A conjectura de Zariski para a multiplicidade

Abstract

In his retiring Presidential address to the American Mathematical Society in 1971, Zariski proposed some questions in the Theory of Singularities. One of them concerns the topological invariance of the multiplicity of complex hypersurfaces. In more accurate terms, Zariski asked: if two complex hypersurfaces are homeomorphic as embedded varieties, then are their multiplicities at the origin the same? The multiplicity of a complex hypersurface at the origin is the number of points of intersection of the hypersurface with a generic complex line passing close to the origin, but not through it. The problem still remains unsolved. However, there are some special cases which were answered affirmatively, such as the case of homeomorphic hypersurfaces by a bilipschitz homeomorphism. This work aims at understanding the main results settled for the problem. In the present dissertation, we will make a precise concept of multiplicity of a complex hypersurface and we will give special emphasis to C1-invariance of the multiplicity, bilipschitz invariance and quasihomogeneous hypersurfaces. Besides having great importance by themselves, these cases bring their own interpretations of multiplicity helping us to understand better such an object.