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

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.