Hyperbolic Geometry and Moduli of Real Curves of Genus Three

TL;DR

This paper demonstrates that each of the six connected components of the moduli space of smooth real plane quartic curves admits a real hyperbolic structure, linking geometric properties to hyperbolic lattices over Gaussian integers.

Contribution

It establishes a hyperbolic structure on all components of the moduli space and provides explicit descriptions for the maximal real quartic component, including a Coxeter diagram.

Findings

Each component admits a real hyperbolic structure.

The components correspond to real forms of a hyperbolic lattice over Gaussian integers.

Explicit Coxeter diagram for the maximal real quartic component.

Abstract

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to maximal real quartic curves we obtain a more explicit description. We construct a Coxeter diagram that encodes the geometry of this component.