are the Hodge stabilizer, direct centralizer, congruence group, and geometric monodromy separated?	Yes; see Section 85.7.3.
Is equality of these groups stated as a theorem already proved?	No; it is the required arithmetic and monodromy identification theorem (Problem 85.24).
Is abstract lattice isometry distinguished from labeled embedding rigidity?	Yes; see Section 85.5.4 and the labeled-rigidity warning (Warning 85.92).
Are period-domain inclusions distinguished from maps of arithmetic quotients?	Yes; see Section 85.7.6 and the domain/quotient warning (Warning 85.93).
Is the direct K3 construction required to factor through the Enriques construction?	Yes; this is Target theorem B in Section 85.17.3.
Is an \(O(A)\) orbit computation used as an integral cusp theorem?	No; see Section 85.8.3 and the finite-shadow warning (Warning 85.28).
Are the possible Enriques Type II cusp images read with the correct single/double rectangle convention?	Yes; see Section 85.8.5 and the AEGS Figure 4 border-type warning (Warning 85.98).
Are wall slices called Coxeter diagrams?	No; see Section 85.9.5 and the wall-slice warning (Warning 85.99).
Is the corrected thick edge \(\alpha_{20}\)–\(\alpha_{21}\) recorded?	Yes; see Section 85.9.4.
Are maximal parabolic lists required to be exhaustive?	Yes; see the Coble Vinberg/root-completeness theorem (Problem 85.34) and the parabolic-exhaustion warning (Warning 85.100).
Is an affine involution treated as automatically algebraic?	No; see Section 85.11.
Is the dlt quotient identified with the stable model before relative Proj?	No; see Section 85.13 and the dlt/stable-model warning (Warning 85.106).
Is restriction of an ambient semifan treated as formal?	No; see Section 85.16.4.
Is the autonomous direct strategy presented as already proving the Heegner theorem?	No; the two target theorems are separated in Section 85.17.
Are computational scripts assigned explicit mathematical limits?	Yes; see Section 85.22.

Two residual points are intentionally left as research problems rather than silently resolved.

1. The exact descent category of the degree-\(2\) class on the singular Coble model—Cartier line bundle, reflexive sheaf, or numerical class in special families—must be fixed as part of the direct moduli theorem. The generic projective model supports the stated numerical and ampleness properties.

2. The family-theoretic construction of the canonical cover and simultaneous resolution is not used as a formal black box. It is isolated in the cover–singular equivalence theorem and in the root-marked universal-family lemma of Strategy A.

85.25 Source map

The following references indicate where the standard theories used in this report may be consulted.

• AEGS (Alexeev et al. 2025) is the controlling model for the projective diagram, unique embedding chain, cusp folding, integral-affine quotient models, generalized Coxeter semifans, and the finite/no-further-coarsening step.

• Nikulin (Nikulin 1980) is the primary source for even \(2\)-elementary lattices, discriminant forms, and primitive embeddings.

• Namikawa (Namikawa 1985) and Sterk (Sterk 1991) control the Enriques period and cusp theory used in the Heegner strategy.

• Dolgachev–Kond=o (Dolgachev and Kondō 2013) treat orthogonal modular descriptions of classical Coble surfaces.

• Vinberg (Vinberg 1972, 1975) supplies the reflection-group algorithms and Coxeter theory.

• Alexeev–Engel and related work (Alexeev and Engel 2022, 2023; Alexeev et al. 2021, 2023, 2022) supplies recognizable divisors, integral-affine models, and semitoroidal normalizations of K3 moduli.

• Koll'ar, Koll'ar–Mori, Koll'ar–Xu, and Birkar (Kollár and Mori 1998; Kollár 2023; Kollár and Xu 2020; Birkar 2023) supply the MMP and KSBA framework.

• Kulikov, Persson–Pinkham, Friedman–Scattone, Gross–Hacking–Keel, Engel, and Engel–Friedman (Kulikov 1977; Persson and Pinkham 1981; Friedman and Scattone 1986; Gross et al. 2015a, 2015b; Engel 2018; Engel and Friedman 2021) supply the degeneration and integral-affine background.

• The Stacks Project (The Stacks Project Authors 2026) is a general reference for relative spectra, finite morphisms, normalization, descent, and family-theoretic language.

85.26 Conclusion

The Coble compactification problem now has a precise mathematical ontology. The open geometry is governed by invariant nodal \((4,4)\) curves, a nodal K3 cover with two commuting involutions, the lattice \(T_{\mathrm{Co}} \cong \langle2\rangle\oplus U(2)\oplus E_8(2)\), and the singular stable pair \((V^\sharp,\epsilon R^\sharp)\).

On the resolution, the degree-\(2\) class is a quasipolarization, \(|-2K|\) consists generically of one rational \((-4)\)-curve, and the crepant dlt boundary is

[ C+R_{V}. ]

The principal unresolved issues are not the local quotient calculation or the basic lattice type. They are the labeled embedding rigidity, equality with geometric monodromy, integral cusp lifting, complete Vinberg chambers, algebraic realization of the Coble affine package, intrinsic relevance of walls, and the comparison between the autonomous compactification and the normalized Enriques Heegner closure.

The autonomous strategy organizes these issues into direct Coble analogues of the AEGS theorem chain. The Heegner strategy organizes them into arithmetic normalization, universal-family, and no-further-coarsening statements. A complete proof may combine the two, but it must eventually identify their lattices, groups, divisors, cusp structures, semifans, and KSBA families as the same mathematical objects.

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