  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.
