UC Berkeley

Splitting loci are certain degeneracy loci that arise from the data of a family of vector bundles on P1. They can be viewed as closed substacks of the stack of vector bundles on P1, and are closely related to the Brill-Noether theory of general k-gonal curves, as well as opposite Schubert varieties in the affine Grassmannian for the general linear group. We construct a modular resolution of singularities for splitting loci using a relative flag Quot construction, and use this resolution to study their singularities. We identify the singular locus of Σ_→e, prove that tame splitting loci have rational singularities, and characterize exactly which splitting loci are Gorenstein or Q-Gorenstein. The proof for rational singularities requires a cohomology vanishing result for tautological bundles on certain Quot schemes Quotr,d P1 (O⊕N ), which we push further into a Borel-Weil-Bott type theorem.