Isolated hypersurface singularity

Summary

An isolated hypersurface singularity is the germ at a point of the zero set of a holomorphic function $f$ of $n + 1$ variables, see the definition. The gradient of $f$ vanishes at the point but nowhere else nearby. It is the basic object of the local theory of singularities of complex spaces. Its topology is described by the Milnor fibration, the link and the Milnor number. It is a special kind of singular point of a complex analytic variety.