Enumerative and asymptotic analysis of a moduli space

TL;DR

This paper analyzes the combinatorial structure of the Hilbert series of the cohomology ring of moduli spaces of genus zero curves, deriving asymptotic formulas and exact values for its coefficients using integral operator identities.

Contribution

It introduces an integral operator identity for the Hilbert series and explores its asymptotic behavior, connecting algebraic geometry with combinatorics and classical enumeration problems.

Findings

Derived asymptotic behavior of Hilbert series coefficients.

Obtained exact values for certain coefficients.

Connected the total dimension's asymptotics to the Lambert W function.

Abstract

We focus on combinatorial aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. We show its graded Hilbert series satisfies an integral operator identity. This is used to give asymptotic behavior, and in some cases, exact values, of the coefficients themselves. We then study the total dimension, that is, the sum of the coefficients of the Hilbert series. Its asymptotic behavior involves the Lambert W function, which has applications to classical tree enumeration, signal processing and fluid mechanics.