Globally generated vector bundles on the Segre threefold with Picard number two (Q2805372)

The authors classify the vector bundles on \({\mathbb P}^1 \times{\mathbb P}^2\) with Chern class \(c_1=(1,1)\), or \((1,2)\), or \((2,1)\). The classification of (globally generated) vector bundles of rank \(\geq 2\) is related, via the Serre-Hartshorne correspondence, with the classification of (smooth) curves in \({\mathbb P}^1 \times{\mathbb P}^2\). Taking into account the (simple) intersection theory in \({\mathbb P}^1 \times{\mathbb P}^2\), one analyzes smooth curves \(C \subset {\mathbb P}^1 \times{\mathbb P}^2\) such that \({\mathcal I}_C\), suitable twisted, is globally generated. As the cases \(c_1=(0,1), \text{ or } c_1=(1,0)\) are easy, by this, the classification of globally generated vector bundles on \({\mathbb P}^1 \times{\mathbb P}^2\) with Chern classes \(c_1=(a,b)\), \(a+b \leq 3\) is settled.