Del Pezzo surfaces as hyperplane sections

Abstract

Let $L$ be a very ample line bundle on an $n$ -dimensional complex projective manifold $X$ . In this article we classify pairs (X, $L$ ) as above with some smooth $A\in|L|$ being del Pezzo, i.e., with a smooth $A\in|L|$ such that $-K_{A}=(n-2)H$ for some ample line bundle $H$ on $A$ . We assume that $n\geqq 3$ since otherwise we are in the completely understood case when $A$ is an elliptic curve.