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

Johannes Schmitt

Paper

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

Abstract

For the pure

-class intersection numbers

on the moduli space

of stable curves, we determine for which choices of

the value of

becomes extremal. The intersection number is minimal for powers of a single

-class (i.e. all

but one vanish), whereas maximal values are obtained for balanced vectors (

for all

). The proof uses the nefness of the

-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).