IJPAM: Volume 65, No. 1 (2010)

A FINITENESS RESULT ON THE SETS
COMPUTING -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 -rank of 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 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.