Past exam of the geometry course of the University of Oxford / 2009 / B / 6 / ii

Apply the preceding argument to $q+r$ to obtain a nonconstant meromorphic function $g$ with simple poles exactly at $q,r$. Both $f,g$, and the constant function one, belong to $L(p+q+r)$.

If $a+bf+cg=0$, examination at $p$, where only $f$ has a pole, forces $b=0$. Examination at $r$, where only $g$ has a pole, forces $c=0$, and then $a=0$. Thus the three functions are linearly independent. Since the Riemann-Roch theorem gives $\ell (p+q+r)=3$,