Hyperbolic geometry and moduli of real cubic surfaces

TL;DR

This paper demonstrates that each component of the moduli space of smooth real cubic surfaces admits a hyperbolic structure, constructed via quotients of hyperbolic space by arithmetic groups, and explores related topological properties.

Contribution

It establishes hyperbolic structures on all components of the moduli space of smooth real cubic surfaces and describes the associated lattices and topological features.

Findings

Each of the five components admits a hyperbolic orbifold structure.

Explicit lattices in PO(4,1) are described for each component.

The moduli space of stable real cubic surfaces also admits a hyperbolic structure with a nonarithmetic lattice.

Abstract

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.