For $g = 0$, smooth curves of genus 0 are conics. Over an algebraically closed field, there is a unique curve: $\mathbb{P}^1$. But $\mathrm{Aut}(\mathbb{P}^1) = \mathrm{PGL}_2$, which is 3-dimensional. Thus $\mathcal{M}_0 = [*/\mathrm{PGL}_2] = B\mathrm{PGL}_2$, the classifying stack. The coarse moduli space is a point.

For $g = 1$, we need a marked point to get the moduli of elliptic curves $\mathcal{M}_{1,1}$. Without the marking, $\mathcal{M}_1$ is again a classifying-type stack. The stack $\mathcal{M}_{1,1}$ is a Deligne-Mumford stack with coarse moduli the $j$-line $\mathbb{A}^1_j$.

The generic elliptic curve has $\mathrm{Aut}(E) = \lbrace \pm 1 \rbrace \cong \mathbb{Z}/2\mathbb{Z}$. Special curves have larger automorphism groups:

• $j = 1728$ ($y^2 = x^3 - x$): $\mathrm{Aut}(E) \cong \mathbb{Z}/4\mathbb{Z}$
• $j = 0$ ($y^2 = x^3 - 1$): $\mathrm{Aut}(E) \cong \mathbb{Z}/6\mathbb{Z}$

Every curve of genus 2 is hyperelliptic. The moduli stack $\mathcal{M}_2$ has dimension $3 \cdot 2 - 3 = 3$. A genus 2 curve over a field of characteristic $\neq 2$ can be written as $y^2 = f(x)$ where $f$ has degree 5 or 6 with distinct roots. The hyperelliptic involution $\iota: (x, y) \mapsto (x, -y)$ is always an automorphism.

The moduli space can be described via Igusa invariants $(J_2, J_4, J_6, J_{10})$ modulo weighted projective equivalence.

| Genus $g$ | $\dim \mathcal{M}_g$ | Description | |-----------|---------------------|-------------| | 0 | $-3$ (empty/point) | Only $\mathbb{P}^1$ | | 1 | 0 (but $\mathcal{M}_{1,1}$ has dim 1) | Elliptic curves: 1 parameter ($j$) | | 2 | 3 | Hyperelliptic: 6 branch points modulo $\mathrm{PGL}_2$ | | 3 | 6 | Non-hyperelliptic: plane quartics | | 4 | 9 | Intersection of quadric and cubic in $\mathbb{P}^3$ | | 5 | 12 | Intersection of 3 quadrics in $\mathbb{P}^4$ |

A stable curve of genus 2 can be:

• A smooth genus 2 curve.
• Two elliptic curves $E_1, E_2$ meeting at a single node (genus $1 + 1 = 2$).
• An elliptic curve with a rational component attached at two points, forming a node each.
• A rational curve with 3 nodes (genus $0 + 3 = 3$... but this has genus 3, not 2).

For genus 2, the boundary $\overline{\mathcal{M}}_2 \setminus \mathcal{M}_2$ consists of two irreducible divisors:

• $\Delta_0$: irreducible curves with one node (a smooth genus 1 curve with a self-intersection).
• $\Delta_1$: two elliptic curves joined at a node.

The boundary $\overline{\mathcal{M}}_g \setminus \mathcal{M}_g$ is a normal crossing divisor with irreducible components $\Delta_0, \Delta_1, \ldots, \Delta_{\lfloor g/2 \rfloor}$ where:

• $\Delta_0$: irreducible nodal curves (normalizing the node gives a genus $g-1$ curve).
• $\Delta_i$ for $1 \leq i \leq \lfloor g/2 \rfloor$: curves consisting of a genus $i$ component and a genus $g-i$ component meeting at one node.

The stack $\overline{\mathcal{M}}_{0,n}$ is actually a smooth projective scheme (fine moduli!) for $n \geq 3$, since pointed rational curves with $\geq 3$ marked points have no automorphisms ($\mathrm{PGL}_2$ acts sharply 3-transitively on $\mathbb{P}^1$).

Dimensions: $\dim \overline{\mathcal{M}}_{0,n} = n - 3$.

