Some remarks on Weierstrass points

Abstract:

The author proves that, at a point $P$ on a closed Riemann surface of genus $g$, if $h$ is the first nongap at $P$ and $k$ is relatively prime to $h$, then $k$ is a gap if $g > \tfrac {1}{2}(h - 1)(k - 1)$. A consequence is that at the Weierstrass points of a closed Riemann surface, if the first nongap is a prime, the situation mirrors that in the hyperelliptic case, at least in a limiting sense.