Rational Cuspidal Curves

Abstract

A classical problem in algebraic geometry is the question of how many and what kind of singularities a plane curve of a given degree can have. In this thesis we investigate this problem for rational curves where all the singularities are cusps.

How many and what kind of cusps can a rational cuspidal curve have?

In this thesis we present some of the most recent results on this problem, and we give an overview of rational cuspidal curves. One very important tool in the investigation of rational cuspidal curves is Cremona transformations. We will therefore give a thorough definition of Cremona transformations and use them to construct some rational cuspidal curves of low degree. Moreover, a rational cuspidal curve in the plane can be viewed as a resulting curve of a projection of a curve in a higher-dimensional projective space. This represents a new and interesting way to approach such curves. In this thesis we will therefore also investigate the rational cuspidal curves from this point of view.