Hyperbolic geometry and real moduli of five points on the line

TL;DR

This paper demonstrates that the moduli space of stable real binary quintics can be modeled as a complete hyperbolic orbifold, linking algebraic geometry with hyperbolic geometry through arithmetic quotients.

Contribution

It establishes a geometric structure on the moduli space of real binary quintics using hyperbolic geometry, revealing its orbifold nature and explicit quotient description.

Findings

Moduli space is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane.

The metric extends to a complete hyperbolic orbifold structure.

The space is a quotient by a non-arithmetic triangle group.

Abstract

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the moduli space of stable real binary quintics. This turns the moduli space of stable real binary quintics into the quotient of the real hyperbolic plane by the non-arithmetic triangle group of angles $\pi/3, \pi/5$ and $\pi/10$.