Past exam of the algebraic curves course of the University of Oxford / 2023 / 3 / b

Choose a nonzero meromorphic differential on a Riemann surface $\eta$ and write $K=(\eta )$. Multiplication defines a linear mapIt takes values in the holomorphic differential forms because $(f\eta )=(f)+K\ge 0$. It is injective since $\eta \ne 0$. It is surjective because, for any holomorphic differential form $\omega$, the ratio $\omega /\eta$ is a meromorphic function and has $(\omega /\eta )+K=(\omega )\ge 0$. Thus this map identifies $\mathrm{HD}(C)$ with the canonical Riemann-Roch space.

Furthermore, $\mathcal{L}(0)=\mathbb{C}$: a meromorphic function with no poles is holomorphic on the compact Riemann surface $C$, and hence constant by the maximum modulus principle. Applying the Riemann-Roch theorem with $D=0$ givesTherefore