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Preprint Number 18

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18. Tristram de Piro
A Theory of Branches for Algebraic Curves

Submission date: 27 September 2006


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 the work of 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.

Mathematics Subject Classification: 03-xx/14-xx

Keywords and phrases : Zariski Structures, Branches, Italian School of Algebraic Geometry, Singularities of Curves.

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