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 $g$ 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:

$\Phi$ is a bijection from $F_g(\operatorname{Spec} \C)$ onto the set of $\C$ -points of $M_g$ , and
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 $T_g$ 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 $M_g$ .

Tag	`RREV63H509`
URI	`m2://RREV63H509`
Kind	definition
Writing	W0 stub
MSC	14H10, 14H15, 32G15
Defines	Moduli of algebraic curves
Changed	2 November 2026

Moduli of algebraic curves

Definition

Remarks

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