Extremal descendant integrals on moduli spaces of curves: An inequality discovered and proved in collaboration with AI

TL;DR

This paper identifies extremal values of pure ψ-class intersection numbers on moduli spaces of curves, using nefness and log-concavity, and highlights collaboration with AI in mathematical proof discovery and documentation.

Contribution

It determines which intersection numbers are extremal on moduli spaces and demonstrates AI-assisted proof methods and transparent authorship documentation.

Findings

Minimal intersection numbers occur for powers of a single ψ-class.

Maximal intersection numbers occur for balanced vectors of exponents.

AI models contributed significantly to proof discovery and drafting.

Abstract

For the pure $\psi$-class intersection numbers $D(\textbf{e})=\langle \tau_{e_1} \cdots \tau_{e_n} \rangle_g$ on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves, we determine for which choices of $\textbf{e}=(e_1, \ldots, e_n)$ the value of $D(\textbf{e})$ becomes extremal. The intersection number is minimal for powers of a single $\psi$-class (i.e. all $e_i$ but one vanish), whereas maximal values are obtained for balanced vectors ($|e_i - e_j| \leq 1$ for all $i,j$). The proof uses the nefness of the $\psi$-classes combined with Khovanskii--Teissier log-concavity. Apart from the mathematical content, this paper is also meant as an experiment in collaborations between human mathematicians and AI models: the proof of the above result was found and formulated by the AI models GPT-5 and Gemini 3 Pro. Large parts of the paper were drafted by Claude Opus 4.5, and a part of the argument was formalized in Lean with the help of Claude Code and GPT-5.2. The paper aims for maximal transparency on the authorship of different sections and the employed AI tools (including prompts and conversation logs).