Singularities of Plane Algebraic Curves

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http://urn.fi/URN:NBN:fi:hulib-202109293765
Title: Singularities of Plane Algebraic Curves
Author: Härkönen, Robert Mattias
Other contributor: Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta
University of Helsinki, Faculty of Science
Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten
Publisher: Helsingin yliopisto
Date: 2021
Language: eng
URI: http://urn.fi/URN:NBN:fi:hulib-202109293765
http://hdl.handle.net/10138/334694
Thesis level: master's thesis
Degree program: Matematiikan ja tilastotieteen maisteriohjelma
Master's Programme in Mathematics and Statistics
Magisterprogrammet i matematik och statistik
Specialisation: Matematiikka
Mathematics
Matematik
Abstract: Plane algebraic curves are defined as zeroes of polynomials in two variables over some given field. If a point on a plane algebraic curve has a unique tangent line passing through it, the point is called simple. Otherwise, it is a singular point or a singularity. Singular points exhibit very different algebraic and topological properties, and the objective of this thesis is to study these properties using methods of commutative algebra, complex analysis and topology. In chapter 2, some preliminaries from classical algebraic geometry are given, and plane algebraic curves and their singularities are formally defined. Curves and their points are linked to corresponding coordinate rings and local rings. It is shown that a point is simple if and only if its corresponding local ring is a discrete valuation ring. In chapter 3, the Newton-Puiseux algorithm is introduced. The algorithm outputs fractional power series known as Puiseux expansions, which are shown to produce parametrizations of the local branches of a curve around a singular point. In chapter 4, Puiseux expansions are used to study the topology of complex plane algebraic curves. Around singularities, curves are shown to have an iterated torus knot structure which is, up to homotopy, determined by invariants known as Puiseux pairs.
Subject: singularity
plane curve
Puiseux
torus knot


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