Geodesic
On the birational gonalities of smooth curves
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 1.

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Let C be a smooth curve of genus g. For each positive integer r the birational r-gonality s_r(C) of C is the minimal integer t such that there is L∈^t(C) with h^0(C,L) =r+1. Fix an integer r≥ 3. In this paper we prove the existence of an integer g_r such that for every integer g≥ g_r there is a smooth curve C of genus g with s_r+1(C)/(r+1) gt; s_r(C)/r, i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails. 
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Ballico, E. On the birational gonalities of smooth curves. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 1. http://geodesic.mathdoc.fr/item/AUM_2014_68_1_a2/

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