# Singularities of Plane Algebraic Curves

 dc.contributor Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta fi dc.contributor University of Helsinki, Faculty of Science en dc.contributor Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten sv dc.contributor.author Härkönen, Robert Mattias dc.date.issued 2021 dc.identifier.uri URN:NBN:fi:hulib-202109293765 dc.identifier.uri http://hdl.handle.net/10138/334694 dc.description.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. en 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. dc.language.iso eng dc.publisher Helsingin yliopisto fi dc.publisher University of Helsinki en dc.publisher Helsingfors universitet sv dc.subject singularity dc.subject plane curve dc.subject Puiseux dc.subject torus knot dc.title Singularities of Plane Algebraic Curves en dc.type.ontasot pro gradu -tutkielmat fi dc.type.ontasot master's thesis en dc.type.ontasot pro gradu-avhandlingar sv dct.identifier.urn URN:NBN:fi:hulib-202109293765 dc.subject.specialization Matematiikka fi dc.subject.specialization Mathematics en dc.subject.specialization Matematik sv dc.subject.degreeprogram Matematiikan ja tilastotieteen maisteriohjelma fi dc.subject.degreeprogram Master's Programme in Mathematics and Statistics en dc.subject.degreeprogram Magisterprogrammet i matematik och statistik sv