IJPAM: Volume 91, No. 3 (2014)

OPEN RANK (OR WIDERANK) FOR REDUCIBLE
PROJECTIVE SETS: A CLASSIFICATION
OF AN EXTREMAL CASE

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY


Abstract. Let $X\subset \mathbb {P}^r$, $r\ge 2$, be a reduced projective set such that $\langle X\rangle =\mathbb {P}^r$, where $\langle \ \rangle$ denote the linear span of $X$. For any $P\in \mathbb {P}^r$ the $X$-widerank $w_X(P)$ of $P$ is the minimal integer $t>0$ such that for each closed set $B\subset X$ containing no irreducible component of $X$ there is $S\subset X\setminus B$ with $P\in \langle S\rangle$. Here we classify all $(r,X)$ such that $w_X(P) \ge r+1$ for a general $P$ ($r$ is odd and $X$ is the union of $(r+1)/2$ linearly independent lines). We give conditions on $X$ which imply that $w_X(P) \le r$ (or $w_X(P)\le r+1-\dim (X)$) for every $P\in \mathbb {P}^r\setminus X$.

Received: December 20, 2013

AMS Subject Classification: 14N05, 14Q05, 15A69

Key Words and Phrases: widerank, open rank, symmetric tensor rank, reducible varieties, lines, strange variety

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DOI: 10.12732/ijpam.v91i3.12 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 91
Issue: 3