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On the X-rank with respect to linear projections of projective varieties

Edoardo Ballico 1 Alessandra Bernardi 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (... - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this paper we improve the known bound for the $X$-rank $R_{X}(P)$ of an element $P\in {\mathbb{P}}^N$ in the case in which $X\subset {\mathbb P}^n$ is a projective variety obtained as a linear projection from a general $v$-dimensional subspace $V\subset {\mathbb P}^{n+v}$. Then, if $X\subset {\mathbb P}^n$ is a curve obtained from a projection of a rational normal curve $C\subset {\mathbb P}^{n+1}$ from a point $O\subset {\mathbb P}^{n+1}$, we are able to describe the precise value of the $X$-rank for those points $P\in {\mathbb P}^n$ such that $R_{X}(P)\leq R_{C}(O)-1$ and to improve the general result. Moreover we give a stratification, via the $X$-rank, of the osculating spaces to projective cuspidal projective curves $X$. Finally we give a description and a new bound of the $X$-rank of subspaces both in the general case and with respect to integral non-degenerate projective curves.
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Contributor : Alessandra Bernardi <>
Submitted on : Tuesday, November 29, 2011 - 11:34:06 AM
Last modification on : Monday, October 12, 2020 - 10:27:38 AM

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Edoardo Ballico, Alessandra Bernardi. On the X-rank with respect to linear projections of projective varieties. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2011, 284 (17-18), pp.2133-2140. ⟨10.1002/mana.200910275⟩. ⟨hal-00646117⟩