Classification of Irreducible Parameterized Plane Curve Singularities

Abstract

The aim of the thesis is the classi cation of parametrized plane curve singularities. We give the classi cation of simple, unimodal and bimodal parametrized plane curve singularities over the real numbers and the complex numbers and a classi cation of simple plane curve singularities over an algebraically closed eld of characteristic p > 0. In rst problem we used the results of Hafez and Hernandez [HH11] to give a di erent proof for the classi cation of simple and unimodal singularities of mappings (C; 0) 􀀀! (C2; 0) with respect to A-equivalence done by Bruce and Ga ney [BJ82] resp. Ishikawa and Janeczko [IJ06] and then give the classi cation bimodal singularities. The results are extended to a classi cation over the real numbers R. In the second task we assume that K is an algebraically closed eld of characteristic p > 0. We give the classi cation of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of parametrized plane curve singularities which are not simple such that almost all parametrized plane curves deform to one of those and show that remaining parametrized plane curve singularities are simple.