Weierstrass semigroups on the Giulietti-Korchm\'aros curve

TL;DR

This paper explicitly determines the Weierstrass semigroups at all points of the Giulietti-Korchmárós curve, classifying them into three types based on rationality over finite fields, and confirms a conjecture about their structure.

Contribution

It provides a complete classification of Weierstrass semigroups on the Giulietti-Korchmárós curve, including a proof of a conjecture for points rational over _{q^6}_{q^2}.

Findings

Three types of Weierstrass semigroups identified.

Confirmed conjecture for _{q^6}_{q^2}-rational points.

Set of Weierstrass points equals _{q^6}-rational points.

Abstract

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchm\'aros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is an $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational point. As a corollary we also obtain that the set of Weierstrass points of $\mathcal{X}$ is exactly its set of $\mathbb{F}_{q^6}$-rational points.