Generating stable modular graphs

TL;DR

This paper introduces the Boundary program that systematically generates all stable graphs of a given genus and number of marked points, aiding the study of the moduli space of stable curves.

Contribution

The paper presents a new computational tool for generating all stable graphs of specified genus and marked points, facilitating topological analysis of moduli spaces.

Findings

Successfully generates all stable graphs for given parameters

Enables detailed stratification of moduli space boundary

Provides accessible source code for the community

Abstract

We present the program Boundary, whose source files are available at http://people.sissa.it/~maggiolo/boundary/. Given two natural numbers g and n satisfying 2g+n-2>0, the program generates all genus g stable graphs with n unordered marked points. Each such graph determines the topological type of a nodal stable curve of arithmetic genus g with n unordered marked points. Our motivation comes from the fact that the boundary of the moduli space of stable genus g, n-pointed curves can be stratified by taking loci of curves of a fixed topological type.