Lins Neto's examples of foliations and the Mori cone of blow-ups of $P^2$

TL;DR

This paper uses algebraic foliations introduced by Lins Neto to provide new evidence supporting a conjecture about the Mori cone of blow-ups of the projective plane, which relates to longstanding conjectures in algebraic geometry.

Contribution

It introduces a novel approach using Lins Neto's foliations to gather evidence for a conjecture on the Mori cone, connecting it to Nagata's conjecture.

Findings

Provides new evidence supporting the conjecture about the Mori cone

Links the conjecture to the Harbourne-Hirschowitz and Nagata conjectures

Suggests a potential pathway to prove the conjecture using foliations

Abstract

We use a family of algebraic foliations given by A. Lins Neto to provide new evidences to a conjecture, related to the Harbourne-Hirschowitz's one and implying the Nagata's conjecture, which concerns the structure of the Mori cone of blow-ups of $\mathbb{P}^2$ at very general points.