On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

TL;DR

This paper classifies certain rank-two vector bundles on Hirzebruch surfaces as very ample and uniform, and studies their projectivizations as scrolls within Hilbert schemes, revealing their geometric properties and component structures.

Contribution

It provides a detailed analysis of families of rank-two uniform bundles on Hirzebruch surfaces and their Hilbert scheme components, including cases where they fill entire components or subvarieties.

Findings

Bundles are all very ample and uniform under certain conditions.

Scrolls correspond to smooth points in Hilbert scheme components.

For specific parameters, scrolls fill entire components or subvarieties.

Abstract

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e=0,1 the scrolls fill up the entire component of the Hilbert scheme, while for e=2 the scrolls exhaust a subvariety of codimension 1.