We prove this in several cases.

Case 1: $C$ is smooth. The canonical bundle $\omega_C$ has degree $2g - 2 > 0$, so $\omega_C$ is ample. Any automorphism $\sigma \in \mathrm{Aut}(C)$ preserves $\omega_C$, hence preserves the $n$-canonical embedding $\phi_{n\omega_C}: C \hookrightarrow \mathbb{P}^N$ for $n \geq 3$. Thus $\mathrm{Aut}(C) \hookrightarrow \mathrm{PGL}_{N+1}$ as a closed subgroup fixing the embedded curve $\phi(C)$.

To show finiteness: the tangent space to $\mathrm{Aut}(C)$ at the identity is $H^0(C, T_C)$. Since $T_C = \omega_C^{-1}$ has degree $-(2g-2) < 0$ for $g \geq 2$, we have $H^0(C, T_C) = 0$. Hence $\mathrm{Aut}(C)$ is a 0-dimensional group scheme. Being also of finite type (closed in $\mathrm{PGL}_{N+1}$), it is finite.

In characteristic $p > 0$, one must additionally show the group scheme is reduced. This requires more care; in general, $\mathrm{Aut}(C)$ is a finite etale group scheme when $p > 2g + 1$ or when $C$ has no "wild" automorphisms.

Case 2: $C$ is nodal but irreducible. Let $\nu: \widetilde{C} \to C$ be the normalization, and let $\lbrace p_1, q_1, \ldots, p_\delta, q_\delta \rbrace \subset \widetilde{C}$ be the preimages of the $\delta$ nodes. An automorphism of $C$ lifts to an automorphism of $\widetilde{C}$ that preserves the set $\lbrace (p_i, q_i) \rbrace$ (permuting within each pair or between pairs mapping to the same node). Thus $\mathrm{Aut}(C) \hookrightarrow \mathrm{Aut}(\widetilde{C}, \lbrace p_i, q_i \rbrace)$, a subgroup of the finite group of automorphisms of $\widetilde{C}$ permuting finitely many marked points.

Case 3: $C$ is reducible. Write $C = C_1 \cup \cdots \cup C_r$ as a union of irreducible components. An automorphism either permutes the components or preserves them. The permutation group is finite, and the group of automorphisms preserving each component is finite by the argument for irreducible curves (each component $C_i \cong \mathbb{P}^1$ has at least 3 special points by stability, and $\mathrm{Aut}(\mathbb{P}^1; p_1, p_2, p_3)$ is finite).

The stability condition is crucial: a rational component with only 2 special points would have a $\mathbb{G}_m$-family of automorphisms (fixing 2 points on $\mathbb{P}^1$), giving an infinite automorphism group.

Step 3a: Embedding via pluricanonical maps.

For $n \geq 3$ and a stable curve $C$ of genus $g \geq 2$, the $n$-canonical sheaf $\omega_C^{\otimes n}$ is very ample (Deligne-Mumford). By Riemann-Roch: $h^0(C, \omega_C^{\otimes n}) = n(2g - 2) - g + 1 = (2n-1)(g-1)$ Let $N = (2n-1)(g-1) - 1$. The $n$-canonical embedding gives $C \hookrightarrow \mathbb{P}^N$.

The Hilbert polynomial of $C$ in this embedding is: $P(m) = mn(2g-2) - g + 1$

Step 3b: Hilbert scheme.

Let $\mathcal{H} = \mathrm{Hilb}^P_{\mathbb{P}^N / \mathbb{Z}}$ be the Hilbert scheme parametrizing closed subschemes of $\mathbb{P}^N$ with Hilbert polynomial $P$. By Grothendieck's theorem, $\mathcal{H}$ is a projective scheme over $\mathbb{Z}$.

Let $H \subset \mathcal{H}$ be the locally closed subscheme parametrizing $n$-canonically embedded stable curves. More precisely, $H$ parametrizes closed subschemes $C \subset \mathbb{P}^N$ such that:

1. $C$ is a stable curve of genus $g$.
2. $\mathcal{O}_C(1) \cong \omega_C^{\otimes n}$ (the embedding is $n$-canonical).

Then $H$ is a quasi-projective scheme.

Step 3c: Group action and quotient.

The group $G = \mathrm{PGL}_{N+1}$ acts on $H$ by changing the embedding (i.e., changing coordinates in $\mathbb{P}^N$). Two points of $H$ represent isomorphic curves if and only if they are in the same $G$-orbit. The quotient stack is: $\overline{\mathcal{M}}_g \cong [H / G]$

The projection $H \to [H/G] = \overline{\mathcal{M}}_g$ is a smooth morphism (since $G$ is smooth), giving a smooth atlas for $\overline{\mathcal{M}}_g$.

Step 3d: Etale atlas via level structure.

To obtain an etale atlas, one can add level-$n$ structure (for $n \geq 3$) to kill all automorphisms. Define $H^{[n]} \to H$ to be the scheme parametrizing curves with level-$n$ structure. Since the level structure trivializes automorphisms, the map $H^{[n]} \to \overline{\mathcal{M}}_g$ factors through an etale morphism $H^{[n]} \to \overline{\mathcal{M}}_g^{[n]} \to \overline{\mathcal{M}}_g$, providing an etale atlas.

