Hyperbolic geometry and moduli of real cubic surfaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, p. 69-115

Let 0 be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H 4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO (4,1). We also derive several new and several old results on the topology of 0 . Let s be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to 0 is that just mentioned. The corresponding lattice in PO (4,1), for which we find an explicit fundamental domain, is nonarithmetic.

On note 0 l’espace des modules des surfaces cubiques réelles lisses. Nous montrons que chacune de ses composantes admet une structure hyperbolique réelle. Plus précisément, en enlevant de l’espace hyperbolique réel H 4 certaines sous-variétés totalement géodésiques de dimension inférieure, puis en prenant le quotient par un groupe arithmétique, on obtient une orbifold isomorphe à une composante de l’espace des modules. Il y a cinq composantes. Nous décrivons le réseau de PO(4,1) qui correspond à chacune d’entre elles. Nous démontrons également quelques résultats sur la topologie de 0 , dont certains sont nouveaux. On note s l’espace des modules des surfaces cubiques réelles qui sont stables au sens de la théorie géométrique des invariants. Nous montrons que cet espace admet une structure hyperbolique dont la restriction à 0 est celle évoquée ci-dessus. Nous décrivons un domaine fondamental pour le réseau correspondant de PO(4,1), qui s’avère être non arithmétique.

DOI : https://doi.org/10.24033/asens.2116
Classification:  14J15,  14P99,  20F55,  22E40
Keywords: cubic surface, moduli, real agebraic geometry, hyperbolic geometry, arithmetic groups, Coxeter groups, uniformization
@article{ASENS_2010_4_43_1_69_0,
     author = {Allcock, Daniel and Carlson, James and Toledo, Domingo},
     title = {Hyperbolic geometry and moduli of real cubic surfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {1},
     year = {2010},
     pages = {69-115},
     doi = {10.24033/asens.2116},
     zbl = {1187.14043},
     mrnumber = {2583265},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2010_4_43_1_69_0}
}
Allcock, Daniel; Carlson, James A.; Toledo, Domingo. Hyperbolic geometry and moduli of real cubic surfaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, pp. 69-115. doi : 10.24033/asens.2116. http://www.numdam.org/item/ASENS_2010_4_43_1_69_0/

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