The Hurwitz bound $|\mathrm{Aut}(C)| \leq 84(g-1)$ for smooth curves of genus $g \geq 2$ follows from the Riemann-Hurwitz formula applied to $C \to C/\mathrm{Aut}(C)$. The bound is achieved by the Klein quartic for $g = 3$: $x^3 y + y^3 z + z^3 x = 0 \subset \mathbb{P}^2$ which has $|\mathrm{Aut}(C)| = 168 = 84 \cdot 2 = 84(3-1)$, with $\mathrm{Aut}(C) \cong \mathrm{PSL}_2(\mathbb{F}_7)$.

Every genus 2 curve $C$ is hyperelliptic with involution $\iota$. The full automorphism group satisfies:

• Generic curve: $\mathrm{Aut}(C) = \lbrace 1, \iota \rbrace \cong \mathbb{Z}/2\mathbb{Z}$.
• $y^2 = x^6 - 1$: $\mathrm{Aut}(C) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$, order 12.
• $y^2 = x^5 - x$: $\mathrm{Aut}(C)$ has order 10.
• $y^2 = x^5 - 1$: $\mathrm{Aut}(C) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}$, order 10.
• $y^2 = x^6 - x$: $\mathrm{Aut}(C)$ has order $2 \cdot 24 = 48$ (the maximum for genus 2).

Consider the stable curve $C$ obtained by taking an elliptic curve $E$ and identifying two points $p, q \in E$. Then $\mathrm{Aut}(C)$ includes automorphisms of $E$ fixing $\lbrace p, q \rbrace$ as a set. If $E$ is generic and $p, q$ are generic, then $\mathrm{Aut}(C) = \lbrace \mathrm{id} \rbrace$ (or $\mathbb{Z}/2\mathbb{Z}$ from the involution swapping $p$ and $q$ when such exists). The group is always finite.

Let $C$ be a stable curve of genus $g$ with a single node $p$. Locally at $p$, $C = \mathrm{Spec}\, k[[x,y]]/(xy)$. The universal deformation of the node is $xy = t_1$, and the remaining deformations give a smooth base $k[[t_2, \ldots, t_{3g-3}]]$.

The boundary divisor $\Delta \subset \overline{\mathcal{M}}_g$ is locally cut out by $t_1 = 0$, so $\Delta$ is a smooth divisor in the smooth stack $\overline{\mathcal{M}}_g$.

A maximally degenerate stable curve of genus $g$ has $3g - 3$ nodes (the maximum for a stable curve). Every component is rational ($\cong \mathbb{P}^1$) with exactly 3 special points (nodes). The dual graph $\Gamma$ is a trivalent graph with $2g - 2$ vertices, $3g - 3$ edges, and first Betti number $g$.

At such a point, the deformation space is $k[[t_1, \ldots, t_{3g-3}]]$ where each $t_i$ smooths one node. The boundary of $\overline{\mathcal{M}}_g$ has normal crossings at this point, cut out by $t_1 \cdots t_{3g-3} = 0$.

Consider the family $y^2 = x(x-1)(x-t)$ of elliptic curves over $\mathrm{Spec}\, k[[t]]$. At $t = 0$, the curve $y^2 = x^2(x-1)$ has a node at $(0,0)$. The stable reduction gives a rational curve with a node (the Neron model's special fiber has a node).

For pointed curves ($\mathcal{M}_{1,1}$), the stable limit is uniquely determined. For $t = 1$ (where two branch points collide), the stable limit in $\overline{\mathcal{M}}_{1,1}$ is the rational curve $\mathbb{P}^1$ with $0$ and $\infty$ identified.

Consider the family of genus 2 curves $y^2 = x^6 - t$ over $k[[t]]$. At $t = 0$, the fiber $y^2 = x^6$ has a non-stable singularity (a cusp of sorts). After the base change $t = s^6$, we get $y^2 = x^6 - s^6 = (x^3 - s^3)(x^3 + s^3)$, which can be resolved to a stable curve. The stable limit is a curve with two rational components meeting at three points.

For $g = 0$, $\mathcal{M}_0$ is a point (just $\mathbb{P}^1$), trivially irreducible. For $g = 1$, $\mathcal{M}_{1,1}$ is irreducible (the $j$-line). The first nontrivial case is $g = 2$, where $\mathcal{M}_2$ is a 3-dimensional irreducible variety.

• $g = 2$: $\overline{M}_2$ is a 3-dimensional projective variety. Every genus 2 curve has $\mathrm{Aut}(C) \supseteq \mathbb{Z}/2\mathbb{Z}$, so the coarse moduli has quotient singularities everywhere.
• $g = 3$: $\overline{M}_3$ is a 6-dimensional projective variety. The non-hyperelliptic curves form an open dense subset. A generic genus 3 curve has $\mathrm{Aut}(C) = \lbrace \mathrm{id} \rbrace$, so $\overline{M}_3$ is generically smooth.
• $g \geq 3$: The generic curve has trivial automorphism group, so $\overline{M}_g$ is generically a manifold.

Harris-Mumford (1982) and Eisenbud-Harris (1987) proved:

• For $g \geq 24$, $\overline{M}_g$ is of general type (i.e., $\kappa(\overline{M}_g) = 3g - 3$).
• For $g \leq 16$, $\overline{M}_g$ is unirational (rational or close to rational).
• The cases $17 \leq g \leq 23$ remain largely open (Kodaira dimension unknown for most of these).

The key technique is computing the canonical class $K_{\overline{M}_g}$ in terms of the tautological classes $\lambda$ and $\delta_i$: $K_{\overline{M}_g} = 13\lambda - 2\delta_0 - 3\delta_1 - 2\delta_2 - \cdots - 2\delta_{\lfloor g/2 \rfloor}$ and showing it is ample (big) for $g \geq 24$.

For $g \geq 3$, the Picard group of $\mathcal{M}_g$ is: $\mathrm{Pic}(\mathcal{M}_g) \otimes \mathbb{Q} \cong \mathbb{Q} \cdot \lambda$ where $\lambda = c_1(\mathbb{E})$ is the first Chern class of the Hodge bundle. For the compactification: $\mathrm{Pic}(\overline{\mathcal{M}}_g) \otimes \mathbb{Q} \cong \mathbb{Q} \cdot \lambda \oplus \bigoplus_{i=0}^{\lfloor g/2 \rfloor} \mathbb{Q} \cdot \delta_i$ where $\delta_i = [\Delta_i]$ are the boundary divisor classes.

On $\overline{\mathcal{M}}_g$, key intersection numbers include:

• $\lambda^{3g-3} = 0$ on $\overline{M}_g$ (since $\lambda$ is not ample on the boundary).
• $\int_{\overline{\mathcal{M}}_{1,1}} \lambda = \frac{1}{24}$ (as a rational number, reflecting the stacky structure).
• $\int_{\overline{\mathcal{M}}_{2}} \lambda^3 = \frac{1}{240}$.

Stable curves have finite automorphism groups. If we allowed semistable curves (nodal curves where rational components meet the rest in $\geq 2$ points instead of $\geq 3$), automorphism groups could be infinite (e.g., a rational bridge with 2 nodes has $\mathbb{G}_m$ automorphisms). This is why the stability condition is essential for the DM property.