Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z subset of X be a smooth surface, which is the zero locus of a section of an ample vector bundle epsilon of rank dim X - 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 (P-2 O-P2 (4)) as its minimal adjunction theoretic reduction. Triplets (X, S, H) as above are discussed and classified. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.