Moduli of Curves - Key Properties

The moduli stacks $\mathcal{M}_{g,n}$ and $\overline{\mathcal{M}}_{g,n}$ have rich geometric structure that has been intensively studied. Understanding their properties is fundamental to algebraic geometry.

TheoremIrreducibility and Smoothness

For $2g - 2 + n > 0$ :

$\mathcal{M}_{g,n}$ is a smooth irreducible Deligne-Mumford stack of dimension $3g - 3 + n$
$\overline{\mathcal{M}}_{g,n}$ is a proper Deligne-Mumford stack with normal crossing boundary
The locus of curves with automorphisms has codimension at least 1 for $g \geq 3$

The smoothness is fundamental for intersection theory and deformation theory of curves.

DefinitionBoundary Stratification

The boundary $\partial \overline{\mathcal{M}}_{g,n} = \overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}$ admits a stratification by topological type: $\partial \overline{\mathcal{M}}_{g,n} = \bigcup_{\Gamma} \mathcal{M}_\Gamma$ where $\Gamma$ ranges over dual graphs. Each stratum $\mathcal{M}_\Gamma$ is a product of moduli stacks of lower genus or fewer marked points.

ExampleGenus 2 Boundary

For $\overline{\mathcal{M}}_2$ , the boundary consists of:

$\Delta_0$ : Irreducible curves with one node (genus 1 curve with 2 points identified)
$\Delta_1$ : Reducible curves (two genus 1 curves meeting at a node)

The boundary divisor is $\Delta = \Delta_0 + \Delta_1$ , and its class is explicitly computed in the Picard group.

TheoremPicard Group

For $g \geq 2$ : $\text{Pic}(\overline{\mathcal{M}}_g) \otimes \mathbb{Q} = \mathbb{Q}[\lambda, \delta_0, \ldots, \delta_{[g/2]}]$ where $\lambda$ is the Hodge class and $\delta_i$ are boundary classes. Relations come from pullbacks under gluing morphisms.

DefinitionHarel Classes

The $\psi$ -classes $\psi_i \in A^1(\overline{\mathcal{M}}_{g,n})$ are defined as: $\psi_i = c_1(\mathbb{L}_i)$ where $\mathbb{L}_i$ is the cotangent line bundle to the universal curve at the $i$ -th marked point. These classes satisfy: $\psi_i|_{\mathcal{M}_\Gamma} = \text{(boundary terms)}$ and generate much of the tautological ring.

TheoremMumford's Conjecture

The stable cohomology ring $H^*(\mathcal{M}_g, \mathbb{Q})$ as $g \to \infty$ is a polynomial algebra: $H^*(\mathcal{M}_{\infty}, \mathbb{Q}) = \mathbb{Q}[\kappa_1, \kappa_2, \kappa_3, \ldots]$ where $\kappa_i$ are MMumford-Morita-Miller classes. This was proved by Madsen-Weiss using techniques from homotopy theory and is one of the deepest results about moduli of curves.

DefinitionHodge Bundle

The Hodge bundle $\mathbb{E} \to \overline{\mathcal{M}}_{g,n}$ has fiber $H^0(C, \omega_C)$ over a curve $C$ . Its Chern classes: $\lambda_i = c_i(\mathbb{E}) \in A^i(\overline{\mathcal{M}}_{g,n})$ are fundamental cohomology classes satisfying numerous relations (Faber's conjectures).

Remark

The relationship between $\psi$ -classes, $\lambda$ -classes, and boundary classes is governed by complex combinatorics of stable graphs. The tautological ring $R^*(\overline{\mathcal{M}}_{g,n})$ is conjecturally finite-dimensional in each genus, but this remains open for $g \geq 24$ .

These properties make $\overline{\mathcal{M}}_{g,n}$ one of the most studied objects in algebraic geometry, with connections to topology, number theory, and mathematical physics.