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Real cubic surfaces and real hyperbolic geometry. (English) Zbl 1055.14057
In their previous paper [J. Algebraic Geom. 11, No. 4, 659–724 (2002; Zbl 1080.14532)], the authors introduced a hyperbolic structure on the moduli space of complex cubic surfaces. In the present paper, the authors describe the moduli space of stable (i.e., smooth or nodal) real cubic surfaces as a quotient of the real hyperbolic four-space by a discrete non-arithmetic group. On the other hand, it can be built out of the five components of the moduli space of smooth cubic surfaces, glued along the facets, corresponding to nodal surfaces. Here, each component appears to be a quotient of the hyperbolic four-space by some arithmetic group. Computing Euler characteristics, the authors calculate volumes of all these components of the moduli space in the metric of constant curvature \(-1\), as well as the total volume equal to \(37\pi^2/1080\).

14P05 Real algebraic sets
14J10 Families, moduli, classification: algebraic theory
Full Text: DOI
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