Complements of hyperplane sub-bundles in projective space bundles over the projective line

TL;DR

This paper characterizes the isomorphism types of complements of ample hyperplane sub-bundles in projective space bundles over the projective line, showing they depend solely on the self-intersection number and revealing a torsor structure over an affine line with a double origin.

Contribution

It proves that the isomorphism type of these complements is determined solely by the r-fold self-intersection of the hyperplane sub-bundle, independent of the ambient bundle or specific sub-bundle.

Findings

Isomorphism type depends only on the r-fold self-intersection.

Complements have a structure of a non-trivial torsor under a vector bundle.

The torsor structure is over an affine line with a double origin.

Abstract

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.