Moduli of Curves - Core Definitions

The moduli space of curves is one of the most important objects in algebraic geometry, parametrizing algebraic curves up to isomorphism. The stack formulation is essential for capturing automorphisms.

DefinitionModuli Stack of Curves

For integers $g, n \geq 0$ with $2g - 2 + n > 0$ , the moduli stack $\mathcal{M}_{g,n}$ is defined by: $\mathcal{M}_{g,n}(S) = \{\text{smooth proper curves } C \to S \text{ of genus } g \text{ with } n \text{ sections}\}$

A morphism in $\mathcal{M}_{g,n}(S)$ is an isomorphism of families compatible with the sections. This is a smooth Deligne-Mumford stack of dimension $3g - 3 + n$ over $\mathbb{Z}$ .

The automorphism group of a genus $g \geq 2$ curve is finite, making $\mathcal{M}_{g,n}$ Deligne-Mumford. For genus 0 and 1, additional marked points ensure finite automorphisms when $2g - 2 + n > 0$ .

DefinitionStable Curves

A stable curve of genus $g$ is a connected, nodal curve $C$ such that:

The arithmetic genus is $g$
Every smooth rational component meets the rest of the curve in at least 3 points
Every elliptic component (smooth genus 1) meets the rest in at least 1 point

These conditions ensure finite automorphism groups.

Remark

The stability condition is designed to eliminate curves with infinite automorphism groups. A smooth rational curve has $\text{Aut} = PGL_2$ , which is infinite. Requiring 3 special points forces finite automorphisms.

DefinitionDeligne-Mumford Compactification

The compactified moduli stack $\overline{\mathcal{M}}_{g,n}$ parametrizes stable curves of genus $g$ with $n$ marked points. It is a proper Deligne-Mumford stack containing $\mathcal{M}_{g,n}$ as an open dense substack. The boundary $\overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}$ consists of singular stable curves.

ExampleLow Genus Cases

$\mathcal{M}_{0,n}$ for $n \geq 3$ : Configuration space of $n$ points on $\mathbb{P}^1$ modulo $PGL_2$
$\mathcal{M}_{1,1}$ : Elliptic curves, a smooth stack of dimension 1
$\overline{\mathcal{M}}_{0,4} = \mathbb{P}^1$ : Cross-ratios of 4 points
$\overline{\mathcal{M}}_{1,1} = [\mathbb{A}^1/(\mathbb{Z}/2 \times \mathbb{Z}/3)]$ (with stacky points)

DefinitionUniversal Curve

Over $\mathcal{M}_{g,n+1}$ , there is a universal curve $\mathcal{C}_{g,n+1} \to \mathcal{M}_{g,n+1}$ obtained by forgetting the last marked point. This gives a morphism: $\pi: \mathcal{C}_{g,n+1} \to \mathcal{M}_{g,n+1} \to \mathcal{M}_{g,n}$

The universal curve is fundamental for constructing tautological classes and for Gromov-Witten theory.

DefinitionTautological Classes

The tautological ring $R^*(\overline{\mathcal{M}}_{g,n})$ is the $\mathbb{Q}$ -subalgebra of the Chow ring generated by:

$\psi_i = c_1(L_i)$ where $L_i$ is the cotangent line bundle at the $i$ -th marked point
$\lambda_i = c_i(E)$ where $E = \pi_*\omega_{\mathcal{C}/\mathcal{M}}$ is the Hodge bundle
Boundary classes $\delta_{\Gamma}$ for dual graphs $\Gamma$

These classes encode geometric information about curves and satisfy many relations (Mumford's conjecture, Faber-Zagier relations).

The moduli of curves is central to algebraic geometry, string theory, and mathematical physics, providing a rich testing ground for stack theory.