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.
Categories: math.AG