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27  Overview

Remark 27.1 A Coble surface is a smooth projective rational surface \(S\) with \(\abs{-K_S} = \emptyset\) but \(\abs{-2K_S}\neq \emptyset\). Such surfaces arise from the work of (Coble 1919) and (Coble 1929) on Cremona transformations of \(\PP^2\) preserving an irreducible rational sextic \(C\) with ten nodes – the blowup \(S\) along these nodes yields a Coble surface. (Coble 1919) shows that the Cremona class of such a curve \(C\) can be written as a union of finitely many projective equivalence classes, and that if \(C\) is sufficiently general then \(\Aut(S) \cong W(E_{10})\), the Weyl group of an infinite root system of type \(E_{10}\). Moreover, he shows that modulo \(\Aut(S)\), there are only finitely many smooth rational negative curves on \(S\).

Coble surfaces occur as degenerations of Enriques surfaces and were ultimately classified in (Dolgachev and Zhang 1999). As such, they are intimately tied to the theory of algebraic K3 surfaces equipped with a nonsymplectic involution, which were classified by (Nikulin 1979). For a reduced plane sextic \(C\), the double cover of \(S\) branched along the proper transform of \(C\) is a K3 surface \(X\) which can be realized as a degeneration of the universal double cover of an Enriques surface, where \(X\) acquires an \(A_1\) singularity fixed by the Enriques involution. The resulting quotient has a quartic singularity whose resolution is an irreducible smooth rational curve \(\tilde C\) satisfying \(\tilde C^2 = -4\), and thus by (Nuer 2015, sec. 3) is a Coble surface.

We will be interested in Coble surfaces in the strong sense, where \(\abs{-2K_S} = \theset{C_1 + \cdots + C_n}\) is a single reduced divisor comprised of \(n\) disjoint smooth rational curves referred to as boundary components of \(S\). It is known that \(1\leq n\leq 10\), and for each such \(n\), there is a moduli space \(F_{\Co, n}\) of Coble surfaces with \(n\) boundary components. When \(n=1\), the moduli space \(F_{\Co} \definedas F_{\Co, 1}\) of Coble surfaces can be described as a boundary divisor \(\mch_{-2}\) in the 10-dimensional moduli space \(F_{\En}\) of unpolarized Enriques surfaces, and thus \(F_{\Co}\) is 9-dimensional. Moreover, \(F_{\Co}\) was shown to be rational in (Dolgachev and Kondō 2013) via a comparison to a moduli space of cuspidal plane quintics. A natural question is whether or not \(F_{\Co}\) admits a geometrically meaningful, modular compactification \(\overline{F_{\Co}}\), and if so, if the boundary \(\partial \overline{F_{\Co}}\) can be described and classified. Toward this end, we turn to the stable pair compactifications of Kollár, Shepherd-Barron, and Alexeev (Kollár and Shepherd-Barron 1988; Alexeev 1996; Kollár 2023).

Remark 27.2 The search for modular compactifications of moduli spaces is a central problem in algebraic geometry. The prototypical example stems from the work of (Deligne and Mumford 1969; Knudsen and Mumford 1976; Knudsen 1983) on the Deligne-Mumford-Knudsen compactification \(\overline{\Mgn}\) of the moduli space of stable pointed curves \(\Mgn\). By (Mumford 1965; Namikawa 1976; Alexeev and Nakamura 1999; Alexeev 2002), a similarly modular compactification \(\overline{\Ag}\) of the moduli space \(\Ag\) of principally polarized abelian varieties via stable pairs exists, the normalization of which coincides with a particular choice of toroidal (Ash et al. 2010) compactification by the work of (Kollár and Shepherd-Barron 1988) and (Alexeev 1996). By (Piatetski-Shapiro and Shafarevich 1971), the coarse moduli space \(F_{2d}\) of primitively polarized K3 surfaces of degree \(2d\) is isomorphic to an arithmetic quotient \(D_{L_{2d}}/\Gamma_{L_{2d}}\) of a Type IV Hermitian symmetric domain associated to a lattice \(L_{2d}\), where \(D_{L_{2d}}\) is a period domain classifying Hodge structures of K3 type and \(\Gamma_{L_{2d}}\) is an arithmetic subgroup of the orthogonal group \(\Orth(L_{2d})\). By (Baily and Borel 1966), such domains admit a Baily-Borel compactification which is a projective variety. For \(F_{2d}\) and related lattice-polarized moduli spaces \(F_S\) of K3 surfaces, the boundary of the compactification consists of a configuration of 0-cusps (points) and 1-cusps (curves), see (Scattone 1987). See Figure 27.1. By (Ash et al. 2010), there additionally exist infinitely many toroidal compactifications described by a choice of fans at each cuspidal point of the Baily-Borel compactification. The work of (Looijenga 1985) on semitoroidal compactifications simultaneously generalizes the Baily-Borel and toroidal compactifications, retaining the advantage that the boundary can be understood in terms of semifans and studied using toric geometry.

