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

Abstract

We characterize components of the locally free locus $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ 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 $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ is connected, and we give an explicit bound for when $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ 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-S\"oderberg cone are Betti diagrams of actual modules.