1. On the X-rank with respect to linear projections of projective varieties
- Author
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Alessandra Bernardi, Edoardo Ballico, University of Trento [Trento], Geometry, algebra, algorithms (GALAAD), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), European Project: 252367,EC:FP7:PEOPLE,FP7-PEOPLE-2009-IEF,DECONSTRUCT(2010), Edoardo Ballico, Alessandra Bernardi, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Nice Sophia Antipolis (... - 2019) (UNS)
- Subjects
Linearly normal curve ,MAT/03 Geometria ,General Mathematics ,[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC] ,010103 numerical & computational mathematics ,Rank (differential topology) ,Rational normal curve ,Commutative Algebra (math.AC) ,01 natural sciences ,Projection (linear algebra) ,Combinatorics ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Projective variety ,Mathematics ,Rank ,Secant Varieties ,Rational Normal Curves ,Projections ,010102 general mathematics ,Mathematics - Commutative Algebra ,Linear subspace ,SYMMETRIC TENSORS ,rank ,14N05, 14H50 ,Tangential varietie ,SECANT VARIETIES ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Element (category theory) ,Subspace topology ,Osculating circle - 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., 10 pages
- Published
- 2009
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