Curves in low dimensional projective spaces with the lowest ranks
Author
Ballico, Edoardo
Full text
http://revistas.ufro.cl/ojs/index.php/cubo/article/view/247110.4067/S0719-06462020000300379
Abstract
Let \(X\subset {\mathbb P}^r\) be an integral and non-degenerate curve. For each \(q\in {\mathbb P}^r\) the \(X\)-rank \(r_X(q)\) of \(q\) is the minimal number of points of \(X\) spanning \(q\). A general point of \({\mathbb P}^r\) has \(X\)-rank \(\lceil (r+1)/2\rceil\). For \(r=3\) (resp. \(r=4\)) we construct many smooth curves such that \(r_X(q) \le 2\) (resp. \(r_X(q) \le 3\)) for all \(q\in {\mathbb P}^r\) (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.