Stable curves and chromatic polynomials

TL;DR

This paper introduces a new class of intersection numbers on moduli spaces of stable curves, linking them to graph theory and chromatic polynomials, with proofs using algebraic geometry and hyperplane arrangements.

Contribution

It establishes a closed-form formula connecting intersection numbers to chromatic polynomials, providing two different proofs and exploring related combinatorial questions.

Findings

Derived a simple closed formula for intersection numbers in terms of chromatic polynomials.

Provided two proofs: one via intersection theory and another via hyperplane arrangements.

Discussed new conjectures related to chromatic polynomials of directed graphs.

Abstract

The intersection numbers of moduli spaces of stable curves $\overline{\mathcal{M}}_{g,m}$ are well-studied and are known to have rich combinatorial structure. We introduce a natural class of these intersection numbers $\omega_{G,g,m}$ indexed by finite simple graphs $G=(V,E)$. In genus zero, these numbers are closely related to several previously-studied quantities, including maximum likelihood degrees in algebraic statistics, counts of regions of certain hyperplane arrangements, and Kapranov degrees. We give two proofs of a simple closed formula $\omega_{G,g,m}=(-1)^{\left\lvert V \right\rvert}\chi_G(-(2g-2+m)),$ where $\chi_G$ is the chromatic polynomial of $G$ -- one proof via intersection theory on moduli spaces of stable curves, and the other using the theory of hyperplane arrangements. We discuss several related questions and speculations, including new candidates for the chromatic polynomial of a directed graph.