IJPAM: Volume 93, No. 6 (2014)

ON THE STRATIFICATION BY X-RANKS OF A PROJECTIVE
SPACE (MAXIMAL AND SUBMAXIMAL X-RANKS)

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


Abstract. Let $X\subset \mathbb {P}^N$ be an integral and non-degenerate variety. For each $P\in \mathbb {P}^N$ the X-rank $r_X(P)$ of $P$ is the minimal cardinality of a subset of $X$ whose linear span contains $P$. For each $x>0$ set $A_X(x) := \{P\in \mathbb {P}^N: r_X(P) =x\}$. We prove that if $r_{\m}$ is the maximum of all $r_X(P)$ and $a:= \dim (X) \le N-2$, then $\dim (A_X(r_{\m} -1)) \ge \max \{1+a, \dim (A_X(r_{\m} ))+2\}$.

Received: February 25, 2014

AMS Subject Classification: 14N05, 15A69

Key Words and Phrases: X-rank, open rank, secant variety

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DOI: 10.12732/ijpam.v93i6.5 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: 93
Issue: 6
Pages: 799 - 802

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).