• $\overline{\mathcal{M}}_{0,3} = \mathrm{Spec}\, k$ (a point: unique 3-pointed $\mathbb{P}^1$).
• $\overline{\mathcal{M}}_{0,4} \cong \mathbb{P}^1$ (the cross-ratio).
• $\overline{\mathcal{M}}_{0,5}$ is the del Pezzo surface of degree 5 (blowup of $\mathbb{P}^2$ at 4 points).

This parametrizes stable 1-pointed genus 1 curves. The compactification adds a single boundary point: the rational nodal curve (a $\mathbb{P}^1$ with $0$ and $\infty$ identified) with the smooth marked point. So $\overline{\mathcal{M}}_{1,1}$ is a DM stack with coarse moduli $\overline{M}_{1,1} \cong \mathbb{P}^1$ (the $j$-line compactified).

An etale atlas for $\mathcal{M}_g$ can be constructed using the Hilbert scheme. Choose $n \gg 0$ so that $\omega_C^{\otimes n}$ embeds $C \hookrightarrow \mathbb{P}^N$ where $N = n(2g-2) - g$. Let $H \subset \mathrm{Hilb}_{\mathbb{P}^N}$ be the locally closed subscheme parametrizing $n$-canonically embedded smooth curves of genus $g$. Then $\mathrm{PGL}_{N+1}$ acts on $H$, and $\mathcal{M}_g \cong [H / \mathrm{PGL}_{N+1}].$ The morphism $H \to \mathcal{M}_g$ is a smooth surjection, providing a smooth atlas. To get an etale atlas, one can use the rigidification provided by level structures.

On $\overline{\mathcal{M}}_{0,4} \cong \mathbb{P}^1$, the class $\psi_1$ is the class of a point. The integral $\int_{\overline{\mathcal{M}}_{0,4}} \psi_1 = 1$. More generally, the Witten-Kontsevich theorem gives: $\int_{\overline{\mathcal{M}}_{0,n}} \psi_1^{a_1} \cdots \psi_n^{a_n} = \binom{n-3}{a_1, \ldots, a_n}$ when $a_1 + \cdots + a_n = n - 3$.

The Hodge bundle $\mathbb{E} = \pi_* \omega_{C/S}$ is a rank-$g$ vector bundle on $\overline{\mathcal{M}}_g$. The $\lambda$-classes are $\lambda_i = c_i(\mathbb{E})$. The Mumford relation gives: $c(\mathbb{E}) \cdot c(\mathbb{E}^{\vee}) = 1$ in $A^*(\overline{\mathcal{M}}_g)$, where $c$ denotes the total Chern class.

Consider the family of elliptic curves $y^2 = x^3 - t$ over $\mathrm{Spec}\, k[[t]]$. At $t = 0$, this degenerates to the cuspidal cubic $y^2 = x^3$, which is not stable (cusp is not a node). After a base change $t = s^6$ and appropriate birational modifications, one obtains a stable reduction. The stable limit is a rational curve with a node (the Kodaira fiber of type $II$).

Consider $y^2 = (x^2 - t)(x^2 - 2t)(x^2 - 3t)$ as a family of genus 2 curves over $\mathrm{Spec}\, k[[t]]$. At $t = 0$, the six branch points $\pm\sqrt{t}, \pm\sqrt{2t}, \pm\sqrt{3t}$ all collide at the origin. After base change $t = s^2$, the branch points become $\pm s, \pm s\sqrt{2}, \pm s\sqrt{3}$, and a semistable model can be constructed with the stable limit being two rational curves meeting at three nodes (a curve of genus 2 with compact type degeneration).

Define the generating function $F(t_0, t_1, \ldots) = \sum_{g \geq 0} \sum_{n \geq 0} \frac{1}{n!} \sum_{a_1, \ldots, a_n \geq 0} \left( \int_{\overline{\mathcal{M}}_{g,n}} \prod_{i=1}^n \psi_i^{a_i} \right) \prod_{i=1}^n t_{a_i}$ Then $e^F$ is a $\tau$-function for the KdV hierarchy. Equivalently, the intersection numbers satisfy the Virasoro constraints: $L_n \cdot e^F = 0 \quad \text{for } n \geq -1$ where $L_n$ are specific differential operators forming a representation of (half of) the Virasoro algebra.