---
tag: ZW5A88PCPK
kind: concept
name: Regular and singular points of an analytic set
msc: [32B10, 32C25, 32S05]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

A point of a [[complex analytic variety|GB7M582M5Y]] is regular when the set is a complex submanifold near that point, and singular otherwise.
The regular points form an open dense subset of the variety, and the singular points form a nowhere dense analytic subset.
Singular points are where the local geometry is interesting, and their study is the subject of singularity theory.

## Explanations

### L1 human

Near a regular point, an analytic set looks like a flat piece of $\C^d$, bent smoothly.
At a singular point something goes wrong: several pieces cross, or the set has a sharp tip.
The curve $\set{z_1 z_2 = 0}$ is singular at the origin, where two lines cross.
The cusp $\set{z_1^2 = z_2^3}$ is a single piece, but it is singular at the origin, where it has a sharp point.
All other points of these two curves are regular.