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47  Constructions of the moduli space

We summarize the relevant moduli spaces:

\[ \begin{aligned} \fen &= \bD(\ten)/\Orth^+(\ten)^* & \fentwo &= \bD(\ten)/\gent \\ \fco &= \bD(T_\Co) / \Orth^+(T_\Co)^* & F_{\Co, 2} &= \bD(T_\Co)/\Gamma_{\Co, 2} \end{aligned} \]

where

\[ \begin{aligned} \gent &= \Orth(\ten) \intersect \Orth(\tdp) \iscontainedin \Orth(\lkt) \\ \Gamma_{\Co, 2} &= \mathrm{Stab}_{\Orth(\ten)}(T_\Co) \iscontainedin \Orth(\ten) \\ F_{\Nod} &= \bD(T_\Nod)/\Orth^+(T_\Nod)^* \\ F_{\Nod, 2} &= ??? \end{aligned} \]

where \(\tdp\) is described in (Alexeev et al. 2025). Note that we implicitly use the embedding \(\eta: T_\Co \injects \ten\) of the primitive-embedding lemma (Lemma 48.1).

Question 47.1 Degree 2 polarized Coble surfaces do not seem to appear in previous literature, merely (unpolarized?) Cobles with \(n\) boundary components. So I can not yet determine a more explicit description of \(\Gamma_{\Co, 2}\).

We note, as in (Dolgachev and Kondō 2013), that \(T_{\Co} \cong v^{\perp \ten}\) for some \(v^2=-2\), which implies that there is a birational isomorphism \(\fco \birationaliso \mch_{-2}/\Orth^+(\ten)^*\).

The tower relates the degree-\(2\) numerically polarized Coble and Enriques moduli \(F_{\Co,2}\), \(\fentwo\), their unpolarized quotients \(\fco\), \(\fen\), and the degree-\((2,2,0)\) K3 moduli space \(\fttz\injects F_4\). The map \(F_{\Co,2}\to \fco\) marked \(?\) is the one not yet determined, for the reason above: \(\Gamma_{\Co,2}\) has no explicit description.

47.1 The GIT construction

Remark 47.2 (GIT construction) Following (Dolgachev and Kondō 2013), by varying the coefficients of \(p_i\) in the planar blowup construction, one can construct \(\fco\) as a locally closed subvariety of \((\PP^2)^{10}/\PGL_3\), which is of dimension

\[ \dim (\PP^2)^{10}/\PGL_3 = \dim (\PP^2)^{10} - \dim \PGL_3 = 2\cdot 10 - (3^2-1) = 12 . \]

A posteriori, the number of moduli for a Coble surface is 9, which means that there should be 3 conditions imposed upon the configuration of 10 points. These 3 conditions are precisely the discriminant conditions described in (Coble 1919, sec. 2, Prop. (10)). Letting \(D\) be the corresponding discriminant locus, we can identify \(\fco\) as an open subset of \(\qty{ (\PP^2)^{10} \sm D }/\PGL_3\) at the level of coarse moduli spaces.

47.2 The Horikawa model

Remark 47.3 (Horikawa's construction) Alternatively, Horikawa (Horikawa 1977) and more recently (Alexeev et al. 2025) consider the following: let \(Y\definedas \PP^1\times \PP^1\) and define an involution \(\tau(x,y) \definedas (-x,-y)\). Letting \(B\in\abs{-2K_Y}^\tau\) be a \(\tau\)-invariant anti-bicanonical curve in \(Y\), if \(B\) passes through a \(\tau\)-fixed point \(x,y\in \theset{0, \pm \infty}\), then the corresponding double branched cover branched over \(B\) is a nodal K3 surface \(X\) with \(A_1\) singularities and covering involution \(\idp\) such that \(Y = X/\generators{\idp}\). Letting \(\ien\) be a lift of \(\tau\), the quotient \(Z\definedas X/\generators{\ien}\) is a Coble surface. The case in which \(B\) does not pass through a \(\tau\)-invariant point yields an Enriques surface \(Z\), and an analysis of the corresponding moduli is carried out in (Alexeev et al. 2025).

