IJPAM: Volume 70, No. 5 (2011)

POINTS OF A PROJECTIVE SPACE WITH
A PRESCRIBED NUMBER OF SUBSETS
COMPUTING THEIR $X$-RANK

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract. Let $X\subset \mathbb {P}^n$ be an integral non-degenerate subvariety. For any $P\in \mathbb {P}^n$ the $X$-rank of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$. Let $\mathcal {S}(X,P)$ be the set of all $S\subset X$ computing the $X$-rank of $P$. Here we construct smooth curves $X\subset \mathbb {P}^n$ and $P\in \mathbb {P}^n$ such that $\sharp (\mathcal {S}(X,P))$ is a prescribed integer.

Received: December 11, 2010

AMS Subject Classification: 14N05

Key Words and Phrases: ranks, $X$-rank, border rank, secant variety

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 5