Define what is meant by a divisor on a compact Riemann surface, the degree of a divisor, and a linear equivalence between divisors. For a divisor $D$, define $\ell(D)$ and show that if a divisor $D^{\prime}$ is linearly equivalent to $D$ then $\ell(D)=\ell\left(D^{\prime}\right)$. Determine, without using the Riemann-Roch theorem, the value $\ell(P)$ in the case when $P$ is a point on the Riemann sphere $S^{2}$.

[You may use without proof any results about holomorphic maps on $S^{2}$ provided that these are accurately stated.]

State the Riemann-Roch theorem for a compact connected Riemann surface $C$. (You are not required to give a definition of a canonical divisor.) Show, by considering an appropriate divisor, that if $C$ has genus $g$ then $C$ admits a non-constant meromorphic function (that is a holomorphic map $C \rightarrow S^{2}$ ) of degree at most $g+1$.