It is well-known that there consequently exists a Coble surface \(S\) and a blowdown \(S\to X/\iota\) along the strict transform of \(B\). In this way, one realizes the moduli space \(\fco\) of unpolarized Coble surfaces as a divisor in \(\fen\), the 10-dimensional moduli space of unpolarized Enriques surfaces.

47.3 Period domains

By passing to the K3 cover, one can embed \(\fco\) into an arithmetic quotient of a 9-dimensional Hermitian symmetric domain of type \(\rm{IV}\). Let \(\lkt = U^3 \oplus E_8^2\) be the canonical K3 lattice. We recall that for any primitively embedded lattice \(S\injects \lkt\), letting \(T \definedas S^{\perp \lkt}\), there is a Hodge-theoretic description of the coarse moduli space \(F_S\) of \(S\)-polarized K3 surfaces given by \[F_S \definedas D_T/\Orth^+(T)^*\]

where \(\Orth^+(T)^* \definedas \Orth^+(T) \intersect \ker\qty{\Orth(T) \to \Orth(q_T)}\), the group \(\Orth^+(T)\) is the index-two subgroup of \(\Orth(T)\) preserving the component \(D_T\), and \(D_T\) is a connected component of

\[ \Omega_T \definedas \theset{[v]\in \PP(T_\CC) \st v^2=0,\, v\bar v > 0} . \]

Proposition 47.4 (Dimension of a type IV domain) Let \(T\) be a lattice of rank \(r\) and signature \((2, r-2)\). Then the type IV domain \(D_T\) has complex dimension \(r - 2\).

Proof. The complexification \(T_\CC\) has dimension \(r\), so \(\PP(T_\CC)\) has dimension \(r-1\). The equation \(v^2 = 0\) is a single nondegenerate quadratic equation and cuts out a smooth quadric hypersurface of dimension \(r-2\) inside \(\PP(T_\CC)\). The condition \(v\bar v > 0\) is open, so it selects an open subset of that quadric, and \(D_T\) is one of its connected components.

Corollary 47.5 (Dimension of $M$-polarized K3 moduli) Let \(M\injects\lkt\) be a primitive embedding of a lattice of signature \((1, \rank(M) - 1)\) with orthogonal complement \(T = M^{\perp\lkt}\). Then \[ \dim_\CC D_T = \rank(T) - 2 = 20 - \rank(M) . \] For the Coble lattices \(\rank(S_\Co) = 11\), so \(\dim_\CC D_{T_\Co} = 9\).

Proof. The K3 lattice has rank \(22\), so \(\rank(T) = 22 - \rank(M)\), and \(T\) has signature \((2, 20 - \rank(M))\); apply the type-IV dimension proposition (Proposition 47.4).

Letting \(E_{10} \definedas U \oplus E_8\), one can similarly consider the Enriques lattices \(\sen := E_{10}(2)\) with \(\ten = U \oplus E_{10}(2)\). Letting

\[ \mch_{-2d} = \Union_{\delta^2 = -2d} \delta^{\perp D_T} , \]

be the union of the hyperplane sections cut out by the vectors \(\delta\) of norm \(-2d\), as in Figure 47.1, one can thus present

\[ \begin{aligned} \fen &= \qty{D_{\ten} \sm \mch_{-2}} / \Orth^+(\ten)^* \\ \fco &= \mch_{-2} / \Orth^+(\ten)^* \\ F_{\Nod} &= \qty{\mch_{-4} \sm \mch_{-2}} / \Orth^+(\ten)^* \end{aligned} \]

where surfaces along the divisor \(\mch_{-2}\) in \(\fen\) correspond precisely to Coble surfaces and those along \(\mch_{-4}\sm \mch_{-2}\) correspond to Enriques surfaces with \(A_1\) singularities.

Figure 47.1: The period domain \(D_{\ten}\) together with the hyperplanes \(v_i^{\perp}\) cut out by vectors \(v_i\) of fixed negative norm. The Coble surfaces are precisely the periods lying on the hyperplanes of \(\mch_{-2}\).

