Globally generated vector bundles on the Segre threefold with Picard number two

TL;DR

This paper classifies globally generated vector bundles with small first Chern class on the Segre threefold, using the Hartshorne-Serre construction to analyze associated smooth curves.

Contribution

It provides a classification of such vector bundles on the Segre threefold with low first Chern class, a novel application of the Hartshorne-Serre construction in this context.

Findings

Complete classification for c_1 = (a,b) with a+b ≤ 3

Identification of associated smooth curves via Hartshorne-Serre construction

New insights into vector bundle structures on Segre threefold

Abstract

We classify globally generated vector bundles on $\mathbb{P}^1 \times \mathbb{P}^2$ with small first Chern class, i.e. $c_1= (a,b)$, $a+b \leq 3$. Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne-Serre construction.