Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z ⊂ X be a smooth surface, which is the zero locus of a section of an ample vector bundle E of rank dimX – 2 ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H Z to Z is a very ample line bundle and assume that (Z, H Z ) is a Bordiga surface, i.e., a rational surface having (P2, O P 2 (4)) as its minimal adjunction theoretic reduction. Triplets (X, E, H) as above are discussed and classified. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)