Alternatively, one can construct the period domain for \(\fco\) directly. Following (Dolgachev and Kondō 2013 Prop. 3.1) almost verbatim, let \(S\) be the blowup of \(\PP^2\) along 10 \(A_1\) singularities of a rational sextic curve \(C\). The double branched cover \(f:X\to S\) branched along the proper transform of \(C\) contains the pullback of ten exceptional classes \(E_1,\cdots, E_{10}\) over the ten \(A_1\) singularities, as well as \(E_0\), the pullback of a hyperplane class. Since \(E_i^2 = -2\) for \(1\leq i \leq 10\), and one can show that \(E_0^2 = 2\), the \(E_i\) generate a lattice isometric to

\[ S_\Co \definedas \generators{2}\oplus \generators{-2}^{10} = \latI_{1, 10}(2) = (11, 11, 1)_1 \cong \generators{-2} \oplus E_{10}(2) . \]

This follows from the fact that \(S_{\Co}\) is a 2-elementary lattice of signature \((1, 10)\) with discriminant group \((\bZ/2\bZ)^{11}\) and thus uniquely determined up to isometry by its invariants \((r,a,\delta)\). Similarly, its orthogonal complement \(T_{\Co}\) in the K3 lattice \(\lkt\) is a 2-elementary lattice of signature \((2, 9)\) and satisfies \(q_{S_{\Co}} = -q_{T_{\Co}}\). The isomorphism class of \(q_{T_{\Co}}\) determines \(T_\Co\), and thus one has

\[ T_\Co \definedas S_\Co^{\perp \lkt} = (11, 11, 1)_2 \cong \generators{2} \oplus E_{10}(2) . \]

Alternatively, this follows immediately from the mirror move \(S\leadsto T\) of (Alexeev and Engel 2022 Thm. 5.10) applied to \(S_\Co =(11, 11, 1)_1\). We obtain \(\fco\) as an open subset of the period domain \(D_{T_\Co}/\Orth^+(T_\Co)^*\), a normal quasiprojective variety of dimension 9, by (Dolgachev and Kondō 2013 Prop. 3.2) and an application of the Torelli theorem for algebraic K3 surfaces from (Piatetski-Shapiro and Shafarevich 1971). We note that (Dolgachev and Kondō 2013) shows that \(\fco\) is rational by relating it to a codimension one subvariety of a moduli space of certain \(A_2\)-singular quintics in \(\PP^2\).

47.4 The sextic moduli construction

Remark 47.6 (Orientation) The constructions above start from the surface: from the Coble surface \(S\) and its K3 cover, or from the Enriques period domain in which the Coble locus is a Heegner divisor. A third route starts from the branch curve and never mentions \(S\): one takes the moduli space of plane sextics of a fixed singularity type, and maps it to an arithmetic quotient by the periods of the associated K3. Yu–Zheng–Zhong carry this out for every singular type \(T\) of plane sextic with simple singularities (Yu et al. 2025), and the Coble case is the type \(T = 10A_1\). What this route supplies that the two above do not is a compactification of the open moduli space by geometry of the curves, namely the geometric invariant theory quotient, together with an identification of that compactification in arithmetic terms.

Notation 47.7 (Moduli of sextics of a fixed singular type) For a singular type \(T\) with root lattice \(R\), write \(\mcv_T\) for the space of sextic curves \(Z\subset\PP^2\) with singularities exactly of type \(T\), and \[ \mcm_T \definedas \PP\mcv_T \modmod \SL_3 \] for the moduli space of such curves, the quotient being taken in the sense of geometric invariant theory (Yu et al. 2025, sec. 2.1). Let \(\wh X\) be the double cover of \(\PP^2\) branched along \(Z\), let \(X\) be its minimal resolution — a K3 surface — and let \(H\in\Pic(X)\) be the pullback of the line class, so \(H^2 = 2\). The exceptional curves of \(X\to\wh X\) span a copy \(L\iscontainedin\Pic(X)\) of the root lattice \(R\), with the exceptional classes as a base \(\Delta\). Write \(P\) for the primitive hull of \(\generators H\oplus L\) in \(H^2(X;\ZZ)\) and \(Q \definedas P^{\perp}\), of signatures \((1,\rank R)\) and \((2, 19 - \rank R)\) (Yu et al. 2025, sec. 2.3). For a general member of \(\mcv_T\) one has \(\Pic(X) = P\) (Yu et al. 2025, sec. 3.3).

