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

Given $f$, choose the hyperplane divisors as in (a). The ratios $f_Q$ form a linear subspace of $L(D)$ by (d), and its dimension equals $md+1-g=\ell (D)$ by (b). A linear subspace of a finite-dimensional vector space with the same dimension is the whole space. Hence $f=f_Q$ for some homogeneous polynomial $Q$ of degree $m$, givingThe zero function corresponds to $Q=0$. Thus every allowed meromorphic function is a ratio of homogeneous polynomials of equal degree, with a denominator that can be chosen as a product of linear forms.