On the Geometry of the Moduli Space of Real Binary Octics

http://dx.doi.org/10.4153/CJM-2011-026-1
Canad. J. Math. 63(2011), 755-797

  Published:2011-04-30
 Printed: Aug 2011

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ 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 $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.