CW-Decompositions of Plane Algebraic Curves and Milnor Fibers of Non-Isolated Quasi-Ordinary Singularities
This is a brief abstract that outlines the topics and contents of this work. The reader interested in a more detailed overview can skip directly to the introduction. The braid monodromy is an invariant of algebraic curves that encodes strong information about their topology. Let C be an affine algeb...
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| Tipo de recurso: | tesis doctoral |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/11037 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/11037 |
| Access Level: | acceso abierto |
| Palabra clave: | 514.142(043.2) Algebraic geometry Affine algebraic Geometría algebraica Geometría afín Geometria algebraica 1201.01 Geometría Algebraica |
| Sumario: | This is a brief abstract that outlines the topics and contents of this work. The reader interested in a more detailed overview can skip directly to the introduction. The braid monodromy is an invariant of algebraic curves that encodes strong information about their topology. Let C be an affine algebraic plane curve, defined by a polynomial function f, and having a generic projection on the x axis of C². The braid monodromy of C can be presented as a homomorphism ρ : π1(C\{x1, . . . ,xₘ}) → βₙ, where x1, . . . ,xₘ are the values of x on which f(x, y) have multiple roots, and βₙ denotes the braid group of n strands. If we see the curve as the image of a multivalued function g, the image under ρ of a given loop is determined by the paths in C² that (x, g(x)) follows when x runs along the loop. The braid monodromy has a long story and its development and applications has passed through the works of Zariski ([44, 45]), van Kampen ([16]), Moishezon and Teicher ([26, 27, 28, 29, 30, 31]), and Carmona ([9]) among many others ([11, 10, 19, 37, 2, 18, 3]). A result by Carmona ([9]) shows that the braid monodromy of a curve C determines the topology of the pair (P², C). He also provided a program that calculates the braid monodromy of a curve from its equation. However, it remained an open problem to find what this topology actually is. This is, given the braid monodromy of C, to find a description for the topology of (C², C) or (P², C). In this work we provide such a presentation for the affine case. It consists of a regular CW decomposition of the pair (D,C∩D), where D is a large enough polydisc in C². The construction uses the presentation of the braid monodromy in the form of local braids and conjugating braids. In this presentation the local braids must be given as an ordered set of independent sub-braids, associated with different preimages of a critical value of a generic projection. The main theorem concerning the algebraic curves states the good definition of this decomposition (Theorem 1.18)... |
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