Geometry of plane curves via Tschirnhausen resolution tower

Abstract

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at infinity. For instance, we discribe the singularities at infinity (up to equisingular deformation) for curves of genus 0,1 and 2. From this follows an extension of the Abhyankar-Moh-Suzuki theorem to genus 1 and 2.