Curves in low dimensional projective spaces with the lowest ranks

  • Edoardo Ballico Department of Mathematics, University of Trento, 38123 Povo (TN), Italy.
Keywords: X-rank, projective curve, space curve, curve in projective spaces

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.

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Published
2020-12-07
How to Cite
[1]
E. Ballico, “Curves in low dimensional projective spaces with the lowest ranks”, CUBO, vol. 22, no. 3, pp. 379–393, Dec. 2020.