---
tag: FJZRW9P0K3
kind: concept
name: Moduli space
msc: [14D22, 14D20]
sources:
  - url: https://en.wikipedia.org/wiki/Moduli_space
same_as:
  wikidata: Q845677
  wikipedia: Moduli space
---

## Summary

A moduli space is a geometric space whose points correspond to the isomorphism classes of objects of some kind.
The objects may be curves, vector bundles or subvarieties, for example.
A fine moduli space represents the moduli functor and carries a universal family.
A coarse moduli space only has the right points and is universal for maps out of the functor.
Moduli spaces turn classification problems into geometry, and their geometry, such as dimension and singularities, reflects the deformations of the objects.
Examples are the [[Grassmannian|CYPA882RB9]], the [[Hilbert scheme|XK866CAZHE]] and the [[moduli of algebraic curves|NK9ATH4WB6]].

## Explanations

### L1 human

Some families of objects can be described by a few continuous parameters.
Lines through the origin in the plane are described by one angle, so they form a circle, the projective line.
Elliptic curves over the complex numbers are classified by one complex number, the [[j-invariant|9538W1K1EX]], so they form a line.
A moduli space is the space of these parameters, built so that nearby points are nearby objects.