Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 24F / b / iii / Solution

For $N>2g-2$, the divisor $K_X-Np$ has negative degree, so $\ell (K_X-Np)=0$. The Riemann-Roch theorem givesStarting from $\ell (0)=1$ and using part (ii), each integer $1\le n\le N$ contributes one to $\ell (Np)$ exactly when it is not a gap. ThereforeComparison with the preceding formula gives $|G(p)\cap \{1,\dots ,N\}|=g$. Moreover every $n>N$ is a nongap because the Riemann--Roch formula makes every subsequent dimension jump equal to one. Thus the Weierstrass gap theorem says