---
tag: VTGYEES3VS
kind: concept
name: Isolated hypersurface singularity
msc: [32S25, 32S05, 14B05]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_map
---

## 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|4KAWK0FD4V]].
The gradient of $f$ vanishes at the point but nowhere else nearby.
It is the basic object of the local theory of [[singularities|42V5XGZS4F]] of complex spaces.
Its topology is described by the [[Milnor fibration|KTFG1S2SMG]], the [[link|C831AN3RFM]] and the [[Milnor number|1ZE51P7GRQ]].
It is a special kind of [[singular point|ZW5A88PCPK]] of a [[complex analytic variety|GB7M582M5Y]].

## Explanations

### L1 human

A hypersurface in $\C^{n+1}$ is the set where one holomorphic function is zero.
At most points it is smooth, like a curved sheet.
At an isolated singularity it has a single bad point, such as the tip of a cone, while every nearby point is smooth.
The cone $z_0^2 + z_1^2 + z_2^2 = 0$ in $\C^3$ is the simplest example: its only singular point is the origin.