Algebraic Curves and Compact Riemann Surfaces

Every compact Riemann surface arises as an algebraic curve, establishing a profound connection between complex analysis and algebraic geometry.

Algebraic Curves

Definition10.4Algebraic curve

A (smooth) algebraic curve over $\mathbb{C}$ is the zero locus $C = \{(z, w) \in \mathbb{C}^2 : P(z, w) = 0\}$ of an irreducible polynomial $P \in \mathbb{C}[z, w]$ , possibly completed in $\mathbb{CP}^2$ . After resolving singularities, $C$ carries a natural Riemann surface structure.

ExampleExamples of algebraic curves

Rational curve: $w = z$ (genus $0$ , isomorphic to $\hat{\mathbb{C}}$ ).
Elliptic curve: $w^2 = z^3 + az + b$ ( $\Delta = -16(4a^3 + 27b^2) \neq 0$ ), genus $1$ .
Hyperelliptic curve: $w^2 = \prod_{i=1}^{2g+2}(z - a_i)$ has genus $g$ .
Fermat curve: $z^n + w^n = 1$ has genus $(n-1)(n-2)/2$ .

Meromorphic Functions and Divisors

Definition10.5Divisor

A divisor on a compact Riemann surface $S$ is a formal finite sum $D = \sum n_p \cdot p$ where $p \in S$ and $n_p \in \mathbb{Z}$ . The degree of $D$ is $\deg D = \sum n_p$ . The divisor of a meromorphic function $f$ is

$\text{div}(f) = \sum_p \text{ord}_p(f) \cdot p$

where $\text{ord}_p(f)$ is the order of the zero (positive) or pole (negative) of $f$ at $p$ .

Theorem10.2Degree of a principal divisor

For any non-zero meromorphic function $f$ on a compact Riemann surface $S$ :

$\deg(\text{div}(f)) = 0.$

That is, a meromorphic function has equally many zeros and poles (counted with multiplicity). This is a consequence of the residue theorem applied to $f'/f$ .

ExampleDivisors on $\hat{\mathbb{C}}$

On the Riemann sphere, $f(z) = (z-1)^2/(z+i)$ has $\text{div}(f) = 2\cdot(1) - 1\cdot(-i) - 1\cdot(\infty)$ . The degree is $2 - 1 - 1 = 0$ .

The Riemann-Roch Theorem

Theorem10.3Riemann-Roch theorem

Let $S$ be a compact Riemann surface of genus $g$ and $D$ a divisor on $S$ . Let $\ell(D) = \dim L(D)$ where $L(D) = \{f \text{ meromorphic} : \text{div}(f) + D \geq 0\} \cup \{0\}$ . Then

$\ell(D) - \ell(K - D) = \deg D - g + 1$

where $K$ is a canonical divisor (the divisor of any meromorphic $1$ -form).

RemarkSignificance of Riemann-Roch

The Riemann-Roch theorem is one of the most important results in algebraic geometry. It computes the dimension of spaces of meromorphic functions with prescribed poles. Special cases include:

$D = 0$ : $\ell(0) = 1$ (only constants), confirming $g = \ell(K)$ (the number of independent holomorphic $1$ -forms equals the genus).
$\deg D > 2g - 2$ : $\ell(K-D) = 0$ , so $\ell(D) = \deg D - g + 1$ .

Abel-Jacobi Theory

Definition10.6Jacobian variety

The Jacobian of a compact Riemann surface $S$ of genus $g$ is the complex torus

$\text{Jac}(S) = \mathbb{C}^g / \Lambda$

where $\Lambda$ is the period lattice of the $g$ independent holomorphic $1$ -forms. The Abel-Jacobi map $\mu: S \to \text{Jac}(S)$ sends $p \mapsto \left(\int_{p_0}^p \omega_1, \ldots, \int_{p_0}^p \omega_g\right) \mod \Lambda$ .

RemarkAbel's theorem

A divisor $D$ of degree $0$ on $S$ is the divisor of a meromorphic function if and only if its image under the Abel-Jacobi map is zero in $\text{Jac}(S)$ . This connects the algebraic structure of divisors to the analytic structure of period integrals.