Proposition 47.8 (The Coble lattices are the lattices of the type $10A_1$) Let \(Z\) be an irreducible sextic of type \(T = 10A_1\), so that \(Z\) is rational by the ten-nodal-sextic lemma (Lemma 34.1), and \(X\) is the K3 cover of the Coble surface \(S = X/\iota\). Then \[ P = \generators{H}\oplus A_1^{\oplus 10} \cong \latI_{1,10}(2) = S_\Co, \qquad Q = P^{\perp\lkt} = T_\Co . \] In particular \(\dim D(Q) = 9\).

Proof. The saturation of \(\generators H\oplus L\) inside \(H^2(X;\ZZ)\) is \((\ZZ/2)^{l'-1}\), where \(l'\) is the number of irreducible components of \(Z\) (Yu et al. 2025, sec. 5.3); for an irreducible sextic \(l' = 1\), so no saturation occurs and \(P = \generators H\oplus L\). With \(H^2 = 2\) and \(L = A_1^{\oplus 10} = \generators{-2}^{\oplus 10}\) this is \(\generators 2\oplus\generators{-2}^{\oplus 10} = \latI_{1,10}(2)\), which is \(S_\Co\) by the Coble invariant-lattice proposition (Proposition 32.4). Taking orthogonal complements in \(\lkt\) gives \(Q = T_\Co\), and the polarized-K3 dimension corollary (Corollary 47.5) gives the dimension.

Remark 47.9 (The involution acts trivially on the root lattice) For a general singular type the covering involution \(\iota\) acts on \(L\) by \(-w_0(L)\), where \(w_0\) is the longest element of \(W(L)\) (Yu et al. 2025, sec. 5.2), and this action folds the root lattice. For \(L\) of type \(A_1\) one has \(-w_0 = \id\), so in the type \(10A_1\) the involution fixes \(L\) pointwise and no folding occurs; the sublattice of \(H^2(S;\ZZ)\) spanned by the exceptional classes of \(S\to\PP^2\) then becomes \(L\) after scaling by \(2\) (Yu et al. 2025, sec. 5.3). That is the same twist by \(2\) recorded geometrically in the K3-cover twist remark (Remark 32.5): the Coble Picard lattice is the blowup lattice \(\latI_{1,10}\) scaled by \(2\), with no folding correction.

Theorem 47.10 (The occult period map and its image) Let \(\Gamma_T\) be the image of \[ \Orth\bigl(H^2(X;\ZZ),\, \Delta,\, H\bigr) \too \Orth(Q) , \] restricted to the subgroup preserving a chosen component \(D(Q)\); it is arithmetic of finite index in \(\Orth(Q)\), and contains every isometry of \(Q\) acting trivially on \(A_Q\) (Yu et al. 2025, sec. 3.1). The occult period map \[ \per_T\colon \mcm_T \too \Gamma_T\backslash D(Q) \] is an algebraic open embedding with image \(\Gamma_T\backslash\bigl(D(Q) - \mch_T\bigr)\), where \(\mch_T\) is the arrangement of hyperplanes \(r^{\perp}\) for roots \(r\) orthogonal to \(H\) and not lying in \(L\). When \(T\) is a nodal type, \(\per_T\) is moreover an isomorphism of orbifolds onto \(P\Gamma_T\backslash(D(Q) - \mch_T)\) (Yu et al. 2025, sec. 1, §3.2, §3.3, §6.2). The description of the image restates, at the level of moduli spaces, the equisingular deformation theory of Urabe (Urabe 1988).

