Past exam of the geometry course of the University of Oxford / 2008 / B / 8 / i / Solution

A divisor on an algebraic curve is a finite formal sum $D=\sum n_{p}p$ with integer coefficients. Its degree is $\mathrm{deg}D=\sum n_p$. The principal divisor of a nonzero meromorphic function is $(f)=\sum {\mathrm{ord}}_p(f)p$. For a nonconstant $f$, zeros and poles are fibers over zero and infinity of the same map to the sphere, each of total multiplicity equal to its degree; hence $\mathrm{deg}(f)=0$. A nonzero constant has zero divisor.

Write $L(D)=\{f:(f)+D\ge 0\}\cup \{0\}$ and $\ell (D)=\dim L(D)$. The Riemann-Roch theorem stateswhere $K_C$ is a canonical divisor. Taking $D=p_1+\cdots +p_{g+1}$ gives $\ell (D)=2+\ell (K_C-D)\ge 2$. Constants span only one dimension, so there is a nonconstant $f\in L(D)$. It has no poles outside these points and at most a simple pole at each. Thus the required nonconstant function exists, with actual poles at some nonempty subset because a holomorphic function on a compact connected curve is constant.