Del Pezzo surfaces as hyperplane sections

Let L be a very ample line bundle on a complex projective manifold X of dimension n. We classify pairs (X,L) as above with some smooth A \in |L| being del Pezzo, i.e., -K_A=(n-2)H for some ample line bundle H on A. If H=L_A, then the problem reduces to the classification of del Pezzo manifolds, which has been done by Fujita in the more general setting of ample divisors. However, there are several examples showing that H \not= L_A can occur, which suggests the developement of a detailed structure theory in which both Fujita's theory (in the very ample setting) and all known examples fit. This is exactly the content of the paper.