We prove $\overline{\mathcal{M}}_{g,n}$ is a proper Deligne-Mumford stack for $2g - 2 + n > 0$.

Step 1: Definition of the Moduli Functor

Define $\overline{\mathcal{M}}_{g,n}: \textbf{Sch}^{\text{op}} \to \textbf{Grpd}$ by: $\overline{\mathcal{M}}_{g,n}(S) = \{\pi: \mathcal{C} \to S \text{ flat proper with $n$ sections, fibers stable curves of genus } g\}$

Morphisms are isomorphisms of families preserving marked sections. We must show this is an algebraic stack.

Step 2: Representability of the Diagonal

Given two families $\mathcal{C}_1 \to S$, $\mathcal{C}_2 \to S$ with sections, the isom-functor: $\text{Isom}(\mathcal{C}_1, \mathcal{C}_2) \to S$ must be representable by schemes (or algebraic spaces).

By stability, automorphism groups are finite, so isomorphism schemes are finite over $S$. The key is:

• Stable curves have finite automorphisms (by stability condition)
• The Isom-scheme is proper over $S$ (curves are proper)
• Properness + finite type = finite

Thus the diagonal is representable, separated, and quasi-compact.

Step 3: Boundedness

We show the moduli problem is bounded: stable curves embed in $\mathbb{P}^N$ with bounded invariants.

For genus $g$ stable curve $C$:

• The dualizing sheaf $\omega_C$ is ample
• For $m \geq 3$, $\omega_C^{\otimes m}$ is very ample
• This embeds $C \hookrightarrow \mathbb{P}^N$ where $N = (2m-1)(g-1) - 1$
• The Hilbert polynomial is determined by $g, m$

By Grothendieck's Hilbert scheme theory, curves with fixed Hilbert polynomial form a bounded family. Thus $\overline{\mathcal{M}}_{g,n}$ parametrizes a bounded family of polarized curves.

Step 4: Construction via Hilbert Scheme

Consider the Hilbert scheme $\mathcal{H}$ parametrizing subschemes of $\mathbb{P}^N$ with the appropriate Hilbert polynomial. There is a universal family $\mathcal{C}_{\mathcal{H}} \to \mathcal{H}$.

Let $\mathcal{H}^{st} \subset \mathcal{H}$ be the open subscheme where fibers are stable curves. The locus where marked points can be placed consistently is an open subset. Form: $U = (\mathcal{C}_{\mathcal{H}}^{st})^n / \text{(diagonals)}$

The group $PGL_{N+1}$ acts on $U$, and: $\overline{\mathcal{M}}_{g,n} = [U/PGL_{N+1}]$

This quotient stack is a Deligne-Mumford stack by general theory of quotient stacks by finite stabilizers.

Step 5: Étaleness of the Atlas

The natural morphism $U \to \overline{\mathcal{M}}_{g,n}$ is étale, not just smooth. This follows from:

• Deformations of stable curves are unobstructed (automorphisms are finite)
• The automorphism groups act freely on the tangent spaces
• Infinitesimal lifting is unique when automorphisms are discrete

Thus $\overline{\mathcal{M}}_{g,n}$ is Deligne-Mumford.

Step 6: Properness

We verify the valuative criterion for properness. Let $R$ be a DVR with fraction field $K$, and suppose we have:

• A stable family $\mathcal{C}_K \to \text{Spec}(K)$ of genus $g$ with $n$ sections
• We must extend to $\mathcal{C}_R \to \text{Spec}(R)$

Stable Reduction Theorem: Every smooth curve over $K$ admits a stable model over $R$ after finite base change $R' \to R$. The stable model is unique up to unique isomorphism.

This provides the extension, showing $\overline{\mathcal{M}}_{g,n} \to \text{Spec}(\mathbb{Z})$ is proper.

Step 7: Smoothness

The smoothness of $\overline{\mathcal{M}}_{g,n}$ follows from deformation theory:

• The tangent space at a stable curve $C$ is $H^1(C, T_C)$
• The obstruction space is $H^2(C, T_C) = 0$ for curves
• The dimension is $3g - 3 + n$ by Riemann-Roch

Thus $\overline{\mathcal{M}}_{g,n}$ is smooth of the expected dimension.

Conclusion

We have shown:

1. $\overline{\mathcal{M}}_{g,n}$ is a stack with representable diagonal
2. An étale atlas exists (via Hilbert scheme quotient)
3. It is proper (by stable reduction)
4. It is smooth (by deformation theory)

Therefore $\overline{\mathcal{M}}_{g,n}$ is a proper smooth Deligne-Mumford stack.