Surfaces of section for Seifert fibrations

TL;DR

This paper classifies surfaces of section for flows on 3-manifolds with Seifert fibrations, exploring their relationships via branched coverings and algebraic curves in weighted projective planes.

Contribution

It provides a comprehensive classification of surfaces of section in Seifert fibrations and links them to algebraic geometry through branched coverings and complex projective planes.

Findings

Classification of global surfaces of section for Seifert fibrations

Connections between surfaces of section and algebraic curves in weighted projective planes

Relationships via branched coverings between different Seifert fibrations

Abstract

We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings -- one way or the other -- between surfaces of section for the Hopf flow and those for any other Seifert fibration of the 3-sphere, and we relate these surfaces of section to algebraic curves in weighted complex projective planes.