On the Geometry of the Moduli Space of Real Binary Octics

Abstract

Abstract The moduli space of smooth real binary octics has five connected components. They para- metrize the real binary octics whose defining equations have 0, … , 4 complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of 5- dimensional real hyperbolic space by the action of an arithmetic subgroup of Isom. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.