A multiplicidade de interseção de curvas planas singulares

Abstract

This work’s main objective is to present the notion of multiplicity of intersection between two singular plane curves. For that, we consider the ring of formal power series. We present the Weierstrass Preparation Theorem and some of its useful consequences for the development of the work. From this result, we can associate formal power series with pseudopolynomials, which helps us in studies with these series. Furthermore, we introduce the Puiseux parameterization, which is a useful tool when dealing with intersection multiplicity. When talking about the multiplicity of intersection between two curves, we cover its definition, properties and some examples. We also present results that show different ways of obtaining this multiplicity. Subsequently, we discuss the resolution of singularities by Blow-up. We define Blow-up and present examples where we can visualize the resolution of singularities process. Finally, we enunciate and prove Noether’s Formula, which relates the multiplicity of intersections of curves with the multiplicity of intersections of their Blow-ups.