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# On the Geometry of the Moduli Space of Real Binary Octics

Published:2011-04-30
Printed: Aug 2011
• Kenneth C. K. Chu,
Department of Mathematics, University of Utah, Salt Lake City, Utah, USA
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## Abstract

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.
 Keywords: real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm
 MSC Classifications: 32G13 - Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15] 32G20 - Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] 14D05 - Structure of families (Picard-Lefschetz, monodromy, etc.) 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}