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

TL;DR

This paper investigates the uniqueness of Weierstrass points with specific semigroups on smooth curves, establishing conditions under which such semigroups occur at most once and providing bounds related to genus.

Contribution

It proves the uniqueness of Weierstrass semigroups <a;b> on certain curves and generalizes the result to bounds on genus where such semigroups are unique.

Findings

Weierstrass semigroup <a;b> occurs at most once under specified conditions.

The genus of the curve with semigroup <a;b> is (a-1)(b-1)/2.

A lower bound on genus g ensures the uniqueness of semigroups containing <a;b>.

Abstract

Assume $a$ and $b=na+r$ with $n \geq 1$ 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 $r \neq a-1$ and $b \neq a+1$ we prove $C$ has no other point $Q \neq P$ 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.