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

Feiyang Lin; Theodore Lysek

Paper

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

Abstract

We characterize components of the locally free locus

of the Quot scheme associated to any vector bundle on

. 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

is connected, and we give an explicit bound for when

is irreducible. The key ingredient is a combinatorial criterion for when a triple of vector bundles on

arises in a short exact sequence. As a consequence, we prove that in codimension

, all integral lattice points in the Boij-Söderberg cone are Betti diagrams of actual modules.