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

Adrien Dubouloz (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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.

Mots clés: Danilov-Gizatullin Theorem; hyperplane sub-bundles; affine-linear bundles; Danilov-Gizatullin Theorem.

Domaines: Géométrie algébrique [math.AG]

Adrien Dubouloz. Complements of hyperplane sub-bundles in projective space bundles over the projective line. 2011. ⟨hal-00618025⟩