IJPAM: Volume 65, No. 1 (2010)

A FINITENESS RESULT ON THE SETS
COMPUTING $X$-RANK AND SPANNING
A PRESCRIBED LINEAR SPACE

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 and non-degenerate variety defined over an algebraically closed field $\mathbb {K}$ such that $\mbox{char}(\mathbb {K})=0$. For each $P\in \mathbb {P}^n$ the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$. Let $\mathcal {S}(X,P)$ denote the set of all subsets $S\subset X$ such that $\sharp (S)=r_X(P)$ and $P\in \langle S\rangle$. Let $\mathcal {L}(X,P)$ the subset of the Grassmannian $G(r_X(P)-1,n)$ parametrizing all linear spaces $\langle S\rangle$, $S\in \mathcal {S}(X,P)$. For each $M\in \mathcal {L}(X,P)$ set $\mathcal {S}(M,X,P):= \{S\in \mathcal {S}(X,P): M =\langle S\rangle \}$. Here we prove that every $\mathcal {S}(M,X,P)$ is finite.

Received: September 23, 2010

AMS Subject Classification: 14N05

Key Words and Phrases: $X$-rank, symmetric tensor rank

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 1