The locally free locus of Quot schemes on ${\mathbb{P}}^1$

We characterize components of the locally free locus ${\mathrm{Quot}}_{{\mathbb{P}}^1}^{n,d}(\mathcal{O}(\vec{e}){)}^{\circ }$ of the Quot scheme associated to any vector bundle on ${\mathbb{P}}^1$. Specifically, we show that the components are in bijection with certain combinatorial objects which we call strongly stable pairs. Using our explicit understanding of the components, we prove that ${\mathrm{Quot}}_{{\mathbb{P}}^1}^{n,d}(\mathcal{O}(\vec{e}){)}^{\circ }$ is connected, and we give an explicit bound for when ${\mathrm{Quot}}_{{\mathbb{P}}^1}^{n,d}(\mathcal{O}(\vec{e}){)}^{\circ }$ is irreducible. The key ingredient is a combinatorial criterion for when a triple of vector bundles on ${\mathbb{P}}^1$ arises in a short exact sequence. As a consequence, we prove that in codimension $2$, all integral lattice points in the Boij-Söderberg cone are Betti diagrams of actual modules.