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A hyperbolic structure on the real locus of the moduli space of marked cubic surfaces. (English) Zbl 1062.14045
From the introduction: The moduli space \(M\) of marked cubic surfaces admits a complex hyperbolic structure [D. Allcock, J. Carlson and D. Toledo, C. R. Acad. Sci., Paris, Sér. I, Math. 326, 49–54 (1998; Zbl 0959.32035) and J. Carlson and D. Toledo, Duke Math. J. 97, No. 3, 621–648 (1999; Zbl 0978.14007)]. It enjoys a bi-regular action of the group \(G\cong W(E_6)\), the Weyl group of type \(E_6\), and has a smooth \(G\)-equivariant compactification \({\mathcal C}\) [I. Naruki, Proc. Lond. Math. Soc. III. Ser. 45, 1–30 (1982; Zbl 0508.14005)]. The hypersurface \({\mathcal C}-M\) has two types of irreducible components, the (three-dimensional) Terada model and \(\mathbb{P}^1 \times \mathbb{P}^2\times \mathbb{P}^1\); the components of each type form a \(G\)-orbit. The divisors of the latter type do not intersect with each other and can be blown-down to points; let us denote the resulting variety by \(\overline M\), and the complement of these points in \(\overline M\) by \(\overline M^\diamondsuit\).
On the other hand, consider the principal congruence subgroup \(\Gamma(1-\omega):= \{g\in\Gamma\mid g\equiv I_5\,\text{mod}\,(1-\omega)\} \)/center, \(\omega=\exp 2\pi i/3\) with level \((1-\omega)\), of the modular group \(\Gamma:=\{g\in GL_5 ({\mathcal E}) \mid^t \overline ghg=h\}\)/center, \({\mathcal E}=\mathbb{Z} [\omega]\). These groups act on the complex 4-ball and D. Allcock et al. [loc. cit.] proved the isomorphism \(\overline M^\diamondsuit\cong B^4/\Gamma(1-\omega) \).
Since the varieties in question are defined over \(\mathbb{R}\), we can speak about their real loci, which will be denoted by \(M_\mathbb{R}\), \(\overline M_\mathbb{R}\), and so on. The main result is that the variety \(\overline M_\mathbb{R}\) admits a hyperbolic structure. This is done by constructing an isomorphism \(\overline M_\mathbb{R}^\diamondsuit \cong B_\mathbb{R}^4/ \Lambda (3)\) involving the real 4-ball and the real part \(\Lambda(3)\) of \(\Gamma(1-\omega)\).

14J10 Families, moduli, classification: algebraic theory
14P05 Real algebraic sets
14J26 Rational and ruled surfaces
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