Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z be a smooth surface which is the zero locus of an ample vector bundle E of rank dim(X)-2 \geq 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(4)) as its minimal adjunction theoretic reduction. Triplets (X,E,H) as above are discussed and classifie