Linearly dependent subsets of Segre varieties

Abstract

We study the linear algebra of finite subsets S of a Segre variety X. In particular we classify the pairs (S, X) with S linear dependent and $$\#(S)\le 5$$ . We consider an additional condition for linear dependent sets, i.e. that no two of their points are contained in a line of X, and get far better lower bounds for $$\#(S)$$ in term of the dimension and number of the factors of X. In this discussion and in the classification of the case $$\#(S)= 5$$ , $$X\cong {\mathbb {P}}^1\times {\mathbb {P}}^1\times {\mathbb {P}}^1$$ we use the rational normal curves contained in X.