# A symplectic dynamics proof of the degree-genus formula

## TL;DR

This paper provides a symplectic dynamics proof of the degree-genus formula for complex projective curves, utilizing global surfaces of section and an elementary degeneration process inspired by symplectic field theory.

## Contribution

It introduces a novel symplectic dynamics approach to prove the degree-genus formula, connecting Reeb flows, global surfaces of section, and holomorphic degeneration techniques.

## Key findings

- Classification of global surfaces of section for Reeb flows on the 3-sphere
- Elementary degeneration process for complex projective curves
- Proof of the degree-genus formula using symplectic methods

## Abstract

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.