# Singularities of Plane Algebraic Curves

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http://urn.fi/URN:NBN:fi:hulib-202109293765
 Julkaisun nimi: Singularities of Plane Algebraic Curves Tekijä: Härkönen, Robert Mattias Muu tekijä: Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta University of Helsinki, Faculty of Science Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten Julkaisija: Helsingin yliopisto Päiväys: 2021 Kieli: eng URI: http://urn.fi/URN:NBN:fi:hulib-202109293765 http://hdl.handle.net/10138/334694 Opinnäytteen taso: pro gradu -tutkielmat Koulutusohjelma: Matematiikan ja tilastotieteen maisteriohjelma Master's Programme in Mathematics and Statistics Magisterprogrammet i matematik och statistik Opintosuunta: Matematiikka Mathematics Matematik Tiivistelmä: 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. Avainsanat: singularity plane curve Puiseux torus knot
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