A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki

TL;DR

This paper presents a new proof of two classical theorems about algebraic curves and embeddings in the complex affine plane using minimal model theory and properties of $ ext{Q}$-acyclic surfaces.

Contribution

It introduces a novel proof approach for the Lin-Zaidenberg and Abhyankar-Moh-Suzuki theorems, connecting them through minimal models and $ ext{Q}$-acyclic surface properties.

Findings

Proof of Lin-Zaidenberg theorem via minimal models

Alternative proof of Abhyankar-Moh-Suzuki theorem from $ ext{Q}$-acyclic surfaces

Unified approach linking algebraic curves and surface properties

Abstract

Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin-Zaidenberg which says that every topologically contractible algebraic curve in the complex affine plane has equation $X^n=Y^m$ in some algebraic coordinates on the plane. This gives also a proof of the theorem of Abhyankar-Moh-Suzuki concerning embeddings of the complex line into the plane. Independently, we show how to deduce the latter theorem from basic properties of $\mathbb{Q}$-acyclic surfaces.