We need to show that the deformation functor of any stable curve $C$ is unobstructed.

Tangent space computation. For a stable curve $C$ over $k$, the first-order deformations are classified by $\mathrm{Ext}^1(\Omega^1_C, \mathcal{O}_C)$. Using the exact sequence for the cotangent complex of a nodal curve: $0 \to \Omega^1_C \to \omega_C \to \bigoplus_{p \in \mathrm{Sing}(C)} k(p) \to 0$ (where $\omega_C$ is the dualizing sheaf), we get the long exact sequence: $\mathrm{Hom}(\omega_C, \mathcal{O}_C) \to \mathrm{Hom}(\Omega^1_C, \mathcal{O}_C) \to \bigoplus_p \mathrm{Ext}^1(k(p), \mathcal{O}_C)$ $\to \mathrm{Ext}^1(\omega_C, \mathcal{O}_C) \to \mathrm{Ext}^1(\Omega^1_C, \mathcal{O}_C) \to 0$

Now:

• $\mathrm{Hom}(\omega_C, \mathcal{O}_C) = H^0(C, \omega_C^{-1}) = 0$ for $g \geq 2$ (since $\omega_C$ is ample on a stable curve).
• $\mathrm{Ext}^1(k(p), \mathcal{O}_C) \cong k$ for each node $p$ (the smoothing parameter).
• $\mathrm{Ext}^1(\omega_C, \mathcal{O}_C) = H^1(C, \omega_C^{-1}) \cong H^0(C, \omega_C^{\otimes 2})^{\vee}$ by Serre duality on the stable curve (using the dualizing sheaf).

So $T^1 = \mathrm{Ext}^1(\Omega^1_C, \mathcal{O}_C)$ has dimension $3g - 3$ (it equals $\delta + \dim H^0(C, \omega_C^2) - \text{correction}$, but the total is always $3g - 3$ by Riemann-Roch-type arguments).

Obstruction space. The obstructions lie in $T^2 = \mathrm{Ext}^2(\Omega^1_C, \mathcal{O}_C)$. For a curve ($\dim C = 1$), Ext groups vanish in degree $\geq 2$ for locally free sheaves. For the non-locally-free $\Omega^1_C$ on a nodal curve, one computes: $\mathrm{Ext}^2(\Omega^1_C, \mathcal{O}_C) = 0$ This follows from the local analysis: at a node $xy = 0$, the module $\Omega^1$ has a presentation allowing the computation of $T^2 = 0$.

Since $T^2 = 0$, the deformation functor is unobstructed, and $\overline{\mathcal{M}}_g$ is smooth.

By the valuative criterion for properness of stacks, we need: given a DVR $R$ with fraction field $K$ and a stable curve $C_K$ over $K$, there exists (after a finite base change $R \to R'$) a unique extension to a stable curve $\mathcal{C}$ over $R'$.

Existence is the Stable Reduction Theorem. The proof proceeds as follows:

1. Semistable reduction: By resolution of singularities for surfaces (Abhyankar, Lipman), one can find a regular model $\mathcal{C}_0 \to \mathrm{Spec}\, R'$ (after base change) whose special fiber is a reduced normal crossings divisor. This gives a semistable model.

2. Stabilization: From the semistable model, contract all unstable components:

   • Rational components meeting the rest of the curve in only 1 point (these can be blown down).
   • Rational components meeting the rest in exactly 2 points (rational bridges with $\mathbb{G}_m$ automorphisms — these are contracted by identifying the two attachment points).

   After finitely many contractions, one arrives at a stable model.

Uniqueness follows from the separatedness of $\overline{\mathcal{M}}_g$ (the diagonal is proper), which in turn follows from: if two stable curves over $R$ are isomorphic over $K$, then they are isomorphic over $R$ (because the isomorphism, being a rational map between proper schemes over $R$, extends by normality/regularity arguments).

Over $\mathbb{C}$: The classical Teichmuller space $\mathcal{T}_g$ is homeomorphic to a ball in $\mathbb{R}^{6g-6}$ (Teichmuller's theorem), hence connected. Since $\mathcal{M}_g(\mathbb{C}) = \mathcal{T}_g / \mathrm{Mod}_g$ (quotient by the mapping class group), and the mapping class group is connected as a discrete group acting on a connected space, the quotient is connected. As $\mathcal{M}_g$ is also smooth, it is irreducible.

For the compactification $\overline{\mathcal{M}}_g$: the boundary $\overline{\mathcal{M}}_g \setminus \mathcal{M}_g$ has codimension 1, and adding a codimension-1 locus to an irreducible variety preserves irreducibility (by the smoothness of $\overline{\mathcal{M}}_g$).

Over $\overline{\mathbb{F}}_p$: Since $\overline{\mathcal{M}}_g$ is smooth over $\mathbb{Z}$, each fiber $\overline{\mathcal{M}}_g \otimes \overline{\mathbb{F}}_p$ has the same number of irreducible components as the generic fiber. (A smooth proper morphism with connected generic fiber has connected special fibers, by Zariski's connectedness theorem applied to the Stein factorization.) Since the generic fiber (over $\overline{\mathbb{Q}}$) is irreducible, so is every special fiber.