Theorem 47.11 (The GIT compactification is a Looijenga compactification) Let \(\Lambda_1 \definedas H^{\perp\lkt}\), of signature \((2,19)\) with \(A_{\Lambda_1}\cong\ZZ/2\), and let \(\Gamma_1\) be the arithmetic group of the degree-\(2\) K3 moduli space \(\Gamma_1\backslash D(\Lambda_1)\). The roots of \(\Lambda_1\) fall into two \(\Gamma_1\)-orbits, of divisibility \(1\) and \(2\), cutting out arrangements \(\mch_\Delta\) and \(\mch_\infty\); the roots of \(L\) all have divisibility \(1\), so they contribute to \(\mch_\Delta\) only (Yu et al. 2025, sec. 4.1, §4.2). Set \[ \mch^{*}_T \definedas \mch_\infty \intersect D(Q) \iscontainedin \mch_T . \] Then \(\per_T\) extends to an isomorphism \[ \wh{\mcm}_T \;\cong\; \overline{\Gamma_T\backslash D(Q)}^{\,\mch^{*}_T} \] between the GIT compactification of \(\mcm_T\) and the Looijenga compactification (the Looijenga-compactification definition (Definition 54.11)) of \(\Gamma_T\backslash(D(Q) - \mch^{*}_T)\), compatibly with the corresponding statement of Shah and Looijenga for the whole space of sextics, \(\overline{\mcm}\cong\overline{\Gamma_1\backslash D(\Lambda_1)}^{\,\mch_\infty}\) (Shah 1980; Looijenga 2002); the two vertical maps of the resulting square are normalizations onto their images (Yu et al. 2025, sec. 4.2).

Remark 47.12 (The arithmetic group as a normalizer) \(\Gamma_T\) admits a second description: it is the restriction to \(Q\) of the normalizer of the Weyl group \(W(L)\) inside \(\Gamma_1\) (Yu et al. 2025, sec. 4.2). For \(T = 10A_1\) this reads \[ W(L) = \theset{\pm 1}^{10}, \qquad \Gamma_{10A_1} = \Gamma_W|_{Q}, \qquad \Gamma_W = N_{\Gamma_1}\bigl(\theset{\pm1}^{10}\bigr) , \] and the normalizer contains the permutations of the ten nodes, so the induced action on \(A_P\cong(\ZZ/2)^{11}\) is through \(\mathfrak S_{10}\). This is a description of an arithmetic group acting on \(D(T_\Co)\) obtained from the ambient degree-\(2\) K3 group rather than from the Enriques side, and it is therefore independent of the description of \(\Gamma_\Co\) as a stabilizer and centralizer inside \(\Orth(\ten)\) recorded in the Open Problems section.

Question 47.13 (Which quotient of $D(T_\Co)$ is $\fco$?) \(\fco\) is defined above as an open subset of \(D(T_\Co)/\Orth^+(T_\Co)^*\), a quotient by the stable orthogonal group, whereas \(\mcm_{10A_1}\) is a quotient by \(\Gamma_{10A_1}\), which contains \(\Orth^+(T_\Co)^*\) and may be strictly larger. Identifying the two constructions of the Coble moduli space therefore requires the comparison \[ \Orth^+(T_\Co)^* \;\iscontainedin\; \Gamma_{10A_1} \;\iscontainedin\; \Orth(T_\Co) , \] together with the comparison of both against the polarized group \(\Gamma_{\Co,2}\). Until the first inclusion is shown to be an equality, \(\mcm_{10A_1}\) and \(\fco\) are two arithmetic quotients of the same period domain by two different groups, not the same space under two names.

Remark 47.14 (The other singular types) The same theorems hold for every singular type of plane sextic with simple singularities, of which the root lattices have been classified by Urabe (Urabe 1988) and Yang (Yang 1996): the maximal rank is \(19\), and the number of root lattices of rank \(19\), \(18\), \(17\), \(16\) is \(519\), \(987\), \(975\), \(782\) respectively (Yu et al. 2025, sec. 1). Each such type therefore carries both an arithmetic model of its moduli space and a Looijenga model of its GIT compactification. The Coble families with \(n\) boundary components are indexed by the number of irreducible components of the branch sextic ([[coble-lattice-table|the Coble lattice table]]) rather than by its singular type, so which type \(T\) carries which family is a question this section does not settle.

47.5 KSBA spaces

Remark 47.15 By the Baily–Borel extension lemma (Lemma 48.3), there are morphisms \(\overline{\fco}^{\bb} \to \overline{\fen}^{\bb}\) and \(\overline{\fco}^{\bb} \to \overline{\fttz}^{\bb}\) which induce correspondences between the boundary cusps.

Remark 47.16 We set up the moduli space of KSBA stable pairs for Coble surfaces, possibly using the ramification divisor of the K3 involution (which is in this case not fixed-point free). The above embeddings should allow us to take closures of stable pairs in already existing moduli spaces.