Let X ? ?r be an integral and non-degenerate curve. For each q ∈ ?r the X-rank r X (q) of q is the minimal number of points of X spanning q. A general point of ?r has X-rank ?(r 1)/2?. For r = 3 (resp. r = 4) we construct many smooth curves such that r X (q) ≤ 2 (resp. r X (q) ≤ 3) for all q ∈ ?r (the best possible upper bound). We also construct nodal curves with the same properties and almost all geometric genera allowed by Castelnuovo’s upper bound for the arithmetic genus.
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