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


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.


B. Adlandsvik, “Joins and higher secant varieties”, Math. Scand., vol. 62, pp. 213–222, 1987.

G. Blekherman, Z. Teitler, “On maximum, typical and generic ranks”, Math. Ann., vol. 362, no. 3-4, pp. 1231–1031, 2015.

J. Buczyn ́ski, K. Han, M. Mella, Z. Teitler, “On the locus of points of high rank”, Eur. J. Math., vol. 4, pp. 113–136, 2018.

I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge, 2012.

J. P. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York, 1978.

J. Harris (with D. Eisenbud): Curves in projective space, Les Presses de l’Université de Montréal, Montréal, 1982.

R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin–Heidelberg–New York, 1977.

J. M. Landsberg, Z. Teitler, “On the ranks and border ranks of symmetric tensors,” Found. Comput. Math., vol. 10, pp. 339–366, 2010.

P. Piene, “Cuspidal projections of space curves”, Math. Ann., vol. 256, no. 1, pp. 95–119, 1981.

A. Tannenbaum, “Families of algebraic curves with nodes”, Composition Math., vol. 41, no. 1, pp. 107–126, 1980.

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