Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 23H / Solution

A divisor on a compact connected Riemann surface is a finite formal sum $D={\sum }_p{n}_{p}p$ with integer coefficients; its degree is ${\sum }_p{n}_p$. A canonical divisor is the divisor of zeros and poles of a nonzero meromorphic differential. Two divisors are linearly equivalent when their difference is the divisor of a nonzero meromorphic function. Such principal divisors have degree zero, but equal degree does not generally imply linear equivalence of divisors.

For example take distinct points $p,q$ on a genus-one surface. If $p-q$ were principal, its function would define a degree-one meromorphic map to the sphere, since it has just one simple pole. A degree-one map between compact Riemann surfaces is a biholomorphism, contradicting their different genera. Thus these degree-one divisors are not equivalent.

Define $L(D)=\{f\text{ meromorphic}:(f)+D\ge 0\}\cup \{0\}$ and $\ell (D)={\dim }_{\mathbb{C}}L(D)$. The Riemann-Roch theorem statesSince all holomorphic functions on the compact surface are constant, $\ell (0)=1$. Setting $D=0$ gives $\ell (K)=g$. Multiplication by a fixed meromorphic differential identifies $L(K)$ with holomorphic differentials, so their dimension is $g$. Setting $D=K$ then givesFor $g=2$, choose a nonzero holomorphic differential, whose effective canonical divisor has degree two. Since $\ell (K)=2$, $L(K)$ contains a nonconstant meromorphic function. Its pole divisor is bounded by that effective $K$, proving existence of a nonconstant function with total pole multiplicity at most two.