Optimal bend-and-break for foliations

TL;DR

This paper establishes that the optimal bend-and-break constant for tangent rational curves on a foliation of rank r is r+1, using advanced methods and AI tools.

Contribution

It proves a precise bound for tangent rational curves in foliations, combining classical and AI-assisted techniques.

Findings

The bend-and-break constant equals r+1 for rank r foliations.

The proof integrates Bogomolov--McQuillan and bend-and-shatter methods.

Generative AI significantly contributed to the proof process.

Abstract

We show that for every foliation $\mathcal{F}$ of rank $r$ on a normal projective variety, the optimal constant in the bend-and-break inequality for tangent rational curves is $r+1$. The proof combines the method of Bogomolov--McQuillan and the bend-and-shatter method developed by Jovinelly--Lehmann--Riedl. The proof of the main result of this paper substantially uses generative AI, particularly the Rethlas system.