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

Every degree-three monomial evaluated at $(1,f,g)$ is $f^a{g}^b$ with $a+b\le 3$. Its possible pole orders at $p,q,r$ are at most $a$, $a+b$, $b$, respectively, and hence at most three each. Therefore $Q(1,f,g)\in L(3p+3q+3r)$.

The vector space of homogeneous degree-three polynomials in three variables has dimension $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$. The target Riemann-Roch space has dimension nine, because its divisor on an algebraic curve has degree nine and genus is one. Thus the evaluation linear maphas nontrivial kernel. Consequently