Hyperbolic geometry and moduli of real cubic surfaces

Daniel Allcock; James A. Carlson; Domingo Toledo

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 1, page 69-115
  • ISSN: 0012-9593

Abstract

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Let ℳ 0 ℝ 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 ℳ 0 ℝ . Let ℳ s ℝ 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 ℳ 0 ℝ is that just mentioned. The corresponding lattice in PO ( 4 , 1 ) , for which we find an explicit fundamental domain, is nonarithmetic.

How to cite

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Allcock, Daniel, Carlson, James A., and Toledo, Domingo. "Hyperbolic geometry and moduli of real cubic surfaces." Annales scientifiques de l'École Normale Supérieure 43.1 (2010): 69-115. <http://eudml.org/doc/272149>.

@article{Allcock2010,
abstract = {Let $\mathcal \{M\}_0^\mathbb \{R\}$ 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 $\{\rm PO\}(4,1)$. We also derive several new and several old results on the topology of $\mathcal \{M\}_0^\mathbb \{R\}$. Let $\mathcal \{M\}_s^\mathbb \{R\}$ 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 $\mathcal \{M\}_0^\mathbb \{R\}$ is that just mentioned. The corresponding lattice in $\{\rm PO\}(4,1)$, for which we find an explicit fundamental domain, is nonarithmetic.},
author = {Allcock, Daniel, Carlson, James A., Toledo, Domingo},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {cubic surface; moduli; real agebraic geometry; hyperbolic geometry; arithmetic groups; Coxeter groups; uniformization},
language = {eng},
number = {1},
pages = {69-115},
publisher = {Société mathématique de France},
title = {Hyperbolic geometry and moduli of real cubic surfaces},
url = {http://eudml.org/doc/272149},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Allcock, Daniel
AU - Carlson, James A.
AU - Toledo, Domingo
TI - Hyperbolic geometry and moduli of real cubic surfaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 1
SP - 69
EP - 115
AB - Let $\mathcal {M}_0^\mathbb {R}$ 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 ${\rm PO}(4,1)$. We also derive several new and several old results on the topology of $\mathcal {M}_0^\mathbb {R}$. Let $\mathcal {M}_s^\mathbb {R}$ 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 $\mathcal {M}_0^\mathbb {R}$ is that just mentioned. The corresponding lattice in ${\rm PO}(4,1)$, for which we find an explicit fundamental domain, is nonarithmetic.
LA - eng
KW - cubic surface; moduli; real agebraic geometry; hyperbolic geometry; arithmetic groups; Coxeter groups; uniformization
UR - http://eudml.org/doc/272149
ER -

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