Algebraic Geometry/Group Theory
Real cubic surfaces and real hyperbolic geometry
Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 185-188.

The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π2/1080 in the metric of constant curvature −1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components.

L'espace des modules des surfaces cubiques stables et réelles est le quotient de l'espace hyperbolique réel de dimension quatre par un groupe non-arithmétique discret. Le volume de l'espace des modules est 37π2/1080 dans la métrique de courbure constante −1. Chacune des composantes connexes de l'espace des modules peut être décrite comme le quotient de l'espace hyperbolique réel de dimension quatre par un groupe arithmétique spécifique. Nous calculons le volume des composantes.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00287-5

Daniel Allcock 1; James A. Carlson 2; Domingo Toledo 2

1 Department of Mathematics, University of Texas at Austin, University Station C1200, Austin, TX 78712, USA
2 Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA
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Daniel Allcock; James A. Carlson; Domingo Toledo. Real cubic surfaces and real hyperbolic geometry. Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 185-188. doi : 10.1016/S1631-073X(03)00287-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00287-5/

[1] D. Allcock; J. Carlson; D. Toledo Complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris, Ser. I, Volume 326 (1998), pp. 49-54 | arXiv

[2] D. Allcock; J. Carlson; D. Toledo The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom., Volume 11 (2002), pp. 659-724 | arXiv

[3] P. Deligne; G.D. Mostow Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHES, Volume 63 (1986), pp. 5-89

[4] M. Gromov; I. Piatetski-Shapiro Non-arithmetic groups in Lobachevsky space, Publ. Math. IHES, Volume 66 (1988), pp. 93-103

[5] B. Segre The Non-Singular Cubic Surfaces, Oxford, 1942

[6] E.B. Vinberg Some arithmetical discrete groups in Lobacevskii spaces, Discrete Subgroups of Lie Groups and Applications to Moduli, Oxford, 1975, pp. 328-348

[7] M. Yoshida A hyperbolic structure on the real locus of the moduli space of marked cubic surfaces, Topology, Volume 40 (2001), pp. 469-473

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