• $g = 0$: $h^0(\omega_C) = 0$, $\deg K_C = -2$. The curve is $\cong \mathbb{P}^1$ over $\bar{k}$. No nonzero regular differentials.

• $g = 1$: $h^0(\omega_C) = 1$, $\deg K_C = 0$, $\omega_C \cong \mathcal{O}_C$. With a rational point, $C$ is an elliptic curve. The unique differential on $y^2 = x^3 + Ax + B$ is $\omega = dx/(2y)$.

• $g = 2$: $h^0(\omega_C) = 2$, $\deg K_C = 2$. Every genus-2 curve is hyperelliptic: the canonical map $\phi_K : C \to \mathbb{P}^1$ is a degree-2 cover.

• $g = 3$: $h^0(\omega_C) = 3$, $\deg K_C = 4$. The canonical map is $\phi_K : C \to \mathbb{P}^2$. If non-hyperelliptic, this is an embedding as a smooth plane quartic.

• $d = 1$ (line): $g = 0$. $\quad$ $d = 2$ (conic): $g = 0$ ($\cong \mathbb{P}^1$).
• $d = 3$ (cubic): $g = 1$ (elliptic curves). $\quad$ $d = 4$ (quartic): $g = 3$.
• $d = 5$: $g = 6$. $\quad$ $d = 6$: $g = 10$. $\quad$ $d = 10$: $g = 36$.

Missing genera: The values $(d-1)(d-2)/2$ are $0, 0, 1, 3, 6, 10, 15, 21, 28, \ldots$, so genus 2, 4, 5, 7, 8, 9, ... do not occur as smooth plane curves.

For a plane curve of degree $d$ with $\delta$ nodes (ordinary double points):

$p_a(C) = \frac{(d-1)(d-2)}{2}, \quad g(\widetilde{C}) = \frac{(d-1)(d-2)}{2} - \delta$

where $\widetilde{C}$ is the normalization. Each node drops the geometric genus by 1.

• Nodal cubic ($d = 3$, $\delta = 1$): $g = 0$. Normalization is $\mathbb{P}^1$.
• Nodal quartic ($d = 4$, $\delta = 1$): $g = 2$. A hyperelliptic curve.
• 3-nodal quartic ($d = 4$, $\delta = 3$): $g = 0$. Rational.
• Sextic with 4 nodes: $g = 10 - 4 = 6$.

More generally, a singularity of type $P$ contributes its delta-invariant $\delta_P$: $g = \frac{(d-1)(d-2)}{2} - \sum_P \delta_P$.

Curves in $\mathbb{P}^3$ (intersection of two surfaces of degrees $d_1, d_2$):

• $(2, 2)$: degree $4$, $g = 1$. Elliptic (intersection of two quadrics).
• $(2, 3)$: degree $6$, $g = 4$. A canonical curve of genus 4: $\omega_C \cong \mathcal{O}_C(1)$.
• $(2, 4)$: degree $8$, $g = 9$.
• $(3, 3)$: degree $9$, $g = 10$.

Curves in $\mathbb{P}^4$:

• $(2, 2, 2)$: degree $8$, $g = 5$.
• $(2, 2, 3)$: degree $12$, $g = 13$.

• $g = 0$: $C(\mathbb{C}) \cong S^2$ (Riemann sphere). $\chi_{\mathrm{top}} = 2$, simply connected.
• $g = 1$: $C(\mathbb{C}) \cong S^1 \times S^1$ (torus $\mathbb{C}/\Lambda$). $\chi_{\mathrm{top}} = 0$, $\pi_1 \cong \mathbb{Z}^2$.
• $g = 2$: two handles. $\chi_{\mathrm{top}} = -2$, $b_1 = 4$.
• Smooth plane quartic ($g = 3$): three handles. $\chi_{\mathrm{top}} = -4$, $H^1(C, \mathbb{Z}) \cong \mathbb{Z}^6$.

The Hodge decomposition gives $H^1(C, \mathbb{C}) = H^{1,0} \oplus H^{0,1}$ with $h^{1,0} = h^{0,1} = g$.

Hyperelliptic curve $C: y^2 = f(x)$, $\deg f = 2g + 1$ or $2g + 2$, $f$ separable. The $g$ independent regular differentials are:

$\omega_i = \frac{x^{i-1}\,dx}{y}, \quad i = 1, \ldots, g.$

• $g = 1$ ($y^2 = x^3 + Ax + B$): basis $\{dx/y\}$.
• $g = 2$: basis $\{dx/y,\; x\,dx/y\}$.
• $g = 3$ hyperelliptic: basis $\{dx/y,\; x\,dx/y,\; x^2\,dx/y\}$.

Smooth plane curve $F(x, y, z) = 0$ of degree $d$: the differentials are indexed by monomials $x^a y^b z^c$ with $a + b + c = d - 3$, $a, b, c \geq 0$. The count $\binom{d-1}{2} = \frac{(d-1)(d-2)}{2} = g$ confirms the genus-degree formula.

The tangent space to $\mathcal{M}_g$ at $[C]$ is $H^1(C, T_C)$ where $T_C = (\Omega^1_C)^\vee$. By Serre duality, $h^1(T_C) = h^0(K_C^{\otimes 2})$. For $g \geq 2$, Riemann-Roch gives:

$h^0(2K_C) = \deg(2K_C) - g + 1 = 4g - 4 - g + 1 = 3g - 3.$

• $\mathcal{M}_2$ ($\dim = 3$): every genus-2 curve is hyperelliptic $y^2 = f_6(x)$. Six branch points in $\mathbb{P}^1$ modulo $\operatorname{PGL}_2$: $\dim = 6 - 3 = 3$.

