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

A famous and surprising result of Danilov and Gizatullin asserts that the isomorphy type of the complement of an ample section in a Hirzebruch surface depends only its self-intersection, in particular it depends neither on the ambient surface nor on the chosen section. In this talk, we will consider natural higher dimensional analogues where one removes hyperplane sub-bundles in arbitrary projective space bundles over the projective line. We will show in particular that for ample such sub-bundles, the isomorphy type of the complement is again uniquely determined by their top self-intersection.