On the density of foliations without algebraic solutions on weighted projective planes

TL;DR

This paper proves that generic holomorphic foliations on weighted projective planes and Hirzebruch surfaces lack algebraic solutions when their degree is sufficiently large.

Contribution

It establishes the non-existence of algebraic solutions for high-degree foliations on these complex surfaces, extending previous results to weighted projective planes and Hirzebruch surfaces.

Findings

Generic foliations have no algebraic solutions at high degrees.

Results apply to weighted projective planes and Hirzebruch surfaces.

Provides bounds on degree for non-existence of algebraic solutions.

Abstract

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.