Hyperbolic geometry and moduli of real cubic surfaces

2007/07/06 by Daniel Allcock, Allcock, Daniel, James A. Carlson +3
Mathematics · #14D20 #14P25 #20F55 #22E40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #math.AG #math.GR #msc:14D20 #msc:14P25 #msc:20F55 #msc:22E40

paper · pdf · doi:10.48550/arxiv.0707.1058

Major revision, including several new or completely rewritten sections. 56 pages

arxiv created 2009/05/11 · arxiv updated 2009/12/01

Abstract

Let M₀^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 H⁴ 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 M₀^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 M₀^R is that just mentioned. The corresponding lattice in PO(4,1), for which we find an explicit fundamental domain, is nonarithmetic.

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