---
tag: 8CFTJP1WN8
kind: concept
name: Modality of a singularity
msc: [58K40, 32S05, 14B05]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Summary

The modality of a critical point of a function counts continuous parameters, called moduli.
It is the number of moduli needed to describe all nearby critical points up to a change of coordinates.
Germs of modality $0$ are the simple singularities: only finitely many types occur near them.
Arnold introduced the notion and classified the germs of modality $0$, $1$ and $2$.
Modality measures how far the classification of singularities is from being a finite list.

## Explanations

### L1 human

Near some critical points there are only finitely many shapes of critical point, like a short catalogue.
Near others the shapes vary continuously, like the angles in a family of triangles, and no finite catalogue suffices.
The modality is the number of such continuous dials.
For example, the function $x^4 + y^4$ on $\C^2$ vanishes on four lines through $0$.
Four lines through a point have a cross-ratio, which no change of coordinates can alter.
So nearby functions $x^4 + y^4 + a x^2 y^2$ give infinitely many different types, and the modality is $1$.