A Theory of Branches for Algebraic Curves

TL;DR

This paper revisits classical methods of algebraic geometry for algebraic curves, modernizing the arguments for rigor and relevance, and connecting to current research in Zariski structures and non-commutative geometry.

Contribution

It provides a rigorous modern reinterpretation of Italian School techniques for algebraic curves, facilitating understanding of singularities and algebraic surfaces.

Findings

Clarifies classical methods with modern rigor

Links algebraic curves to Zariski structures

Supports research in non-commutative geometry

Abstract

This paper develops some of the methods of the "Italian School" of algebraic geometry in the context of infinitesimals. The results of this paper have no claim to originality, they can be found in Severi, we have only made the arguments acceptable by modern standards. However, as the question of rigor was the main criticism of their approach, this is still a useful project. The results are limited to algebraic curves. As well as being interesting in their own right, it is hoped that these may also help the reader to appreciate their sophisticated approach to algebraic surfaces and an understanding of singularities. The constructions are also relevant to current research in Zariski structures, which have played a major role both in model theoretic applications to diophantine geometry and in recent work on non-commutative geometry.