---
tag: RREV63H509
kind: definition
name: Moduli of algebraic curves
msc: [14H10, 14H15, 32G15]
defines: NK9ATH4WB6
sources:
  - url: https://en.wikipedia.org/wiki/Moduli_of_algebraic_curves
---

## Definition

Let $g \ge 0$ be an integer.
Let $T$ be a scheme over $\C$.
A *family of curves of [[genus|NJ787GZS1N]] $g$* over $T$ is a smooth proper morphism $\pi \colon C \to T$.
Its fibres must be connected algebraic curves of genus $g$.
Let $F_g$ be the functor that sends $T$ to the set of isomorphism classes of families of curves of genus $g$ over $T$.
It sends a morphism $T' \to T$ to the pullback of families.
A *coarse moduli space of curves of genus $g$* is a scheme $M_g$ over $\C$ with a natural transformation $\Phi \colon F_g \to \Hom(-, M_g)$ such that:

1. $\Phi$ is a bijection from $F_g(\operatorname{Spec} \C)$ onto the set of $\C$-points of $M_g$, and
2. for every scheme $N$ and natural transformation $\Psi \colon F_g \to \Hom(-, N)$ there is a unique morphism $u \colon M_g \to N$ with $\Psi = u \circ \Phi$.

## Remarks

Here $F_g(\operatorname{Spec} \C)$ is the set of isomorphism classes of smooth projective connected curves of genus $g$.
A coarse moduli space is unique up to unique isomorphism, and it exists for every $g$.
It is not a fine moduli space, that is, $\Phi$ is not an isomorphism of functors, because some curves have nontrivial automorphisms.
The analytic space underlying $M_g$ is the quotient of the [[Teichmüller space|ZZGC6BZNJ5]] $T_g$ by the mapping class group.

Example: $g = 1$.
Each curve of genus $1$ over $\C$ is an elliptic curve, and the $j$-invariant gives a bijection from isomorphism classes to $\C$.
This bijection comes from a natural transformation as in the definition, and $M_1$ is the affine line $\mathbb{A}^1$.

Non-example: the bare set $F_g(\operatorname{Spec} \C)$ has no scheme structure.
So it is not a coarse moduli space by itself, although it is in bijection with the points of $M_g$.