Plane Algebroid Curves in Arbitrary Characteristic

Abstract

The subject of this Dissertation is the study of germs of plane curves defined over arbitrary algebraically closed fields. Classically, this was performed over the field of complex numbers, by using as a main tool the Newton-Puiseux parametrization, related to the normalization of the curve. The theory was then adapted to arbitrary algebraically closed field using the so-called Hamburger Noether expansions that take track of the entire desingularization process of the curve. In this work, we will use, instead, the notion of contact order among irreducible curves by means of the logarithmic distance introduced by J. Chadzynski and A. Ploski in [CP]. This attack works in arbitrary characteristic and avoids the use of the Hamburger-Noether expansions, making proofs simpler and more elegant. The content of this dissertation is as follows: In Chapter 1, we introduce the notion of algebroid plane curves, their normalization and their intersection theory. We used as a reference for this part the book of A. Seidenberg [Sei] and the survey of A. Hefez [He]. In Chapter 2 and 3, we introduce the notion of semigroup of values of an irreducible plane curve and make a detailed study of their properties, introducing at the end the important notion of Key-polynomials, showing that they are nothing else but some special Apéry polynomials. This part is based on [He] and personal notes of this author. In Chapter 4, we introduce the contact order among irreducible plane curves and study its properties, applying them to deduce some results about irreducible plane curves that have high contact order. The whole theory is used to deduce Merle’s and Granja’s theorems [Me] and [Gr] over arbitrary algebraically closed fields. To conclude the work we present a result due to E. Garcia Barroso and A. Ploski about the relation among the Milnor number of an irreducible power series and the conductor of its semigroup of values. In this part, we used the works of E. Garcia Barroso and A.Ploski[GB-P1]and[GB-P2].