Linear 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 (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