Globally generated vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1 \times \mathbf{P}^1$ with low first Chern classes

TL;DR

This paper classifies globally generated vector bundles with small first Chern classes on the product of three projective lines, using the Hartshorne-Serre correspondence to analyze associated smooth curves.

Contribution

It provides a classification of such vector bundles with low first Chern classes on  imes  imes , advancing understanding of their structure.

Findings

Complete classification for c_1  with a_i   2

Identification of associated smooth curves via Hartshorne-Serre correspondence

New insights into the structure of globally generated vector bundles on product spaces

Abstract

We classify globally generated vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1 \times \mathbf{P}^1$ with small first Chern class, i.e. $c_1= (a_1, a_2, a_3)$, $a_i \le 2$. Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne-Serre correspondence.