PLANE ALGEBROID CURVES IN ARBITRARY CHARACTERISTIC

Milnor number in positive characteristic

Singularities of algebroid curves

Geometria algébrica

Curva algébrica

Singularidades (Matemática)

Garcia, Mahalia Almeida | Posted on:
2016

##### 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]

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##### Document type

Dissertação##### Subject(s)

Singularities in positive characteristicMilnor number in positive characteristic

Singularities of algebroid curves

Geometria algébrica

Curva algébrica

Singularidades (Matemática)

##### License Term

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