Curves in low dimensional projective spaces with the lowest ranks
-
Edoardo Ballico
ballico@science.unitn.it
Downloads
DOI:
https://doi.org/10.4067/S0719-06462020000300379Abstract
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.
Keywords
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 Ìski, 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.
Most read articles by the same author(s)
- Edoardo Ballico, Osculating varieties and their joins: \(\mathbb{P}^1\times \mathbb{P}^1\) , CUBO, A Mathematical Journal: Vol. 25 No. 2 (2023)
- Edoardo Ballico, A characterization of \(\mathbb F_q\)-linear subsets of affine spaces \(\mathbb F_{q^2}^n\) , CUBO, A Mathematical Journal: Vol. 24 No. 1 (2022)
Similar Articles
- Chia-chi Tung, On Semisubmedian Functions and Weak Plurisubharmonicity , CUBO, A Mathematical Journal: Vol. 12 No. 2 (2010): CUBO, A Mathematical Journal
- Ilker Sahin, Mustafa Telci, A Common Fixed Point Theorem for Pairs of Mappings in Cone Metric Spaces , CUBO, A Mathematical Journal: Vol. 16 No. 1 (2014): CUBO, A Mathematical Journal
- Takahiro Sudo, Spectral Rank for ð¶*-Algebras , CUBO, A Mathematical Journal: Vol. 10 No. 2 (2008): CUBO, A Mathematical Journal
- Manoj Bhardwaj, Alexander V. Osipov, Some observations on a clopen version of the Rothberger property , CUBO, A Mathematical Journal: Vol. 25 No. 2 (2023)
- Rinko Shinzato, Wataru Takahashi, A Strong Convergence Theorem by a New Hybrid Method for an Equilibrium Problem with Nonlinear Mappings in a Hilbert Space , CUBO, A Mathematical Journal: Vol. 10 No. 4 (2008): CUBO, A Mathematical Journal
- Hamed M. Obiedat, A Topological Characterization of the Beurling-Björck Space ð”–𜔠Using the Short-Time Fourier Transform , CUBO, A Mathematical Journal: Vol. 8 No. 2 (2006): CUBO, A Mathematical Journal
- V. Renukadevi, On subsets of ideal topological spaces , CUBO, A Mathematical Journal: Vol. 12 No. 2 (2010): CUBO, A Mathematical Journal
- Rubén A. Hidalgo, A sufficiently complicated noded Schottky group of rank three , CUBO, A Mathematical Journal: Vol. 22 No. 1 (2020)
- Anjali Goswami, Special recurrent transformation in an NPR-Finsler space , CUBO, A Mathematical Journal: Vol. 14 No. 1 (2012): CUBO, A Mathematical Journal
- G. S. Saluja, Convergence theorems for generalized asymptotically quasi-nonexpansive mappings in cone metric spaces , CUBO, A Mathematical Journal: Vol. 15 No. 3 (2013): CUBO, A Mathematical Journal
<< < 1 2 3 4 5 6 7 8 9 10 11 12 > >>
You may also start an advanced similarity search for this article.
Downloads
