Coordinate spaces of graphs: approaching interior faces

TL;DR

This paper studies the cell decomposition of the moduli space of real genus two curves, revealing how collapsing graph edges leads to singularities in the mapping to the moduli space.

Contribution

It introduces a graph-based approach to understanding the structure of the moduli space and characterizes the singularities arising from edge collapses.

Findings

Edge collapse causes root-like singularities in the moduli space mapping.

Graph weights encode complex structures on real genus two curves.

The cell decomposition provides insights into the topology of the moduli space.

Abstract

We consider the cell decomposition of the moduli space of real genus two curves with a marked point on the only real oval. The cells are enumerated by certain graphs with their weights describing the complex structure on a curve. We show that collapse of the edge of the graph results in a root like singularity of the natural mapping from the graph weights to the moduli space of curves.