The uniqueness of Weierstrass points with semigroup <a;b> and related subgroups

Abstract

Assume $a$ and $b=na+r$ with $n\ge 1$ and $0<r<a$ are relatively prime integers. In case $C$ is a smooth curve and $P$ is a point on $C$ with Weierstrass semigroup equal to $<a;b>$ then $C$ is called a $C_{a;b}$-curve. In case $r\ne a-1$ and $b\ne a+1$ we prove $C$ has no other point $Q\ne P$ having Weierstrass semigroup equal to $<a;b>$. We say the Weierstrass semigroup $<a;b>$ occurs at most once. The curve $C_{a;b}$ has genus $(a-1)(b-1)/2$ and the result is generalized to genus $g<(a-1)(b-1)/2$. We obtain a lower bound on $g$ (sharp in many cases) such that all Weierstrass semigroups of genus $g$ containing $<a;b>$ occur at most once.

References in corpus (1)

• Galois Weierstrass points whose Weierstrass semigroups are generated by two elements