Description
We study complex surfaces in the complex 3-space locally, in a sufficiently small neighborhood of a singular point of the surface. Our surfaces can be given by a parametrization, whose Jacobian matrix is injective off the origin, and can be given by an equation as well. The concept of the investigation is the study of three different types of 'nearby objects'. The first one is a stable deformation of the parametrization, it takes the singular point to stable pieces, which are complex Whitney umbrellas and triple points. The second one is the associated immersion from the 3-sphere to the 5-sphere, which can be obtained by a restriction of the parametrization. A small modification of the equation provides a non-singular complex surface, which is the so-called Milnor fibre of the singularity, this is the third nearby object. We will compare some invariants of the stabilization and the associated immersion, and construct the boundary of the Milnor fibre, with many examples and illustrations.