Plane Algebroid Curves in Arbitrary Characteristic
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Tipo de documento: | Dissertação |
Idioma: | eng |
Título da fonte: | Repositório Institucional da Universidade Federal Fluminense (RIUFF) |
Texto Completo: | https://app.uff.br/riuff/handle/1/9233 |
Resumo: | 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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Plane Algebroid Curves in Arbitrary CharacteristicCurvas Algebróides Planas em Característica ArbitráriaSingularities in positive characteristicMilnor number in positive characteristicSingularities of algebroid curvesGeometria algébricaCurva algébricaSingularidades (Matemática)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]O assunto dessa dissertação é o estudo dos germes de curvas planas definidas sobre corpos algebricamente fechados arbitrários. Classicamente tal estudo era realizado sobre o corpo dos números complexos, utilizando-se como principal ferramenta para isso as parametrizações de Newton-Puiseux, relacionadas com a normalização da curva. Em seguida, a teoria foi adaptada para corpos algebricamente fechados arbitrários, utilizando-se as chamadas expansões de Hamburger-Noether que levam em conta todo o processo de resolução da singularidade da curva. Neste trabalho, usaremos ao invés a noção de contato entre curvas irredutíveis por meio da distância logarítmica introduzida por J. Chadzynski e A. Ploski em [CP]. Essa abordagem funciona em característica arbitraria e evita o uso das expansões de Hamburger-Noether, tornando as demonstrações mais simples e elegantes. O conteúdo dessa dissertação é o seguinte: No Capítulo 1, introduzimos a noção de curvas algebróides planas, suas normalizações e a sua teoria de interseção. Usamos nessa parte como referência o livro de A. Seidenberg [Sei] e o survey de A. Hefez [He]. Nos Capítulos 2 e 3, introduzimos a noção de semigrupo de valores de uma curva plana irredutível e empreendemos um estudo detalhado de suas propriedades, introduzindo no final a importante noção de polinômios-chave, mostrando que não são nada além de polinômios de Apéry particulares. Nessa parte, baseamo-nos em [He] e em notas pessoais desse autor. No Capítulo 4, introduzimos a ordem de contato entre curvas irredutíveis planas e estudamos as suas propriedades, utilizando-as para deduzir alguns resultados sobre curvas irredutíveis que possuem ordem de contato alta. Toda essa teoria e utilizada para deduzir os teoremas de Merle e de Granja, contidos em [Me] e [Gr], sobre corpos algebricamente fechados arbitrários. Para concluir o trabalho, apresentamos um resultado recente devido a E. Garcia Barroso e A. Ploski sobre a relação entre o número de Milnor de uma série irredutível e o condutor de seu semigrupo de valores. Nessa parte, utilizamos os trabalhos de E. Garcia Barroso e A. Ploski [GB-P1] e [GB-P2] E. Garcia Barroso e A. Ploski [GB-P1] e [GB-P2]51 f.Hefez, AbramoGarcia, Mahalia Almeida2019-04-24T14:55:27Z2019-04-24T14:55:27Z2016info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttps://app.uff.br/riuff/handle/1/9233Aluno de MestradoopenAccesshttp://creativecommons.org/licenses/by-nc-nd/3.0/br/CC-BY-SAinfo:eu-repo/semantics/openAccessengreponame:Repositório Institucional da Universidade Federal Fluminense (RIUFF)instname:Universidade Federal Fluminense (UFF)instacron:UFF2022-11-29T23:05:14Zoai:app.uff.br:1/9233Repositório InstitucionalPUBhttps://app.uff.br/oai/requestriuff@id.uff.bropendoar:21202024-08-19T10:45:45.777843Repositório Institucional da Universidade Federal Fluminense (RIUFF) - Universidade Federal Fluminense (UFF)false |
dc.title.none.fl_str_mv |
Plane Algebroid Curves in Arbitrary Characteristic Curvas Algebróides Planas em Característica Arbitrária |
title |
Plane Algebroid Curves in Arbitrary Characteristic |
spellingShingle |
Plane Algebroid Curves in Arbitrary Characteristic Garcia, Mahalia Almeida Singularities in positive characteristic Milnor number in positive characteristic Singularities of algebroid curves Geometria algébrica Curva algébrica Singularidades (Matemática) |
title_short |
Plane Algebroid Curves in Arbitrary Characteristic |
title_full |
Plane Algebroid Curves in Arbitrary Characteristic |
title_fullStr |
Plane Algebroid Curves in Arbitrary Characteristic |
title_full_unstemmed |
Plane Algebroid Curves in Arbitrary Characteristic |
title_sort |
Plane Algebroid Curves in Arbitrary Characteristic |
author |
Garcia, Mahalia Almeida |
author_facet |
Garcia, Mahalia Almeida |
author_role |
author |
dc.contributor.none.fl_str_mv |
Hefez, Abramo |
dc.contributor.author.fl_str_mv |
Garcia, Mahalia Almeida |
dc.subject.por.fl_str_mv |
Singularities in positive characteristic Milnor number in positive characteristic Singularities of algebroid curves Geometria algébrica Curva algébrica Singularidades (Matemática) |
topic |
Singularities in positive characteristic Milnor number in positive characteristic Singularities of algebroid curves Geometria algébrica Curva algébrica Singularidades (Matemática) |
description |
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] |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 2019-04-24T14:55:27Z 2019-04-24T14:55:27Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://app.uff.br/riuff/handle/1/9233 Aluno de Mestrado |
url |
https://app.uff.br/riuff/handle/1/9233 |
identifier_str_mv |
Aluno de Mestrado |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/br/ CC-BY-SA info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/br/ CC-BY-SA |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da Universidade Federal Fluminense (RIUFF) instname:Universidade Federal Fluminense (UFF) instacron:UFF |
instname_str |
Universidade Federal Fluminense (UFF) |
instacron_str |
UFF |
institution |
UFF |
reponame_str |
Repositório Institucional da Universidade Federal Fluminense (RIUFF) |
collection |
Repositório Institucional da Universidade Federal Fluminense (RIUFF) |
repository.name.fl_str_mv |
Repositório Institucional da Universidade Federal Fluminense (RIUFF) - Universidade Federal Fluminense (UFF) |
repository.mail.fl_str_mv |
riuff@id.uff.br |
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