Singularities of Plane Algebraic Curves

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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. 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. en
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

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