The locally free locus of Quot schemes on ℙ¹

2025/12/15 by Feiyang Lin, Lin, Feiyang, Theodore Lysek +1
Mathematics · #13D02 #14D20 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2512.13386

openalex publication_date 2025/12/15 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

We characterize components of the locally free locus Quot^n,d_ℙ¹(O(e))^∘ of the Quot scheme associated to any vector bundle on ℙ¹. 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 Quot^n,d_ℙ¹(O(e))^∘ is connected, and we give an explicit bound for when Quot^n,d_ℙ¹(O(e))^∘ is irreducible. The key ingredient is a combinatorial criterion for when a triple of vector bundles on ℙ¹ 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.

Citations

• Resolving the Singularities of Splitting Loci
• Betti Numbers of Graded Modules and Cohomology of Vector Bundles

Related

• Irreducibility of some nested Quot schemes
• Hunting Vector Bundles on ℙ¹ × ℙ¹
• Irreducibility of Some Quot Schemes on Nodal Curves
• A Borel--Weil--Bott theorem for Quot schemes on ℙ¹
• Maximal subbundles, quot schemes, and curve counting