---
tag: 4KAWK0FD4V
kind: definition
name: Isolated hypersurface singularity
msc: [32S25, 32S05, 14B05]
defines: VTGYEES3VS
uses: [507C34W0ZT, VZXF7RNFKS, 0YTCMFB4HC]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor_map
---

## Definition

Let $n \ge 0$, let $U$ be an open neighbourhood of $0$ in $\C^{n+1}$ and let $f \colon U \to \C$ be a [[holomorphic function|507C34W0ZT]] with $f(0) = 0$.
The [[germ|VZXF7RNFKS]] $f$ has an *isolated singularity* at $0$ if:

1. $\frac{\partial f}{\partial z_j}(0) = 0$ for $j = 0, \dots, n$, so $0$ is a [[critical point|0YTCMFB4HC]] of $f$;
2. there is a neighbourhood $U' \subseteq U$ of $0$ in which $0$ is the only point where all the partial derivatives $\partial f / \partial z_j$ vanish.

The germ at $0$ of the hypersurface $V = f^{-1}(0)$ is then called an *isolated hypersurface singularity*.

## Remarks

Example: $f(z) = z_0^2 + z_1^2 + \cdots + z_n^2$.
Its gradient is $(2z_0, \dots, 2z_n)$, which vanishes only at $0$, so both clauses hold.

Non-example: $f(z_0, z_1, z_2) = z_0 z_1$ on $\C^3$.
Its gradient is $(z_1, z_0, 0)$, which vanishes on the whole $z_2$-axis, so clause 2 fails.

Let $J$ be the ideal of the ring $\C\{z_0, \dots, z_n\}$ of convergent power series generated by the partial derivatives of $f$.
By the Rückert Nullstellensatz, clause 2 holds if and only if $\C\{z_0, \dots, z_n\} / J$ is finite-dimensional over $\C$.
Its dimension is the Milnor number of $f$.
If $f$ has no repeated factor, the points near $0$ where $V$ is singular are the points of $V$ where the gradient of $f$ vanishes.