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

TL;DR

This paper characterizes the components of the locally free locus of Quot schemes on al, establishing their combinatorial classification, connectedness, and bounds for irreducibility, with implications for Betti diagrams of modules.

Contribution

It provides a combinatorial classification of components of the Quot scheme's locally free locus on al and analyzes their geometric properties.

Findings

Components are in bijection with strongly stable pairs.

The locally free locus is connected.

Explicit bounds for irreducibility are given.

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.