Semitoroidal compactifications are advantageous due to their explicit and combinatorial nature, but do not a priori carry a clear modular interpretation along the boundary strata. Alternatively, letting \((S, R)\) be a pair consisting of a surface and a suitably chosen divisor, the work of (Kollár and Shepherd-Barron 1988; Alexeev 1996; Kollár 2023) yields a compactification obtained by taking the closure of pairs \((S, \varepsilon R)\) in the space of KSBA stable pairs. Although KSBA compactifications admit strong modular interpretations, their boundary strata are generally quite difficult to describe. A recent strategy employed in (Alexeev et al. 2022, 2023, 2025; Alexeev and Engel 2022) is to simultaneously leverage the advantages of both semitoric and KSBA compactifications by finding comparison morphisms between them. This allows the modular boundary of the KSBA compactification to be studied and classified using the combinatorics of toric geometry, lattice theory, and critically, advances in integral affine geometry and mirror symmetry.

The moduli space \(F_\Co\) of Coble surfaces with \(n=1\) boundary components admits a Hodge-theoretic period domain description of the form \(D(T_\Co)/\Gamma_\Co\) where \(T_\Co\) is a fixed lattice and \(\Gamma_\Co \definedas \Orth^+(T_\Co)^*\) is the stable orthogonal group preserving the period domain component (an arithmetic subgroup of finite index in \(\Orth(T_\Co)\)). Its Baily-Borel compactification \(\overline{F_{\Co}}^{\bb}\) contains only one 0-cusp \(p_0\), and thus the combinatorial data of a semitoroidal compactification is determined by a single \(\Gamma_\Co\dash\)invariant semifan associated to a lattice at \(p_0\). A canonical choice one can take is the Coxeter fan, formed by a fundamental domain of the action of the lattice’s Weyl group, along with its reflections; see Figure 27.2. Perhaps more naturally, one can also search for a semifan \(\mcf\) such that the resulting compactification \(\overline{F_\Co}^{\mcf}\) is isomorphic to the KSBA compactification \(\overline{F_\Co}\) for a suitably chosen divisor, and indeed this is what we do in this paper. To this end, we prove the following:

Figure 27.1: Schematic of a Baily–Borel compactification: the shaded arithmetic quotient together with the boundary strata adjoined at its cusps.

Warning 27.3 This schematic is carried over from the earlier draft, where it was drawn but never captioned; the caption above describes only what is drawn. Confirm what the two components and the interior lens are intended to denote.

Figure 27.2: A fundamental polyhedron \(P\) for the Weyl group of a hyperbolic lattice acting on the positive cone \(\mcc\), together with its images under the reflections in the walls of \(P\); the Coxeter fan is the resulting decomposition of \(\mcc\). Such a polyhedron is computed by Vinberg’s algorithm (Vinberg 1975, sec. 1).

Theorem 27.4 There is a semifan \(\mcf\) such that there exists a morphism \[ \Psi: (\overline{F_\Co})^{\nu} \to \overline{F_\Co}^{\mcf} \] from the normalization of the KSBA compactification to the semitoroidal compactification associated with \(\mcf\). The Coxeter fan of \(T_{\Co}\) is a refinement of \(\mcf\), and stable Coble surfaces in the boundary of \(\overline{F_\Co}\) admit explicit descriptions in terms of surfaces associated to sub-Dynkin diagrams of Coxeter diagrams.

Remark 27.5 This result is made possible by recent advances in (Alexeev and Engel 2022; Alexeev et al. 2024) on compactifications of K3 surfaces with nonsymplectic automorphisms, along with the theory of recognizable divisors developed in (Alexeev and Engel 2023). We also critically leverage the related stable pair compactification of the moduli spaces of Enriques surfaces studied in (Alexeev et al. 2025). In particular, we use the folding theory of Coxeter diagrams, their associated integral affine structures, and the theory of \(ADE+BC\) surfaces in order to explicitly describe stable degenerations of Coble surfaces.

27.1 Acknowledgements

I would like to thank my advisor Valery Alexeev for his guidance and support throughout this project. I thank Luca Schaffler and Philip Engel for many useful discussions. I would also like to gratefully acknowledge financial support from the Office of the Graduate School of the University of Georgia and the Research and Training Group in Algebra, Algebraic Geometry, and Number Theory at the University of Georgia.