For $2g - 2 + n > 0$, the moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable $n$-pointed genus $g$ curves exists as a proper smooth Deligne-Mumford stack over $\mathbb{Z}$. Moreover:

1. $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ is open and dense
2. The boundary $\partial \overline{\mathcal{M}}_{g,n}$ is a normal crossings divisor
3. The dimension is $3g - 3 + n$
4. The coarse moduli space $M_{g,n}$ exists as a quasi-projective variety

This was proved by Deligne-Mumford using geometric invariant theory and is the cornerstone of modern curve theory.

Every family of stable curves $\mathcal{C} \to S$ over a smooth base admits a simultaneous resolution: there exists a modification $S' \to S$ and $\mathcal{C}' \to S'$ such that $\mathcal{C}'$ is smooth and the general fiber of $\mathcal{C}' \to S'$ maps isomorphically to the corresponding fiber of $\mathcal{C} \to S$.

This shows that singular stable curves can be resolved in families, essential for studying degenerations.

For $g \geq 24$, the coarse moduli space $M_g$ (and $\overline{M}_g$) is of general type: the canonical bundle $K_{M_g}$ is big. This means: $K^{\dim M_g}_{M_g} > 0$

For smaller genus, $M_g$ has different behavior:

• $g = 2$: $M_2$ is rational (uniruled)
• $3 \leq g \leq 23$: General type question was subtle, resolved affirmatively

This theorem shows moduli spaces have rich birational geometry.

The effective cone of divisors on $\overline{M}_g$ is generated by boundary classes $\delta_i$ and has ray structure determined by slopes: $s(\delta_i) = \frac{i(g-i)}{g}$

Predictions about ample divisors (Fulton's conjecture) and birational models follow from understanding this cone. Many cases remain open, but substantial progress has been made using geometric techniques and modular forms.

The first Chern class of the Hodge bundle satisfies: $\lambda_1 = \frac{1}{12}(\delta_0 - \delta_1)$ on $\overline{M}_2$, with generalizations to all genera. This relates the Hodge bundle to boundary geometry and is fundamental for computing intersections.

The universal curve $\pi: \mathcal{C}_{g,n+1} \to \overline{\mathcal{M}}_{g,n}$ has stable canonical bundle: $\omega_{\mathcal{C}/\mathcal{M}}$ is relatively ample over $\overline{\mathcal{M}}_{g,n}$ for $g \geq 2$. This implies: $\pi_*\omega_{\mathcal{C}/\mathcal{M}}^k \simeq \mathbb{E}^{\otimes k}$ for $k \gg 0$, where $\mathbb{E}$ is the Hodge bundle.

The cohomology groups $H^i(\mathcal{M}_g, \mathbb{Q})$ stabilize as $g \to \infty$: for fixed $i$, they become independent of $g$ for $g \gg i$. The stable cohomology: $H^*(\mathcal{M}_\infty, \mathbb{Q}) = \mathbb{Q}[\kappa_1, \kappa_2, \ldots]$ was computed by Madsen-Weiss, confirming Mumford's conjecture.