---
tag: NK9ATH4WB6
kind: concept
name: Moduli of algebraic curves
msc: [14H10, 14H15, 32G15]
sources:
  - url: https://en.wikipedia.org/wiki/Moduli_of_algebraic_curves
same_as:
  wikidata: Q6889798
  wikipedia: Moduli of algebraic curves
---

## Summary

The moduli space $M_g$ is a geometric space whose points are the isomorphism classes of smooth projective algebraic curves of genus $g$.
Over $\C$ these are the compact [[Riemann surfaces|5K0N8QMV6Q]] of genus $g$, and $M_g$ is the quotient of [[Teichmüller space|ZZGC6BZNJ5]] by the [[mapping class group|JV5NYG913V]].
For $g \ge 2$, $M_g$ is an irreducible quasi-projective variety of dimension $3g - 3$, and it has a compactification $\overline{M}_g$ by stable curves due to Deligne and Mumford.
It is a central object of algebraic geometry, linking geometry, topology and number theory.

## Explanations

### L1 human

Every compact Riemann surface has a genus, the number of holes.
Surfaces of the same genus can still differ in shape, and the moduli space is the space of all these shapes.
For genus $0$ there is only one shape, the Riemann sphere, so $M_0$ is a point.
For genus $1$ the shapes are the tori $\C/\Lambda$, and the $j$-invariant identifies $M_1$ with the complex line.
For genus $g \ge 2$ one needs $3g - 3$ complex numbers to describe the shape, as Riemann already counted.