Boundary complexes of moduli spaces of curves in higher genus

TL;DR

This paper classifies when the intersection properties of boundary divisors in moduli spaces of stable curves of higher genus mirror those in genus zero, extending known results.

Contribution

It provides a complete classification of (g,n) pairs where boundary divisor intersection properties in higher genus moduli spaces behave like in genus zero.

Findings

Identifies all (g,n) pairs with similar intersection properties as genus zero

Extends Giansiracusa's result to higher genus cases

Clarifies the structure of boundary divisor intersections in moduli spaces

Abstract

Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g.