Past exam of the geometry course of the University of Oxford / 2011 / 6 / Solution

The genus-degree formula for a nonsingular plane curve givesA divisor is a finite formal sum $D={\sum }_p{n}_p[p]$ with integer coefficients; its degree is $\mathrm{deg}D={\sum }_p{n}_p$. An effective divisor has all $n_p\ge 0$. A hyperplane divisor is the intersection with a projective line not containing the curve, counted with local intersection multiplicities. Its degree is $d$.

Define $L(D)=\{0\}\cup \{f\text{ meromorphic}:(f)+D\ge 0\}$ and $\ell (D)={\dim }_{\mathbb{C}}L(D)$. For a canonical divisor $K$, the Riemann-Roch theorem statesHere $(f)$ records zero orders positively and pole orders negatively, and $\mathrm{deg}K=2g-2$.