On the Geometry of the Moduli Space of Real Binary Octics

TL;DR

This paper explores the geometric structure of the moduli space of real binary octics, revealing that each component admits a hyperbolic orbifold structure with specific reflection groups, but the entire space does not.

Contribution

It demonstrates that each GIT-stable component of the moduli space forms an arithmetic real hyperbolic orbifold with explicitly computed Vinberg diagrams, and proves the whole space isn't hyperbolic.

Findings

Each component admits a hyperbolic orbifold structure.

Monodromy groups are discrete hyperbolic reflection groups.

The entire moduli space cannot be a hyperbolic orbifold.

Abstract

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are, up to commensurability, discrete hyperbolic reflection groups, and their Vinberg diagrams are computed. We conclude with a simple proof that the moduli space of GIT-stable real binary octics itself cannot be a real hyperbolic orbifold.