 
Summary: HYPERBOLIC GEOMETRY AND MODULI OF REAL
CUBIC SURFACES
DANIEL ALLCOCK, JAMES A. CARLSON, AND DOMINGO TOLEDO
Abstract. Let MR
0 be the moduli space of smooth real cubic
surfaces. We show that each of its components admits a real hy
perbolic 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 orb
ifold 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 MR
0 . Let MR
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 re
striction to MR
0 is that just mentioned. The corresponding lattice
in PO(4, 1), for which we find an explicit fundamental domain, is
