On the cohomology of tautological bundles over Quot schemes of curves

TL;DR

This paper investigates the cohomology of tautological bundles on Quot schemes over the projective line, proving vanishing results and describing global sections through explicit geometric embeddings.

Contribution

It introduces new vanishing theorems and cohomological descriptions for tautological bundles on Quot schemes, utilizing embeddings into Grassmannian products.

Findings

Proved vanishing of higher cohomology for certain tautological bundles.

Described spaces of global sections via tautological constructions.

Constructed resolutions with vanishing cohomology for these bundles.

Abstract

We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions. To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Str{\o}mme. This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers. We further illustrate our approach with a few additional cohomological calculations.