• $\mathcal{M}_3$ ($\dim = 6$): the general curve is a smooth plane quartic. Family dimension: $\dim |\mathcal{O}(4)| - \dim \operatorname{PGL}_3 = 14 - 8 = 6$. The hyperelliptic locus $\mathcal{H}_3$ has $\dim = 5$ (codimension 1).

• $\mathcal{M}_4$ ($\dim = 9$): a general genus-4 curve is a $(2,3)$ complete intersection in $\mathbb{P}^3$.

• $\overline{\mathcal{M}}_g$: boundary has $\lfloor g/2 \rfloor + 1$ irreducible components $\Delta_0, \Delta_1, \ldots, \Delta_{\lfloor g/2 \rfloor}$, where $\Delta_i$ parametrizes nodal curves with components of genera $i$ and $g - i$.

• $g = 2$: bound $84$, actual maximum $48$ (achieved by $y^2 = x^5 - x$).
• $g = 3$: bound $168$, achieved by the Klein quartic $x^3 y + y^3 z + z^3 x = 0$ with $\operatorname{Aut} \cong \operatorname{PSL}_2(\mathbb{F}_7)$.
• $g = 7$: the Macbeath curve attains $|\operatorname{Aut}| = 504 = 84 \cdot 6$.

The bound comes from Riemann-Hurwitz on $C \to C/G$: the minimum positive value of $2g' - 2 + \sum(1 - 1/e_i)$ is $1/42$, giving $|G| \leq 84(g-1)$.

• Genus 2: $y^2 = x^5 + 1$. Six Weierstrass points. $\mathcal{M}_2 = \mathcal{H}_2$ entirely.

• Genus 3 hyperelliptic: $y^2 = x^7 - 1$. Eight Weierstrass points. $\mathcal{H}_3 \subset \mathcal{M}_3$ has codimension 1.

• Genus 3 non-hyperelliptic: $x^4 + y^4 + z^4 = 0$ (Fermat quartic). The canonical map is the identity embedding $C \hookrightarrow \mathbb{P}^2$.

• Genus 4 hyperelliptic: $y^2 = x^9 + x$. $\mathcal{H}_4 \subset \mathcal{M}_4$ has codimension 2.

Curves on $\mathbb{P}^2$: $K_{\mathbb{P}^2} = -3H$, $C \sim dH$, so $2g - 2 = d^2 - 3d$, giving $g = (d-1)(d-2)/2$.

Curves on $\mathbb{P}^1 \times \mathbb{P}^1$ of bidegree $(a, b)$: $K = (-2, -2)$, so $2g - 2 = 2ab - 2a - 2b$, giving $g = (a-1)(b-1)$.

• Bidegree $(2, 2)$: $g = 1$. $\;$ $(2, 3)$: $g = 2$. $\;$ $(3, 3)$: $g = 4$.

On a K3 surface ($K_S = 0$): $2g - 2 = C^2$. A rational curve satisfies $C^2 = -2$.

On an abelian surface ($K_A = 0$): $2g - 2 = C^2 > 0$ for ample $C$, so $g \geq 2$.

The Fermat curve $F_d: x^d + y^d + z^d = 0 \subset \mathbb{P}^2$ has genus $g = (d-1)(d-2)/2$:

• $F_3$: $g = 1$, elliptic with $j = 0$ and CM by $\mathbb{Z}[e^{2\pi i/3}]$.
• $F_4$: $g = 3$, with $|\operatorname{Aut}| = 96$.
• $F_5$: $g = 6$, the Fermat quintic curve.

Automorphisms: $\operatorname{Aut}(F_d) \supseteq (\mathbb{Z}/d)^2 \rtimes S_3$ of order $6d^2$ for $d \geq 4$.

Fermat's Last Theorem: rational points on $F_d$ give solutions to $a^d + b^d = c^d$. By Faltings' theorem ($g \geq 2$ for $d \geq 4$), only finitely many rational points exist.

A curve is trigonal if it admits a $g^1_3$ (degree-3 map to $\mathbb{P}^1$).

• $g = 3$: every non-hyperelliptic curve is a plane quartic; projection from a point gives a $g^1_3$.
• $g = 4$: a non-hyperelliptic curve is a $(2,3)$ complete intersection in $\mathbb{P}^3$. The $g^1_3$ comes from the quadric's ruling.
• $g = 5$: the general curve has gonality 4. The trigonal locus has codimension 1 in $\mathcal{M}_5$.

• $g = 0$: $\operatorname{Jac}(\mathbb{P}^1) = 0$.
• $g = 1$: $\operatorname{Jac}(E) \cong E$ (an elliptic curve is its own Jacobian).
• $g = 2$: $\operatorname{Jac}(C)$ is an abelian surface. The theta divisor $\Theta \cong C$.
• $g = 3$: $\operatorname{Jac}(C)$ is 3-dimensional. For non-hyperelliptic $C$, $\Theta$ has a unique singular point.
• $g = 4$: the Schottky problem: which abelian 4-folds are Jacobians? (Answer: those satisfying the KP equation.)

For $f: C \to D$ of degree $n$: $2g_C - 2 = n(2g_D - 2) + \deg R$, where $R = \sum (e_P - 1) \cdot P$.

• Double cover of $\mathbb{P}^1$ branched at $2r$ points: $g = r - 1$. For $r = 3$: $g = 2$; $r = 4$: $g = 3$.

• Cyclic $d$-fold cover $y^d = (x - a_1) \cdots (x - a_s)$, $d \mid s$: $g = \frac{(s-2)(d-1)}{2}$.

• Unramified double cover of genus $g$: $g_C = 2g - 1$.