Weierstrass semigroups on the Giulietti-Korchmáros curve

2017/08/23 by Beelen, Peter, Montanucci, Maria · 1 citation
#11G20 (Primary) 11R58 #14H05 #14H55 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.06974

Abstract

In this article we explicitly determine the structure of the Weierstrass semigroups H(P) for any point P of the Giulietti-Korchmáros curve X. We show that as the point varies, exactly three possibilities arise: One for the \mathbbF_q²-rational points (already known in the literature), one for the \mathbbF_q⁶ ∖ \mathbbF_q²-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 \mathbbF_q⁶ ∖ \mathbbF_q²-rational point. As a corollary we also obtain that the set of Weierstrass points of X is exactly its set of \mathbbF_q⁶-rational points.

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