On $\mathbb{P}^1$ with coordinate $t$:

• $L(0) = k$ (constant functions): $\ell(0) = 1$.
• $L(P_\infty) = k + k \cdot t$ (functions with at most a simple pole at $\infty$): $\ell(P_\infty) = 2$.
• $L(n \cdot P_\infty) = k + kt + \cdots + kt^n$: $\ell(nP_\infty) = n + 1$.
• $L(-P_0) = 0$ (functions with a zero at $0$ and no poles — only $0$): $\ell(-P_0) = 0$.

If $g = 0$ and $P$ is any point on $C$: $\ell(P) - \ell(K - P) = 1 - 0 + 1 = 2$. Since $\deg K = -2$ and $\deg(K - P) = -3 < 0$, we have $\ell(K - P) = 0$, so $\ell(P) = 2$.

This means there exists a nonconstant function $f$ with a single simple pole at $P$ — equivalently, $f : C \to \mathbb{P}^1$ is an isomorphism. So every genus-0 curve with a rational point is isomorphic to $\mathbb{P}^1$.

Let $E$ be an elliptic curve ($g = 1$) with origin $O$. By Riemann–Roch, $\ell(nO) = n$ for $n \geq 1$ (since $\deg K = 0$ and $K \sim 0$):

| $n$ | $\ell(nO)$ | Basis | New function | |-----|------------|-------|-------------| | 1 | 1 | $\{1\}$ | — | | 2 | 2 | $\{1, x\}$ | $x$ has a double pole at $O$ | | 3 | 3 | $\{1, x, y\}$ | $y$ has a triple pole at $O$ | | 4 | 4 | $\{1, x, y, x^2\}$ | — | | 5 | 5 | $\{1, x, y, x^2, xy\}$ | — | | 6 | 6 | $\{1, x, y, x^2, xy, x^3\}$ | — |

In degree 6, $y^2$ and $x^3$ both have a pole of order 6, so they must be linearly dependent modulo lower-order terms. This gives the Weierstrass equation:

$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$

For a genus-$g$ curve $C$ with $g \geq 2$, the canonical divisor $K$ has $\deg K = 2g - 2$ and $\ell(K) = g$ (by RR: $\ell(K) - \ell(0) = (2g-2) - g + 1 = g - 1$, so $\ell(K) = g$).

If $C$ is not hyperelliptic, the canonical linear system $|K|$ gives an embedding:

$\phi_K : C \hookrightarrow \mathbb{P}^{g-1}.$

This is the canonical embedding. For $g = 3$: $C \hookrightarrow \mathbb{P}^2$ as a smooth quartic curve. For $g = 4$: $C \hookrightarrow \mathbb{P}^3$ as the intersection of a quadric and a cubic.

A divisor $D$ on a genus-$g$ curve $C$ gives an embedding $C \hookrightarrow \mathbb{P}^{\ell(D)-1}$ if and only if:

• $\ell(D - P) = \ell(D) - 1$ for all $P$ (separates points from tangent vectors), and
• $\ell(D - P - Q) = \ell(D) - 2$ for all $P, Q$ (separates points).

By Riemann–Roch: if $\deg D \geq 2g + 1$, then $D$ is very ample ($\ell(K - D + P + Q) = 0$ since $\deg(K - D + P + Q) < 0$). So every curve of genus $g$ embeds in $\mathbb{P}^{g+1}$ via a divisor of degree $2g + 1$.

A curve $C$ of genus $g \geq 2$ is hyperelliptic if there exists a degree-2 map $f : C \to \mathbb{P}^1$ (equivalently, $\ell(D) \geq 2$ for some divisor of degree 2).

By Riemann–Roch: $\ell(D) - \ell(K - D) = 2 - g + 1 = 3 - g$. For $g \geq 3$, $\ell(K - D) \geq g - 3 + \ell(D) \geq g - 1$, so such a $D$ imposes nontrivial conditions.

• $g = 2$: every genus-2 curve is hyperelliptic (since $\ell(K) = 2$ and $\deg K = 2$).
• $g = 3$: a curve is hyperelliptic $\iff$ it is not a smooth plane quartic.
• $g \geq 3$: the hyperelliptic locus is a proper subvariety of the moduli space $\mathcal{M}_g$.

On $\mathbb{P}^2$: $K = -3H$ (where $H$ is a line), $\chi(\mathcal{O}) = 1$, $H^2 = 1$. For $D = dH$:

$\chi(\mathcal{O}(d)) = 1 + \frac{1}{2}(d^2 + 3d) = \frac{(d+1)(d+2)}{2} = \binom{d+2}{2}.$

This matches $h^0(\mathcal{O}(d)) = \binom{d+2}{2}$ for $d \geq 0$ (and higher cohomology vanishes).

A K3 surface $S$ has $K_S = 0$ and $\chi(\mathcal{O}_S) = 2$. For a divisor $D$:

$\chi(\mathcal{O}_S(D)) = 2 + \frac{D^2}{2}.$

If $D$ is effective with $D^2 = 2d - 2$ (e.g., a genus-$d$ curve), then $\chi = d + 1$. For a smooth curve $C \in |D|$ of genus $g = d$, Riemann–Roch on $S$ gives $h^0(\mathcal{O}_S(C)) \geq d + 1$ — a $(d+1)$-dimensional linear system of curves.

For a curve $C$ and a line bundle $\mathcal{L}$ of degree $d$:

• $\operatorname{ch}(\mathcal{L}) = 1 + c_1(\mathcal{L}) = 1 + d \cdot [\mathrm{pt}]$.
• $\operatorname{td}(T_C) = 1 + \frac{1}{2}c_1(T_C) = 1 + (1 - g)[\mathrm{pt}]$.

$\chi(\mathcal{L}) = \int_C (1 + d[\mathrm{pt}])(1 + (1-g)[\mathrm{pt}]) = d + 1 - g. \quad \checkmark$

For a flat family $f: \mathcal{C} \to B$ of genus-$g$ curves, GRR computes the Chern classes of the Hodge bundle $\mathbb{E} = f_*\omega_{\mathcal{C}/B}$ (a rank-$g$ vector bundle on $B$):

$c_1(\mathbb{E}) = f_*\left(\frac{c_1(\omega_{\mathcal{C}/B})^2}{2} + \frac{c_1(\omega_{\mathcal{C}/B})c_1(\Omega^1_f)}{12}\right).$

This leads to the Mumford formula and is foundational for intersection theory on the moduli space $\overline{\mathcal{M}}_g$.