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

The divisor $D={\sum }_i{H}_i$ has degree $md$. The hypothesis gives $\mathrm{deg}(K-D)=2g-2-md<0$. A divisor of negative degree has no nonzero section: if $h\in L(K-D)\setminus \{0\}$, then $(h)+K-D$ would be effective, but its degree would be negative because the divisor of a meromorphic function has degree zero. Thus $\ell (K-D)=0$. The Riemann-Roch theorem now gives