Complements of hyperplane sub-bundles in projective spaces bundles over ${\mathbb {P}}^{1}$

Abstract

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle $H$ of a ${\mathbb {P}}^{r-1}$-bundle ${\mathbb {P}}(E)\rightarrow {\mathbb {P}}^{1}$ depends only on the $r$-fold self-intersection $(H^{r})\in {\mathbb {Z}}$ of $H$. In particular it depends neither on the ambient bundle ${\mathbb {P}}(E)$ nor on the choice of a particular ample sub-bundle with given $r$-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.