Complements of hyperplane sub-bundles in projective spaces bundles over P-1

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle of a -bundle depends only on the -fold self-intersection of . In particular it depends neither on the ambient bundle nor on the choice of a particular ample sub-bundle with given -fold self-intersection. The proof exploits the unexpected property that every such complement comes equipped with the additional structure of an affine-linear bundle over the affine line with a double origin.