Curves in low dimensional projective spaces with the lowest ranks

Abstract

Let X ⊂ P r be an integral and non-degenerate curve. For each q ∈ P r the X -rank r X ( q ) of q is the minimal number of points of X spanning q . A general point of P 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 ∈ P 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. q ∈ P r el -rango X ) q que generan q P r tiene X -rango ⌈ ( r + 1) / 2 ⌉ . Para r = 3 (resp. r = 4) construimos q ) ≤ 2 (resp. r X ( q ) q ∈ P r superior geom´etricos aritm´etico.