Explain what is meant by a divisor $D$ on a compact connected Riemann surface $S$. Explain briefly what is meant by a canonical divisor. Define the degree of $D$ and the notion of linear equivalence between divisors. If two divisors on $S$ have the same degree must they be linearly equivalent? Give a proof or a counterexample as appropriate, stating accurately any auxiliary results that you require.

Define $\ell(D)$ for a divisor $D$, and state the Riemann-Roch theorem. Deduce that the dimension of the space of holomorphic differentials is determined by the genus $g$ of $S$ and that the same is true for the degree of a canonical divisor. Show further that if $g=2$ then $S$ admits a non-constant meromorphic function with at most two poles (counting with multiplicities).

[General properties of meromorphic functions and meromorphic differentials on $S$ may be used without proof if clearly stated.]