Linear dependent subsets of Segre varieties

TL;DR

This paper classifies small linearly dependent subsets of Segre varieties, especially for sets of size up to 5, and explores conditions that improve bounds on their size based on the variety's structure.

Contribution

It provides a classification of linearly dependent finite subsets of Segre varieties with up to five points and introduces bounds under additional geometric conditions.

Findings

Classified pairs (S,X) with S linearly dependent and |S| ≤ 5.

Derived improved lower bounds for |S| based on the variety's dimension and factors.

Utilized rational normal curves in the classification process.

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